Sorry, so we're very happy to have Kristen Lauder here to give her third and final lecture on super-singular misogyny graphs and cryptography. Thanks Jen and let me take this opportunity to thank the organizers. Thank you to Akshay, Bjorn, and Jen, and also to the organizers of PCMI. Organizing these conferences in summer schools is a lot of work and I really appreciate everything you've done and the chance to talk to all of you here. So for my third lecture, if you remember at the end of the second lecture, we started to talk about the quaternion algebras and specifically the quaternion interpretation of the super-singular isogenic graphs. So just to recall, the idea is that for a super-singular elliptic curve, if you associate to it its endomorphism ring, which is a maximal order in a quaternion algebra, you get a different graph which is instead of elliptic curves with isogenes. You have a graph where the vertices are maximal orders in this quaternion algebra and this during correspondence is something that I wanted to explain in more detail but I first needed to start by explaining a little bit about quaternion algebras which is what we started last time. So the quaternion algebras that we'll be working in are quaternion algebras over Q, the rationals, they're rank four vector space over Q and we in particular are interested in definite quaternion algebras which are ramified only at P and infinity and so I started to tell you how we describe and work with these algebras B, P, infinity last time. So we're going to take a basis one i, j, k and it will always be of the form i squared is equal to a, j squared is equal to b and k is equal to i, j which is minus j, i where in general a and b are negative integers and in particular for example if p is congruent to three mod four then a, b can be taken to be minus p minus one and so for other congruence classes of p there's other descriptions of a and b but I didn't write them down here. Okay so then the next thing that we need to know is that we have a norm form on this quaternion algebra. There's a little bit of confusion because in the quaternionic setting we usually talk about the norm well norm and trace actually as reduced norm and reduced trace but throughout the literature in this area you'll see it just referred to as norm so don't be confused by that. Strictly speaking we would call it the reduced norm but throughout this I'm just going to use the term norm which is what everybody says in like for shorthand. So we get the norm by realizing that first of all we have an involution on the quaternion algebra which sends an element which is written in terms of the basis one ijk to basically just negating the ij and k coefficients so I'll call that x star so for example if it's the element if x is the element c plus dj plus fi plus gij then the involution would just be c minus dj minus fi minus gij and then we have a reduced trace and a reduced norm in the space and that is that the reduced trace just takes x plus x star and the norm just takes x minus x star so for example with the basis that I had written went on the previous slide when p is congruent to three mod four then the norm map is just it takes an element that's written in terms of this basis to the element c squared plus d squared plus p times f squared plus g squared so one thing that you should notice which is really important in a lot of other work that I've done in my life is is that p is really large in the cryptographic setting so if you look at this norm form it's you know just c squared plus d squared plus p times f squared plus g squared so if if even if c, d, f, and g are all really small this norm is going to be really large if if f and g are not zero right so what happens is that elements of small norm commute essentially so what what it means is is that if you have an element whose norm is very small compared to p then it means it has no component from the f and g are zero and that means that the the element itself is just an element of a quadratic subfield of the quaternion algebra so that's just kind of an interesting side track that I always like to point out so another thing that's going to be very important for the during correspondence and for the for the relationship between the quaternionic side and the elliptic curve side is that the norm map on quaternions corresponds to the degree map on endomorphisms so I will explain this a little better in the setting of the during correspondence when we go back to that so first of all I think one of the issues is there's a lot there's actually a lot of terminology and a lot of definitions to know and to learn for quaternion algebras and I can't cover all of them here today but just to give you an idea to start with so we say a fractional ideal in the quaternions is actually a rank four z lattice so I've written it as this kind of curly i is going to be the notation for my ideals here so it's a rank four z lattice so it means it's it's written in the form like alpha one z plus alpha two z plus alpha three z plus alpha four z where alpha one alpha two alpha three alpha four form they actually form a basis a q basis for bp infinity so that's what a fractional ideal is but the fact is that a the fractional ideal is not necessarily a and actually a subring of the quaternion algebra and so we say an order oh is a fractional ideal which is also a subring so it's closed under multiplication and I didn't write all this down but like a maximum order is one that's not contained in any other larger order the norm of an ideal is going to be the z module generated by reduced norms of elements in the ideal so another kind of important fact which is very different from the commutative case is that and you can actually you know try to come up with an example of this yourself integral so integral elements do not necessarily form a ring so we have the nice situation like in number fields that if you take the integral elements you get the ring of integers of the ring and that is not necessarily true in the quaternionic case so integral element is it's reduced norm and trace are in z so it's kind of the same definition as in the commutative case but we don't have the same property of the integral elements being closed okay so now i'm going to talk about a very important concept which is the right order of a fractional ideal so if you have a fractional ideal i this curly curly i the right order will be all of the elements in the quaternion algebra bp infinity such that if you multiply on the right if you multiply the ideal on the right by that element alpha you still get another element in the ideal so that's that's the right order and then i didn't write it down but the left order so i'll use the notation o sub r for the right order and o sub l for the left order the left order would be all the elements such that such all the elements alpha such that if you multiply on the left by alpha you would still be in the ideal so now we come to another very important concept which is called the connecting ideal so given two maximal orders o one and o two the connecting ideal is an ideal which has the property that its right ideal is o one and its left ideal is o two so there is a kind of a linear algebra type of procedure for computing a connecting ideal which was as far as i know first written in a paper of kirchmer kirchmer and void in 2008 but i'm i'm pretty sure that it was known before that it was known for example to david kohl and i think it might have even been implemented by david kohl and magma before that i'm not completely certain about that but in the the kind of quaternion algebra package that david wrote in magma i think he might have even already had the connecting ideal in there um i should i should have said at the beginning of the quaternionic quaternion algebra section that kind of the the bible for quaternion algebras is a book by marie france vigneras and from the from the 80s i think and so that's where i learned quaternion algebras from and it's it's written in french and it's very well written and complete um but uh recently people have written things in english that may be a little bit easier to read so john void um has a book uh with a lot of that material and a lot of other things and a lot of algorithms uh but also um there's a couple of references which i can um i can share later i don't think i have them on my slides but uh which give very nice overview so there were some lectures by uh by pete clark that i found that are in in english that have a really nice description quaternion algebra background okay so with some of that uh background in mind and keep in mind i is just not a complete you know course on on quaternion algebras here uh just to give you kind of the basic uh building blocks uh let's go back to what i mentioned at the end of the lecture last time which is the during correspondence so um a super singular j invariant on the elliptic curve side would now be uh associated with a maximal order in a quaternion algebra on the other side and so how what is this correspondence how will it work so a left ideal of oh uh will be will correspond to an isogeny then as follows uh the isogeny called phi sub i will go from the elliptic curve e to an elliptic curve will call e sub i um and it'll be defined by its kernel and the kernel is going to be all of the points p on the elliptic curve which are basically annihilated by the ideal so such that every element of the ideal and remember this is these are elements which are essentially endomorphisms of the elliptic curve every alpha in the ideal takes the point to zero so for all points that that's true they're annihilated by all the elements of the ideal those will be the points in the kernel that's associated to i so that's this is in theory what the um what the correspondence would be now the problem is actually being able to do this in practice to be able to kind of construct the order which is the endomorphism ring of the elliptic curve in a in a constructive way so that you could actually evaluate endomorphisms and see which points are annihilated that's what you would need to do in order to be able to say what this actual isogeny is in practice so the during correspondence is a one-to-one correspondence if the degree is co-prime the degree of the isogeny is co-prime to p in other words this is a separable isogeny determined by its kernel and then um if you take this ideal which the the left ideal i which corresponds to an isogeny the right order of this ideal would then be the endomorphism ring of the kind of target elliptic curve so now what we've done is at least in theory we've built up this other graph which is the kind of the same graph as the super singular isogeny graph described in terms of elliptic curves is just that the elliptic curves are the maximal orders and then the isogenies are these left ideals whose right ideals are the next maximal order or the next vertex in the in the quaternion graph so let's go back to an example so sticking with the p congruent to three mod four where i gave you a specific description of bp infinity and looking at the elliptic curve e zero which is y squared equals x cubed plus x which is super singular for such p it has j invariant 1728 the endomorphism ring of e zero is a particular maximal order which we're going to call it o zero so throughout the rest of this talk o zero is actually going to be this o zero um so for the algorithms that i'm going to describe we're working in this situation where p is three mod four and this is the um what's often called a special extremal um maximal order o zero so if you i'm not sure if you already started on this in the exercises today but if you're using magma or sage you can actually specify p you can specify a quaternion algebra and then you can even just ask for all the maximal orders and it will compute them at least for small p um it will also if you give it a maximal order it should also give you all of the left ideals of that maximal order and then what you could do is you could actually ask for each of those left ideals you could ask for the right order of those ideals and then you'll um get all the kind of neighbors uh of the of the uh order that you started with um okay oh i'm sorry i realized i forgot a kind of an important point in the elliptic curve graph on the elliptic curve side uh we restrict to l isogenes for l equal to some prime which is usually very small like two or three and the um the prime p is extremely large like of cryptographic size so on the quaternion side if we want to build up the that's the that same graph what we need to do is we need to restrict to ideals of norm l so that's the an analog of the degree l isogeny so uh but one of the things that i think is causes a little bit of confusion in this area and we tried to we tried to clarify this um in our in our urocrypt paper in 2018 uh with isentrager haugrin morson and pete is that you can talk about an endomorphism ring of an elliptic curve in a lot of different ways so right now i'm talking about it as being a maximal order in a quaternion algebra so i i wrote down this o zero equals z you know the z module generated by these four elements here one i one plus k over two and i plus j over two but the fact is is that like that's that might not be all that helpful for actually evaluating an endomorphism on the elliptic curve so just below that what i'm showing you is um an actual representation of the endomorphism ring in terms of actual endomorphisms so i think i might have mentioned in my first lecture that um you know we have some endomorphism clearly we have the probanius endomorphism whenever we have an elliptic curve over a finite field um and so that's described here by pi pi is probanius and then in this case we have this special endomorphism which is very nice phi i guess we've forgotten all my greek letters here phi takes x y to minus x comma i y where i squared is minus one so this is an explicit description of the endomorphism ring in terms of actual endomorphisms um and the way to connect it to this o zero i've kind of written written down the map here is like um you know i get sent to phi oh sorry i have a different script here for phi i get sent to phi j gets sent to pi k gets sent to pi phi okay so this is like the best possible scenario where you have a description you have you know kind of all all three things you have an elliptic curve you know what its endomorphism ring is in terms of actual maps in terms of endomorphisms that generate a rank four z module and how they act on the curve and then you also know a description in terms of the quaternion algebra which is a maximal order with its basis and this is a beautiful setup which gives you a lot of advantages in computation but unfortunately or maybe fortunately for the security of our cryptosystems this is not the case for general super singular elliptic curves so i think in the exercises you've been uh you've taken some hopefully done the examples where you you fix a prime and then you use magma or sage to generate all the super singular elliptic curves well you know one way to do that is there's a function that's implemented there so you can just call that function but uh behind the scenes it will um in in the general case it will take and use the Hilbert class polynomial for an imaginary quadratic field where p is inert and then it will take the roots of that polynomial mod p and and or sorry not exactly take the roots just factor it mod p and it'll have linear factors and quadratic factors which correspond to the roots of those correspond to the j invariance of the super singular elliptic curves over fp and fp squared so if you use that function or you follow that procedure and you and you're you got a super singular elliptic curve the problem is you have no idea what its endomorphism ring is you can use that j invariant to write down an equation of the elliptic curve except except for in some very special cases of like characteristic two and three which were which were not in here so you can take the j invariant you can get an elliptic curve and so you'll have some model for your elliptic curve but you have no idea what the endomorphism ring is what the super singular elliptic uh the maximal order in the quaternion algebra or the the the rank four z module in terms of endomorphisms and so that's kind of um uh essentially the very hard problem in this area is is that we don't know how to compute endomorphism rings so um I'll come back to that in a minute um the fact is is that if you knew how to compute endomorphism rings what for general super singular elliptic curves what you could do is you could pass from the elliptic curve description of the graph over to the quaternion side by replacing an elliptic curve in its j invariant with a maximal order in a quaternion algebra and then you could apply an algorithm which we developed and published in ants in 2014 this was with coal and petite and tin yaw which actually can find paths on the quaternion side so in the quaternion in the quaternion version of the graph we can find uh find paths so the way this works is um given two maximal orders 01 and 02 uh you can find the connecting ideal and then um basically at least heuristically you can replace it with you can find a an l power uh an ideal of l power norm which is um which is equivalent to the connecting ideal um so I'm not giving a lot of details here this would take more time to to go into the details but um the connecting ideal uh let me just go back to this reference to Kirchner and Voight um I haven't said exactly how to uh compute this but you can you can compute mu and n such that if you have an ideal of of norm n such that you write the connecting ideal in this form um and then once you have that connecting ideal let's call that i uh you can find an equivalent norm which has um prime prime norm because of the following so if alpha is an element so step three here if alpha is an element of i you can replace the ideal i with an equivalent one which is i times gamma where gamma is alpha star divided by n and so in order to find an element alpha which has prime norm um what we do is uh we take the um the norm form that I described earlier and search through uh all of the past so remember an element alpha in the basis that I gave you for bp infinity is just specified by four coordinates so you're gonna search through a box of um four you know in basically z to the fourth elements that have four coordinates and you're looking for um uh you're looking for solutions to the norm equation such that the norm is prime and so um actually one way to do that is to just use cornaches algorithm like you fix two of the coordinates and then you look for um solutions that will get you what you want for the other two um but in general I guess this is the part which is the heuristic part of the algorithm like we describe our success probability in terms of the heuristics for the distributions of primes and it's just very likely that you that you will find an element of prime norm very quickly and so then um what what we do is we use the um the equivalent ideal i times gamma that I've written here and this so this is an ideal of prime norm and then we use a um kind of strong approximation technique which I'm also not going to describe here to find an equivalent um ideal without power norm and so this corresponds to this is a procedure that works for um the oh for the connecting ideal between um maximum ideal o1 and the special extremo ideal o0 that I um described in the previous slide and so what you do to try to get um to solve this problem between o1 and o2 is you do this procedure twice so you use o0 in the middle and kind of find an l l path back to o0 from o1 and an l path from um back to o0 from o2 and then you concatenate them and then that's what gives you a path from o1 to o2 so um let me just pause here and say that um like like I said this would be very bad news for cryptography if you can actually find the um maximum orders that are associated to elliptic curves and um that's because you would just take two elliptic curves you would find their maximum orders find their endomorphism rings as maximum orders in the quaternion algebra and then you would apply this algorithm the KLPT algorithm and you would find an ideal which is written as a um an ideal which has a norm l to the n and then you would take that ideal and you would break it down into steps and you would find um the path back from uh the the end point of your walk to the beginning so that pulling pulling the path back is also possible if you know both of the orders of the of the two endpoints but the the problem is so computing the endomorphism rings and again there's always this ambiguity when I say endomorphism rings you could mean computing it as the maximum order in the quaternion algebra or you could mean as computing it in terms of the um actual endomorphisms that you can evaluate and even when we're like doing research in this field we're constantly confusing ourselves because people could mean one or the other of those two things or both ideally of course you know both that's the perfect situation where you have for e zero but um in general you don't know either one of those and um so let's talk about how hard it is to find endomorphism rings of of of a super singular elliptic curve so going back to coles thesis in 96 um he essentially said oh okay well here's uh what we call a generic algorithm or like I talked about yesterday a square root algorithm for finding endomorphism rings you just basically walk around the graph until you come back to yourself and you found a cycle this cycle is now an endomorphism you started out at e you ended up at e it's a map from e to e it's it's non-trivial it's a it's a non-trivial endomorphism so if you do this and you can if you could do this once and you could do it twice and so now you'll presumably have two endomorphisms which uh generate the endomorphism ring um and so that was kind of his um you know algorithm in his thesis so unfortunately that's not exactly um correct and so you'll find in the papers of um let me see the win four paper of eisen traeger at all so um I can't sorry I can't remember all the collaborators on that but that was like eisen traeger um park and uh number of co-authors show that it's only going to generate the endomorphism ring if there isn't um any common edge in those two cycles that you found but just assume you found two disjoint cycles and you would actually get the endomorphism ring um and they've also written a follow-up paper on that which is appeared at ants um this past year so it's not quite finding the endomorphism ring but but it's close but anyway it's an exponential time algorithm right so it basically um the birthday paradox tells you that you will in square root of the graph size you know find find this cycle so that's good for cryptography in the sense that we don't have the that doesn't break anything because it's still exponential time um and then so in 2003 I was working on this endomorphism ring finding problem um with Ken McMurdy and um so we wrote down an algorithm with which was essentially the same algorithm that Cervino found at the same time so you'll find two preprints from about 2003 written independently which is another way to kind of build up the correspondence of um endomorphism rings as maximal orders with their the elliptic curves and that is that you can compute um in a maximal order let's say you have um you know sage or magma just spit out a basis for the maximal orders for you and just take one of those maximum orders with its basis now start computing all the elements of norm n for n equals you know one two three four et cetera so you're uh going through and you could see from the way that the norm form looks um if you construct a norm from form from a particular order it will look slightly different than what I gave you for the general algebra so sorry let me go back a few few slides um to this is kind of an important point so here where I gave you the norm uh form for the algebra you can see that you know cd f and g are all integers this is an integer the norm is an integer and also it is uh positive right so because all these things are squared and p is positive and so this is all positive the fact is though that if you look at um the norm form for let me go to the o zero slide that shows you what o zero is so even for this nice uh special extremal order where the basis is one i one plus k over two one plus j over two um if you uh kind of work out the description of the norm form you'll see that there are denominators so now the norm is not necessarily um even an integer and like I said integral elements are those that have integral norm and integral integral trace but in general um it can also be negative because of the way the norm form will look it doesn't look as nice as it did in the in the for the whole algebra but still you can find a way to basically go through um all the possibilities and find all the elements in the quaternion or algebra that have norm n for small n so like I said for one two three four you know like that and those actually correspond to I mentioned this yesterday in the in the q and a those actually correspond to these representation numbers which are the number of elements of norm n correspond to the coefficients of the modular form and the uh the thing that I mentioned yesterday is is that there's a theorem of ser that says that you can determine the order the order is determined by having square root of p of those coefficients okay so this is also a terrible algorithm so p is super gigantic of cryptographic size and now you would need square root of p of these coefficients of the modular form in order to determine the maximum order completely and that is um ridiculous you won't be able to do that in your lifetime um so this is an algorithm which is also exponentially bad and it and it will just compare the number of elements of norm n in a maximal order with the number of isogenes of degree n uh which are actual endomorphisms so again an exponentially bad algorithm but it's something that you can do for very very small p and you can look and you can use this to match up all the maximal orders with all of the um elliptic curves and that is actually what's done in practice and a lot of examples in the papers that we have today um just to see what the graph looks like and how it corresponds to the graph of of elliptic curves so so far as of today we have no classical sub exponential algorithms or polynomial time algorithms for finding paths in elliptic curves and we have no sub exponential or polynomial time algorithms for finding endomorphism rings of super singular elliptic curves so for the future what you have to think about is okay are we going to find any better classical algorithms for the either of these problems or even if we're going to find if uh quantum algorithm experts might find some quantum algorithms for these things so in order for this to be a good um candidate for the um future of post-quantum cryptography um you know both of those problems will continue to need to be studied in the first lecture on uh monday i mentioned like the four main kind of math candidates that are that were considered in the NIST pqc competition these last um four years it's ongoing it's a five-year competition i mentioned code-based cryptography lattice-based cryptography multivariate cryptography and um super singular isogenic graphs and for each of them i put down essentially the date when it was first proposed and you might have noticed that like um you know all of the others were proposed in the 70s or 80s essentially the hard problem was at least whereas super singular isogenic graphs we proposed in 2005 at the hash function competition so you're looking at a system which is roughly 15 years old instead of roughly 40 years old like 30 to 40 years old at least for the others and so what NIST did in the third round of the competition is that they uh specified psych this uh super singular isogenic key exchange as an alternate candidate um that's and stated that it's a candidate that needs um further study so good motivation to keep studying and keep working on these these problems so it's now an alternate for the final for the next stage of the competition okay so in my last um few minutes what i wanted to do was to talk about uh the sickness situation for signature schemes based on super singular isogenes um so we talked i talked about cryptographic hash function in the first lecture talked about key exchange in the second lecture uh both based on the hardness of path finding um so now uh we have another system which is based on the the hardness of path finding um but in 2016 this was proposed by galbraith petit and silva but not really implemented they didn't find a way to uh kind of efficiently implement this and so i wanted to thank yana for pointing out um this paper um sqi sign or i guess some people are saying ski sign from last year appeared at asia crypt 2020 um which is uh kind of a in some sense a very variant of the gps uh scheme but they had to improve the klpt algorithm in order to make this work so with their improvement of the klpt algorithm they proposed um this signature scheme which i'm going to describe to you uh but the the improvement that they made was they were able to find a quaternionic kind of l l path finding algorithm from 01 to 02 for two maximal orders 01 and 02 without going to 0 0 so for the properties for the zero knowledge properties of the um of the signature scheme they didn't uh they couldn't afford to have to go through so they've improved the klpt algorithm so that they can go directly from 01 to 02 uh two maximal orders okay so the setup for their uh system is again they're going to fix some large prime of cryptographic size um there's going to be a um uh and a super singular elliptic curve E0 which is one of the special ones meaning we know what its endomorphism ring is like extremal endomorphism ring 0 0 um just like my previous slide you could like if p is congruent to 3 mod 4 then you could take the one that i gave there um and then you're going to um select an odd smooth not and specify a small prime l as well you're gonna also select an odd a smooth number d which has log p bits so that means um if you're gonna take a an isogeny of degree d that it's going to have to be at least if you translate that into being a walk it means it's going to have to be a walk of at least essentially the diameter of the graph so um the key generation is going to be for someone who wants to prove knowledge of uh something so this is i'm going to describe the isogeny scheme the general signature scheme can be obtained from that identification protocol via the fiat-chamere uh transformation so a prover is going to want to prove knowledge of something so they're going to prove knowledge of an iso a secret isogeny tau so what they're going to do is e is e0 is fixed and known to everyone now they're going to take a random walk and isogeny walk and come up with their elliptic curve ea so this should sound very familiar to you that's like what we did um for the hash function that's like what we did for the first step of the key exchange alice and bob each did this so they're gonna um publish this elliptic curve e sub a and the secret is tau which is the path and so the public key is going to be ea and the secret is tau and now the way that this um commitment scheme works is that the prover generates a random um secret isogeny walk from e0 to e1 and sends e1 to the verifier then the verifier sends the description of a cyclic isogeny um from e1 to e2 of degree d to the prover um so just as an example it could be like degree dd could be l to the power log p that was kind of what i was uh hinting at in the previous slide and then um the response would be that from the from since the prover knows tau prover can construct this isogeny um fee fee composed with psi composed with um tau duo which goes from ea back to e2 and they know that isogeny sorry and then they can construct a new isogeny from ea to e2 of degree d such that fee hat composed with sigma is cyclic and will send sigma uh to the verifier now the verifier will accept if sigma is an isogeny of degree d um from ea to e2 and um fee hat composed with sigma is cyclic and otherwise they will reject it so i think that's a lot of words but let me show you the picture that they have of their um scheme hopefully this will make it a little bit easier to understand so on the left this dotted arrow going down is the secret key isogeny that I said was chosen in the setup and then the commitment isogeny from the prover is the top uh solid blue arrow um and then it goes from e0 to e1 the challenge isogeny is the red arrow down from from the challenger or the verifier and then um the response isogeny is that you can see since um since the prover knows tau and and tau duo the prover can construct this thick blue arrow on the bottom because they can go backwards up the tau arrow and then compose follow the psi arrow and the fee arrow but if they didn't know um if they didn't know tau then they wouldn't be able to construct the isogeny from um ea to e2 and so um being giving the isogeny sigma then the verifier can check um that that has the right properties that were required so this is uh another very nice uh application of the super singular isogeny uh problems so hopefully I've it actually kind of turned out that the structure of the lectures turned out a little better than I had even thought in the sense that we did from the cryptographic point of view we did on the three lectures we did hash function key exchange and then signatures and then from that kind of mathematical point of view uh we talked about um the description of the super singular isogeny graphs also in the second lecture the the properties in terms of an expander graph as specifically a remonaging graph and then um in the third lecture we talked about the description of that same graph as a quaternion graph and the hardness of going back and forth between those two descriptions of the graph so that was uh pretty good I've left out a lot of a lot of other related topics that I could have included but I think that that was a good um set of topics to focus on I thought I would just share with you I've put a couple of links for references in the slides but for the quaternions I actually found what I thought to be a bunch of useful recent write-ups of quaternions in English that I will put on I'll add another slide which I'll I'll add to the slides before sending them today and so I just wanted to end by talking about some of the other graphs that have been investigated and some of the other ongoing work in this field so this is just I guess I'm like a minute or two over but just to end with some of the other things so in our original paper the cryptographic hash functions from expander graphs the cgl paper from 2006 we also proposed other expander graphs for example the lps graphs which are very simple because they can they're kailey graphs in basically sl2fp but cycles were found very quickly in those graphs an algorithm for finding cycles was found by zemer and tillich and in 2008 at Eurocrypt and also that same year in 2008 we extended their algorithm with with pati and kiskate to find pre-images so those graphs are broken from the cryptographic point of view but an interesting thing is that the algorithm that we gave for path finding finding in those graphs is now being used so it's kind of rediscovered in the setting of quantum arithmetic and it's now being used as a way to do efficient quantum arithmetic arithmetic with us for certain quantum infrastructures so even though the lps graphs are essentially broken there are variations that you can take of them that are not yet broken so the morgan stern graph i've mentioned the higher dimensional analogs that we proposed in 2007 but zooming forward now you know almost 15 years there's a lot of ongoing work so other alternate graphs that were have been considered are like the the seaside graph dimension two analogs signatures that i've mentioned here and other variants ongoing work on attacks like pati has an attack that works by using extra the extra information from the knowledge of the images of the torsion points in the key exchange there's ongoing work on understanding the graph structure better so adventures in super singular and paper with yana and sarah arpin and our other co-authors from last year investigating more the structure of the graph and the relationship between the ordinary volcanoes isogenic graphs which are volcanoes and how they embed into the the graph structure for super singular isogenic graphs so sorry for going a couple minutes over time this is just to give you an idea of a large amount of kind of ongoing work in this in this area a lot of other people that are probably not mentioned on this slide so sorry about that if i didn't put your name here so thank you very much for your attention it's been fun talking with you be happy to answer questions if we have time all right well let's start by thanking christen for her very nice lecture and i don't see anything in the chat but if there are further questions we should ask christen now so and rick has a question christen in the chat or are there implementations of k l p t yeah in the chat yes and no so there were definitely experiments in the original k l p t paper which indicated the feasibility and the confirmed like the heuristic analysis of the algorithm um but um actually not exactly sure how does this there's a paper under confidential review for mathcript which has a which has an implementation um i can't remember if it's a honey print already or not so that's why but it will be public soon it should literally be public like within uh two or three weeks if not so that's that's the other one that i know of i'm thinking that the sq sign paper may have an implementation even of their better algorithm which kind of didn't know about until yana pointed that paper out to me so yana i don't know whether you know do you know if they have an implementation um public for a sq sign uh yes i can uh link the could have repository and help so yeah so they do have an implementation for their algorithm available yeah okay good so i would say that's even better hi there are other questions all right i'm not seeing oh i just saw one in chat um this one's pretty long i'm christin do you want to read it are you able to access the chat oh yeah oh this is a really good question this is a really big sticking point in my opinion so um yeah this is kind of interesting okay so i mentioned how you can generate a super singular elliptic curve i think that yana was planning in the in the ta sessions to talk about another easier way to generate a super singular elliptic curve and that is um if the prime is such that like for example p conger to three mind four where we know a super singular elliptic curve it's sitting right there in front of us um there's that one that's a super singular elliptic curve but if you want to hash you have some bit string and you want to hash into the set of super singular elliptic curve so you want to come up with some other super singular elliptic curve based on your bit string can do basically like we do in the hash function which is just to walk from the starting point and now you end up somewhere so that's you know that's one way to do it and that's an excellent way to do it but there's a problem and that is that it leads to the possibility of a trap door so that's what we actually wrote about this in our 2018 eurocrip paper with um pati and eisen trigger haugrin and morrison that um if somebody knows that secret path and they uh now give you a super singular elliptic curve as if it were some random thing but they know that path then they have information which connects them back to that curve that even you know without k lpt because you don't have to you're not in the quaternion side they know a path that goes back to the super singular elliptic curve where you know the endomorphism ring and then they can kind of drag it we have an algorithm for kind of dragging it along so you can break things so if you generate super sync if you rely on that approach to generate super singular elliptic curves then you will be subject to the possibility of this kind of backdoor backdoor attack so instead the kind of theoretical thing that i mentioned which is is that you take um a random could be large imaginary quadratic field where p is inert but it can't be that large because you need to be able to compute the hilbert class polynomial and the hilbert class polynomials are exponentially large although if you use our kind of explicit crt approach you can generate those hilbert class polynomials directly over fp without computing them first with coefficients in z but if you do that then it's true so you know um so you've taken a root of this hilbert class polynomial to be the j invariant and so now you know that that particular super singular elliptic curve you know has cm by that imaginary quadratic field and that could be there is ongoing work looking at the extent to which that can be um that could be problematic it's um some people call it a an orientation that you know on the on the curve so it's possible that that could be a problem all right thanks christin um and yes it is related okay great um all right so i think um that's all the time we have for today um let's thank christin for her wonderful lecture series this week thanks jen thanks everyone and a big thanks to yana who's done a fantastic job on the exercises and the leading the ta session too


This is stock number 11093A. We are here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used Jeep and Jeep Grand Cherokee headquarters. Today we are checking out this super clean 2016 Jeep Grand Cherokee Overland High Altitude Edition. This vehicle has the award-winning 3.6L Pentastar V6 motor. It has been fully-safetyed and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. And this Jeep is 100% ready to go. Red Velvet Pearlcoat is the color. We shoot all of our videos in 1080p, 60 frames per second. So if you have HD capabilities on your computer, tablet, or smartphone device, turn them on right now because it is definitely your best way to check out the quality and condition of the vehicle before seeing it in person. And if you'd like to check out all the photos on this Jeep, in the upper right-hand part of your screen is a link to our website. Click that and check us out there. It comes with the 20-inch Granite Crystal painted alloy rims. You get the Pirelli Scorpion 26550R20 tires. These tires have right around 50% to 60% of the tread left on them. Front fender is in excellent shape. I didn't see any dents or dings on that. This one does have the HID headlamps, the LED running lights, and the projector lamp fog lamps. Front bumper is in really nice shape. Didn't see any major dents, dings, or cracks on that. Kind of get that bronze chrome finish to the grille inserts there, the blacked-out Jeep logo. The hood is in really nice shape. I didn't see any major dents or dings on there. And the headlight lenses are nice and clear. Passengers side front fender is in excellent condition as well. No dents or dings on that. And the passenger side rim, no scuffs or scrapes. As you go down the passenger side here, take note of how clean the body is, how reflective and mirror-like that paint is. We take these HD videos so if you are far away, or even if you're close by, you just cannot make a trip down, but you're still interested in purchasing the vehicle. You can see the vehicle, hear the vehicle, and have confidence in what you were looking at. Before you even get here. So that when you do get here, there's absolutely no surprises. And that is why we offer the Summit Auto FastPass option in the upper right-hand part of your screen. A link right to our website where you can apply for financing, get pre-approved to see a payment quote. Even appraise a trade all from the comfort of your own home. So when you get here, all you got to do is drive the vehicle, love it, you'll be all set to go. Back rim, no scuffs or scrapes. Back tires have just as much tread as the front tires, and it has brand new brakes on the back. Rear quarter is in really nice shape, and as we come around to the back of the vehicle, rear bumper is in just as nice of condition as the rest of the vehicle. You get the dual-tipped exhaust back here, backup parking sensors. The rear gate is in really nice shape. You get the blacked-out Jeep logo, 4x4 logo, and you get an Overland badge back here as well. Also the LED running lights on the tail lamps there. Has the power lift gate. Back storage area is very nice and clean. You do get the subwoofer, part of the Harman Kardon sound system on this vehicle.  Get this gate down, press that button. Has the cover for the receiver hitch. We always keep them inside the vehicles so they don't go missing. Press that button. It'll beep at you three times and then it'll come on down. Down the driver's side, just as clean as that passenger's side. No denser dings on the rear quarter. And for full disclosure, this back rim is in excellent shape as well. Down the rest of the side of the Grand Cherokee, looks pretty good. You do get the heated power mirrors with the blind spot monitoring. Built-in directional signals, they also power fold in. You have the enter and go system, just have to have your key in your pocket. Put your hand in there, it'll automatically unlock. On the doors here we have the power windows, power locks, power mirrors. Memory driver's seat down here, Harman Kardon sound system. Those mirrors do power fold in. It's always like showing that both sides are working, especially on a little bit higher mile vehicles like this one. So you know all the options are in proper working order.  There are no rips, there are no tears on the seats. Side curtain airbags, memory driver's seat, power driver's seat. Both these front seats are heated and cooled and you do get lumbar on the driver's seat.  We'll turn the ignition on here so that you can check everything out. You can see that this Jeep has 115,453 miles. Instrument cluster is very nice and clean. You get the 7-inch LCD display. You also get the compass outside temperature display. This one has the air suspension so it actually can raise up to go into an off-road mode. One and two mode and has an easy exit mode as well. You get the heated leather wrap steering wheel, cruise controls and adaptive cruise controls on the right. Bluetooth and information center controls on the left. It has the paddle shifters right there. This one has the 8.4 3C radio with AM, FM and Sirius XM radio capabilities. It also has the factory navigation system on it. You can do your heated and cooled seat buttons here as well. You can also do your climate controls here. Notice you do get dual climate controls. Pick where it's blowing. You can do everything up there. If you like the more tactile buttons, you can use those. You also have a volume and tune control right there. Back up parking sensors, turn those off and on. Or the front parking sensors as well. Start stop capabilities, you can turn those off. Eco mode, sport mode and stability control right there. Here's your key flop with your remote start right there. And the power lift gate button. And then down here is your Quadra Track 2 system. You get snow, sand, auto, mud and rock. As well as a neutral setting for flat towing. So you can pull this behind a motor home or something. And you can actually raise the suspension right there. With those two buttons right there. You have a four wheel drive low and a downhill assist control. 8 speed automatic transmission. Passenger side seat and floor mat are in excellent condition. No rips or tears on there. Smells very clean inside this Jeep. I don't think it's ever been smoked in. And you do get the power vista roof right there. So that front glass actually power slides. Back is fixed. Home link buttons for your garage door security systems and lane systems, map lights. And then you do get assist and SOS buttons in the rear view mirror. We'll take a quick look at the back seats. And then we'll start it up and check out under the hood. Back seats are just as clean as the front seats. Did want to point out that this one does have the capitalist fuel fill as well. These back seats are heated on the outboard side. No rips, no tears back here. It does have the latch child safety system for any child car seats you may have. Factory floor mats back here as well. Heated seat buttons, two USBs and then a 115 volt 150 watt plug-in. These seats do fold down for extra storage. You can see they go down really nice and flat. And the doors are in really nice shape. You get the child safety locks. Didn't see any corrosion or anything on the bottom of the doors. And we'll start it up and take a quick look under the hood. I would personally like to thank you for checking out the video today. And hopefully from this HD video you will have been able to tell just how clean this grand Cherokee is all the way around inside. Now under the hood we have the award-winning 3.6 liter Pentastar V6 motor. And Jubei is very clean, runs very smooth. And once again this vehicle has been fully safely and inspected by our service shop. Has a fresh oil and filter change. All the fluids have been checked and topped off. And this Jeep is 100% ready to go. And for 115,000 miles I would highly recommend this vehicle on a quality and condition standpoint. And to see more pictures of this Jeep or one of our other 450 new and used cars, trucks, SUVs, minivans, Wranglers, you name it we got it put on our website right there www.SummitAuto.com Full pictures and descriptions of every single vehicle from two locations all at SummitAuto.com. And if you'd like to check out more HD videos you can go to www.youtube.com. Remember to like, subscribe, and share on this video and all the videos that you see there. In fact in a second you will see a link to subscribe to our YouTube channel on your left. Link to more Jeep Grand Cherokee videos like this one on your right if you have not been to our website on the bottom. A link to this vehicle on our website click those check us out. And we're super excited to be offering this really clean 2016 Jeep Grand Cherokee Overland High Altitude and Red Velvet Pro Coat. Thanks again for checking out the video.


Well, here we are back in the car. It's negative. I don't know 12 out today in Minnesota wind shield Maybe a little bit nippier nippier. Yeah, okay. We just saw Eddie the eagle. It doesn't come out for a while So this was a this was a big one for us to get tickets to it was a full house A lot of people are gonna compare it to cool runnings and a lot of people would be correct with that. Definitely You know, we made quite the night of this we saw a movie we went to cub foods did some shopping for you I don't remember what we even picked up, but here we are. We're gonna reminisce a little bit I had no idea what this movie was going in. I don't think you even saw a trailer I haven't seen the trailer. You know, he did send it my way made about as easy as possible Well, I didn't click play for you. So we missed that opportunity This was a pleasant surprise, especially considering it's January when all the shitty movies are out. Yeah, this was this was nice. It's a feel-good movie What camera are you looking at by the way? Are you looking at this one or this one? I was looking in between them. I think This is even more ghetto than usual, but I think if either of it's this one This car side reviews very special. We're using a GoPro as the primary my laptop Which is balancing half on the dash half in your hand And then we have my phone as a light source just blasting me in the face Like a Filipino man usually does Well, there was Christopher Walken Small roll small roll. Yeah, we've got Hugh Jackman Jack and Wolverine. Yeah, Wolverine and then pretending to be an American. Well, he does that often More American than I am really it's more of a man than I am Definitely more than you are. There's no doubt about that. Who's the main guy? I don't even know Eddie I'm not sure but he seemed very similar in looks to the real Eddie the eagle. Yeah, they didn't Michael Edwards No, thank you. They did a good job. Um, you did a good job playing in my guests I don't I don't really know what the real Eddie's like. We thought he had some sort of a disability We looked him up for probably two minutes, which is far more time than I wanted to spend looking up anything Yeah, but I think he's just very quirky. I just have a lot of ticks I mean, you definitely had leg problems yet vision problems. This movie is all about overcoming obstacles This is the textbook Hollywood feel-good film It follows all the staples that you expect, you know, we have the the roaring music Yep, which got a hauling out hauling which you know what the music is the highlight of the I agree It it brought me back to 1988.  Yeah, the music The villains the villain The villain is overcoming obstacles and disabilities and whatnot But there is so many of these little side characters that are just so over the top. I'm talking to his teammate the British Olympic commissioner Like you'll never compete in the Olympics who remind me of Michael or a Pullman Bill Pullman a little bit. Really? I thought so That movie more than anything else made me never want to ski. Oh God take a jump I mean, I like downhill skiing, but I couldn't even do that like 10 meter jump But that time they 10 minute was an option. I think it was a 15. Well, that's the one that pulls you in It might have been 15. Well, what pulled me in was the fantastic Dynamic between Jackman and Eddie the actor that I don't know the name of the essentially Jackman was the John candy You know this movie Do we say that there was a reference to cool running in the film because apparently there was the same year that The Jamaican bobsled team horrifically lost Spoiler alert they don't show up in the movie. Yeah, they don't show up Although you mentioned it would be awesome if there was like Like there was a computer generated John candy that walks by and he's like good luck kid. It was very 90s It was very late 80s very early 90s as far as just the dressing the music the just the beats as you said It plays out so like mighty ducks like the couple montage Yeah, just the beats were beat for a bar fight, of course A little bit of rootiest to it. There was the underdog Well, there was a lot of underdog and I like there's a lot of hugs a lot of hugging to go around and I like that I like when there's hugging in the movie.  You know, I like that. I like the spirit I think we you know, I think we covered do we cover a pheromones or more that needs to be said about a film like this I don't necessarily think so but I think just so that your Viewers know that it was just a fun film when it comes out I wouldn't say you necessarily need to run out to the theaters to see it But whether it's on dvd or otherwise I would recommend checking it out I I was there hasn't been a movie like this for me in a while And so I think that maybe you should go see it in theaters just because you I found myself Smiling almost the entire two hours or hour and 45 minutes while I was watching the movie like at the end scene Yeah, I had a big grin on my face And I only assumed like that if you could look at everybody's face in the theater They also had the same grin people were laughing and smiling and talking when we left the theaters It's not this is not a martian. This is not a you know, Revenant You don't have to think too hard No You're just going and have a good time You're pretty confident on how it's going to play out Especially if you know the background history of eddie, which I did not Now I do I'm glad I know it now But it's so hard to rate these kind of movies. It is run of the mill. It is You know everything you expect I'm gonna go an 8.5 though I'm gonna go higher than I said originally from the eight I like a good true story even if it is only 10 to 15 true and quick time just Quick time quit does that mean nothing recorded? Do you think? I don't know let me I was hoping you're gonna give it an eight because I was gonna give it an eight also For the 88 olympics. Oh, well, I can I can drop down I can scale back for the schtick Review scores really don't matter though. You know an 8.5 today is a nine tomorrow is a 7.5 the next day It's just I got a great feeling from this film. I can easily rewatch it And it's you know, it's a family. It's family friendly minus the part where Hugh Jackman Pretends to have an orgasm for far too long way too long too long But regardless other than that, I thought it was like a well made pg-13 movie I mean you don't see a lot of pg-13 comedies like this anymore. No, all right. That's it Thanks for joining us in the car. We'll see you next time


Free ways the narcissist destroys your health. The narcissist will mess with your mind. They will confuse you. They will gaslight you. They will sow seeds of doubt within you. Making you question your own memory perception and judgment. Some days they won't be so bad.  And then out of nowhere, they will discard you. You will constantly feel like you're always trying to get it right. You're always just trying to be good enough for them. You're always trying to do enough. But just when you think you've got it right, they change the rules. Just when you think you've achieved their desired standards and expectations, they move the goalposts. They change the rules while you are trying to do something. In order to make it difficult for you. In order to gain an advantage over you. They will throw a temper tantrum. They will rant and rave for hours. And then at the end of it, they will come back to you and apologize. When you are around a narcissist, you will always be left feeling of balance. But we always try to rationalize what they're doing. We always try to justify their behavior. Because we want to accommodate the narcissist, our brains quickly learn the rules around us. Our brains quickly learn the way things are or should be done. We were preconditioned in our childhoods to be kind and compassionate. To be forgiving. To always find a way to justify someone else's behavior. And it is our preconditioned minds that keep us stuck in these narcissistic relationships. You may try to justify the narcissist's behavior. You may try to deny what you're really going through. But your body is always going to be honest. It's always going to tell the truth. So even though your mind may be saying that the narcissist isn't that bad, your body will begin to break down. When you are around a narcissist, you develop all kinds of health problems. Even though you may have been healthy before you got involved with them. Being around a narcissist will cause you a lot of stress. And it will affect your physical body. So here are three ways the narcissist destroys your health. One. A constant state of alert. They go back and forth from the idealization to the devaluation. Which can be very confusing and stressful. You're always on edge. You're always walking on eggshells. You're extremely cautious about your words and actions. You're always anticipating. When they're going to find something else to complain about. You're always anticipating. When they're going to demand something from you. When they're going to demand something from you. You're never at rest. You never experience a calm state of mind. The narcissist is like a parasite. They're trying to extract as much energy as they possibly can. And this activates your fight or flight response. It keeps you in a constant state of alert. It affects your ability to heal, grow and rest. It affects your path. You have to keep careful watch for possible danger or difficulties. You're always on edge. Anticipate in the narcissist's next move. And this causes damage to living tissues in the body. Which then results in inflammation. Which is the body's natural reaction against injury and infection. It affects every organ and tissue in the body. It affects the brain. It causes neurological disorders, chronic fatigue, arthritis and fibromyalgia. As well as many other illnesses. When you are involved with a narcissist. It increases your inflammatory response. Which results in inflammatory disease. 2. Metabolic rate. Being around a narcissist will affect your metabolic rate. As the fight or flight response intensifies. Your body will try to defend itself. And it will do this by slowing down your metabolic rate. In an attempt to deal with stress. But this can also result in you feeling weak and tired. It will affect your ability to think. You will feel confused and disorganized. You will find it hard to focus or put your thoughts into words. You will feel like you're not yourself. And you're unable to think clearly. As your metabolic rate decreases. You will also experience weight gain. You will have difficulty losing weight. The combination of the chronic inflammation. And the reduced metabolic rate. Creates a psychological effect. Where it feels like you're under a spell. You begin to lose touch with reality. You begin to lack awareness and sympathy. It keeps you trapped in the narcissist world. Where they then gain the ability to control and influence you. As though they have magical power over you. You lose control of your thoughts and actions. You become so devoted to the narcissist. That it seems like they have you under a spell. This eventually produces a metabolic disorder. Where you continue to gain weight. And you could be at risk of high blood pressure and diabetes. Free. Self worth and self esteem. As your health begins to deteriorate. And you develop more and more health conditions. The narcissist is not going to have any consideration for you. They will only use this opportunity to devalue you. They will take advantage of your health conditions. And use it to further abuse you. They will provoke and challenge you. With insult and remarks. They will put you down. They will try to convince you. That your physical health problems are the result of your own characteristic attributes. As though you are faulty or defective. As though you are not deserving of respect or attention. As though you do not have any attractive or appealing qualities. As though you are incapable of inspiring love or admiration. Which only destroys yourself worth and self esteem. It makes you feel like you're not even worth the time and the energy. That it would take to improve yourself. It causes you to neglect yourself. Empaths tend to ignore what their body needs. They focus on everyone else. They're so focused on everyone else. That they don't realise what's happening to themselves. Until they end up with all kinds of physical health problems. If you are logged into a relationship with a narcissist. Your main focus needs to be on improving your physical health. Try to escape from being under their spell. Thank you for watching. I hope this video has made it with you. Please like, comment, share and subscribe. If you would like to donate. My PayPal link is in the video description. Coaching Inquiries. You can email me at coachnetnarksurvivor.uk Thank you for watching and I'll talk to you soon.


Amen. Alright, keep your place there in Proverbs chapter 12. That'll be kind of our starting point this evening.  The word deceit actually pops up three times in Proverbs chapter 12. So it's basically if you look, it's a chapter in the Bible that's talking a lot about, you know, not lying, telling the truth, things like that. Okay, especially in verses number 22 and number 19 which we'll get to. So keep your place there, kind of mark that in your Bibles and we're going to come back to Proverbs chapter 12. But this evening we're going to talk about lying. We're going to talk about lying is going to be the subject of our sermon tonight. Turn to John chapter 8. John chapter 8. We're going to talk about this idea of telling the truth and not telling the truth. Look at John chapter 8. So the Bible tells us where lies come from and, you know, why people lie. And that's what we're going to look at this evening. Look at John chapter 8 and verse number 44 where the Bible says, year of your father the devil and the lusts of your father you will do. He was a murderer from the beginning and abode not in the truth because there is no truth in him.  Turn to Genesis chapter 3. So the Bible says that Satan himself is the father of all lies and that he was a liar from the beginning. You know, his religion, as we found in the American heresy series, his religion itself is a lie.  You know, the religion of works. Turn to Genesis chapter 3 and look at verse number 1. Genesis chapter 3 and verse number 1. The Bible says this, of course, is the famous story of the serpent beguiling Eve. And in Genesis chapter 3 and verse number 1, the Bible says, now the serpent was more subtle than any beast of the field which the Lord God had made. And he said unto the woman, Ye hath God said, ye shall not eat of every tree of the garden. And the woman said unto the serpent, we may eat of the fruit of the trees of the garden, but of the fruit of the tree, which is in the midst of the garden, God hath said, ye shall not eat of it. So she's telling the serpent what God told her.  And the serpent said unto the woman, ye shall not surely die. Now that was a lie right there. The Satan was lying to Eve. So when we lie, that we are actually obeying Satan and not God when we lie.  Turn back to Proverbs chapter 12. So we talked this morning about coronavirus and no one here has coronavirus. So everyone kind of sat back and ate a nice little bag of gummy worms.  And that we're a bunch of liars. So we're going to look at why that is tonight. Proverbs chapter 12, look at verse number 19. The Bible says, the lip of truth shall be established forever. But a lying tongue is but for a moment. Look at verse number 22. Lying lips are abomination to the Lord. But they that deal truly are His delight. There's that word abomination again. Lying lips, the Bible says, are an abomination to the Lord. That means the Lord hates it. Period. So why would anyone, especially Christians, lie? I mean, everybody knows that it's wrong.   I've never heard anyone out soul winning and all the years that I've been soul winning say that lying is a good thing. Not once. Okay? So why would people admittedly do it? Why would people lie? Especially Christians. So tonight, I want to give you four reasons that Christians lie.  So what you're going to do tonight is as I preach to this sermon, you're just going to check yourself. And you're going to say, is this me? And you say, oh, you know, maybe that is me. Then I'm talking about you. That's how you check yourself with the sermon tonight. Okay?  I hate to break it to you. You know, we're all liars. Turn to 1st Samuel chapter 15 and let's look at the first reason that Christians lie. So we shouldn't do it. I mean, it's an abomination to the Lord. It's of the devil. It's of Satan.  Why in the world? We shouldn't be doing it. We shouldn't be lying, especially as Christians. The first reasons that Christians lie is this. They blame others for things that they have done. They blame others for things that they have done. Look down at 1st Samuel chapter 15. Here we see King Saul, and he's done something wrong, and let's look at how he handles it. All right? Look at 1st Samuel chapter 15, and we'll start reading in verse number seven. So Saul was to go to war against the amicholites, and he was supposed to destroy everything, all the animals, all the people, everything. He was supposed to leave nothing alive. And the Bible says in verse number seven, and Saul smote the amicholites from Havilia until thou comest to sure that is over against Egypt. And he took Agag, the king of the amicholites, alive and utterly destroyed all the people with the edge of the sword. But Saul and the people spared Agag and the best of the sheep, and of the oxen, and of the fatlings, and of the lambs, and all that was good and would not utterly destroy them. But everything that was vile and refuse, that they destroyed utterly. Then came to the word of the Lord unto Samuel, saying, It repented me that I have set up Saul to be king, for he has turned back from following me, and hath not performed to my commandments. And it grieved Samuel, and he cried unto the Lord all night. Here we see Samuel, you know, he's advocating, right? He's upset that Saul has done this. And when Samuel rose to meet Saul in the morning, it was told Samuel, saying, Saul came to Carmel, and behold, he set him up a place, and has gone about, and passed on, and gone down to Gilgall. And Samuel came to Saul, and Saul said unto him, Blessed be thou of the Lord, I have performed the commandment of the Lord. Now, is that true? Was that a true statement? What Saul just said? He said, I have performed the commandment of the Lord. Look back at verse number three. This is what Saul was supposed to do.  He was supposed to kill everything, and he did not do that. Go back to verse number 14. And Samuel said, What meaneth then this bleeding of the sheep in mine ears, and the lowing of oxen which I hear? And Saul said, They have brought them from the Amicholites, for the people spared the best of the sheep, and of the oxen, to sacrifice unto the Lord, thy God, and the rest we have utterly destroyed. Then Samuel said unto Saul, Stay, and I will tell thee what the Lord had said to me in this night. And he said unto him, Say on. And Samuel said, When that was little in thine own sight, was that doubt made head over the tribes of Israel, and the Lord anointed thee king over Israel.  Wherefore, then, didst thou not obey the voice of the Lord, but didst fly upon the spoil, and didst evil in the sight of the Lord? And Saul said unto Samuel, Yea, I have obeyed the voice of the Lord. Again, he lies, and have gone the way in which the Lord sent me, and brought Agag, the king of Amalek, and utterly destroyed the Amicholites. But the people took up the spoil, sheep, and oxen, the chief of the things which should have been utterly destroyed to sacrifice unto the Lord, thy God, in Gilgo. So first of all, Saul did not do what the Lord commanded him to do. That was a lie. And instead of owning up to it, he blamed others. He blamed other people for his shortfall, for his mistakes. Okay? So what is it? I mean, what's the application? What's the takeaway here of this first point? The first point is that you need to own your mistakes in your life. No one can throw you under the bus if you do it first. Look at it that way. At work 18 years ago, 18 years ago, that's how old I am, at work I still remember this like it was yesterday. We were under such pressure at my job that we felt like if we missed a deadline we would all be fired. That's the kind of pressure that we were under. And we were under pressure to get this design done, and it came to a Friday, it was a Friday. I remember because I went out for a walk with my wife that night, and I remember how I felt, but it was a Friday afternoon and everybody had left the office, and I found out that I had made a mistake and we wouldn't be able to ship the design out for two weeks. I made a mistake. It was like a little tiny word in a file that made some simulations run in the wrong way. And I knew that this mistake was going to be a huge deal to people, and I immediately sent out an email and I tried to call people but nobody was available because they had all left.  That's the kind of pressure that that we were under. I thought for sure we're going to be fired. So Monday morning I went in and we always had a meeting every Monday morning and I just went into the group of people and to the manager, and I was I think I was a third year engineer or something. So I mean I wasn't brand new out of school, but I mean I wasn't an expert, but I should have known better. I made a mistake, and I just threw myself on the tail. I said I made a mistake. I've restarted everything over the weekend. That's what I can do, and there it is. Right there, right in front of everybody. And look, nothing happened. The manager of the project was like, okay, thank you for telling us. Let's just do what needs to be done to get it fixed. And nothing happened, but I didn't try to cover it up. And it was the longest weekend of my life. I mean probably not the longest weekend of my life, but I can still remember the stress of it because it was a terribly stressful situation. So just own it immediately. Don't do what Saul did. Okay, otherwise you're just digging yourself a deeper hole. Let's look at see how it worked out for Saul. Look at 1 Samuel 15 and look at verse number 26. Verse number 26. And the Bible says, and Samuel said unto Saul, I will not return with thee, for thou hast rejected the word of the Lord, and the Lord hath rejected thee from being king over Israel. And Samuel turned about to go away, and he laid hold upon the skirt of his mantle, and it rent. And Samuel said unto him, the Lord hath rent the kingdom of Israel from thee this day, and have given it to thy neighbor of thine that is better than thou. So he lost the whole kingdom over this rebellion. That's what Samuel called it. He called it rebellion against the Lord. Turn to Exodus 32. Let's look at another example of this from the text this morning. Turn to Exodus 32. Exodus 32 and look at verse number 22. So this is Aaron. So Aaron has, you know, Moses has come down from the mountain and he's seen Aaron. And Aaron has led the people into this horrible sin of worshiping, you know, this golden calf and committing fornication and whatever else that they were doing. And in verse number 22, this is Aaron's answer to Moses. And Aaron said, Let not the anger of my Lord wax hot. Thou knowest the people that they are set on mischief. So he blames the people too. He just blames the people. For they said unto me, Make us gods which shall go before us. For this Moses, the man that brought us up out of the land of Egypt, we want not what has become of him. And then of course he says that he threw all the things into fire and it just out popped this golden calf, right?  He blames the people just like Saul did. Go back to Genesis chapter 3. First book of your Bible. Go back to Genesis chapter 3. Genesis chapter 3. So Eve is, is, is deceived by Satan and she eats of the tree and then she convinces her husband to eat of the tree as well. And when God asked the man what happened, what does he do? Look at verse number 12 of Genesis 3. And the man said, The woman who gave to me, to be with me, she gave me the tree and I did eat. So he blames the woman and sort of blames God.   He doesn't take any responsibility. Look down at verse number 17. And unto Adam he said, Because thou hast hearkened unto the voice of thy wife and has eaten of the tree, which I commanded thee, saying, Thou shalt not eat of it. Cursed is the ground for thy sake and sorrow shall thou eat of it until the days, in all the days of thy life. So look, when you make excuses and blame other people, it just angers the Lord. It angers God. It makes him mad.  When you're in a leadership role, you own what the people that you're leading do. I mean, Adam, he had better own what he was supposed to be leading his wife. That's what God said in verse number 17. He was supposed to be leading her in that way. Saul was the leader of the people. Aaron was the leader of the people. You can't blame the people that are underneath you when you're the leader. You own those mistakes. So look, there's a lot of people who are leaders. Like anybody who's married, any man who's married is a leader. I mean, whether he knows it or not, but there's very few people who are good leaders. And a good leader would never do this. Would never do this. A good leader knows that although the people beneath him can mess up, that ultimately the failure is his own. Because he was in charge of that situation. Okay? So that's the first reason that Christians lie, is to blame others for mistakes that they should own themselves. The second reason is this. You know, being different, being a different person around different groups of people. Turn your Bibles to Matthew chapter 26.  Turn your Bibles to Matthew chapter 26. Matthew chapter 26. And we'll start reading at verse number 69. Matthew chapter 26. This is the story about Peter denying Jesus. Okay? Jesus has just been arrested. He's standing before the high priest. People are smacking him around. They're starting to get violent with him. They're interrogating him.  And this is the Peter's denial of Jesus. Look at verse number 69 in Matthew chapter 26. Now Peter sat without in the palace and a damsel came unto him saying, thou also was with Jesus of Galilee. But he denied before them all saying, I know not what thou sayest. And when he was gone out into the porch, another maid saw him, and said unto them that were there, this fellow was also with Jesus of Nazareth. And again he denied with an oath, I do not know the man. And after a while came unto him, they that stood by and said to Peter, surely thou also art one of them for thy speech bereithy. Then he began, then began he to curse and to swear saying, I know not the man.  So you see, Peter was trying to literally convince people that he was not who he was through lying. He was trying to show people that this was not who he was. You say, ah, but that was for a good reason. You know, he was, you know, look, is there ever a good reason to deny Christ? I mean, is there ever a good reason for that? Do you say that this was a very specific sin for a specific time and a specific event? Well, what about the Christian who's one person in church and who's another person when they leave church on Sunday night and start their life during the week? What about the person who's one way around brothers and sisters in Christ in another way around another group of friends that they have? What about that person? You don't think that those people exist? What about the person who you couldn't even tell the difference between them and the other people in the office or at work? What about that Christian? It's the same thing. The way they talk, the way you dress, you know, what about around your family? What if you're this one person at church and then around family so you don't offend people, you're another way? It's the same thing.  And look, everyone will know it. There is, look, there is no indication at all in the Bible that these people even believe Peter at all. There's no indication. There's no indication that they thought for a second that he didn't know Jesus. Why do you think they kept asking him? Why do you think people kept saying, yeah, no, it is you. No, it is you. No, we know it's you. And then he starts to, what does he start to do? He starts to curse and swear. He starts to curse and swear so he sounds like everybody else to try to convince them. He overcorrects. I mean, they kept asking him over and over until he literally left. Look at verse 75.  And he went out and wet bitterly. He just left. They didn't believe him. He just left. Think about what those people must have been thinking. Look, first of all, people who are fake are generally unconvincing in my opinion. Peter was overcorrecting. He was trying to curse and swear. People weren't buying it. They weren't buying it. And by the way, this is you.  Think about what those people must have thought about Peter.  They knew it. And he's sitting there and you say, I don't know him. And then he starts cursing and swearing.  Talk about a fair weather friend.  Your testimony will be a joke to these people. You know, at least if you have consistency in your walk, I mean, even if people don't agree with your walk, at least they'll respect your testimony. Look, you know, the idea of a testimony, just that word testimony, that was foreign to me when I became a Baptist. After I got saved and got in a Baptist church, the word testimony was foreign to me. And basically it has two general meanings. First of all, you know, people have a testimony, a lot of old IFB churches. They wanted you to stand up in front of the church when you became a member of the church and tell your testimony to everybody, how you got saved. Right?  That's your testimony. It's your walk. And you know what? That will be the power that you have, you know, against people that don't believe in what you're doing. Is that testimony? Is that walk? Don't ruin it.  That's what these people want to hear.  You know what they want to know? You know what they want to think in their mind? They want to think in their mind. You know what? It's all fake. It's just fake. They're just fake Christians. They're just trying to lift themselves up over everybody. Look, I'm not trying to lift myself up over anybody. I'm trying to do what the Bible says I should do. Period. It's not about being above anybody else, or you know, judgmental about anybody else. I'm just trying to do what the Bible says I should do. And that should be your testimony. But people that, look, the dead Christian, the person that saved going to carpet church or whatever church they're going to, do nothing with their life, that's what they want to hear. They want to think that this is fake to you. They want to think that this is a show. Because guess what?  They don't want to do what you're doing. They don't want to have to change their life. That's why they're in rock concert church. They want, they want to believe, they want you to show them that you're fake. That's what they want. Don't be fake. Be real. The third reason, turn to 2 Samuel 15. This one's harder to recognize. This one's harder to recognize than the first two. The third reason is this. The third reason that Christians lie is to cast shade on others. To cast shade on others. 2 Samuel 15, we see the story of Absalom when he came back into the kingdom. And Absalom subtly cast shade on his father, on his father David, who's the king. Look at 2 Samuel 15. Look at verse number one. And the Bible says, and it came to pass after this, that Absalom prepared him chariots and horses and 50 men to run before him. So he's trying to make himself look super important. He prepared him chariots. Nobody did this for him. Nobody put him in his position. Recognize that. And Absalom rose up early and stood beside the way of the gate. And it was so that when any man had a controversy to come, to came to the king for judgment, then Absalom called unto him and said, of what city art thou? And he said, thy servant is one of the tribes of Israel. So here Absalom, he's sitting in the way of the gate. He's not sitting in the king's gate, but he's sitting in the way of the gate and he's intercepting people that are coming to the king for judgment. In verse number three. And Absalom said unto him, see thy matters are good and right, but there is no man deputies of the king to hear thee. Absalom said, moreover, oh, that I were made judge in the land, that every man which hath any suit or cause might come unto me, and I would do him justice. So in verse number three, he's saying there's no man deputies of the king to hear thee. He's casting shade on the king, on the leadership.  Oh, but only, but I can help you. Because I'll do justice to every man, unlike the king, is what he's saying. He's casting shade on David's leadership. He's implying that David wasn't leading correctly. Look at verse five. And it was so that when any man came nigh unto him to do obeisance, they put forth his hand and took him and kissed him. And on this manor did Absalom to all Israel that came to the king for judgment. So Absalom stole the hearts of the men of Israel. He intercepted the people and he cast shade on his father's leadership behind his back. So let me ask you a question. Was Absalom qualified to give this leadership advice? I mean, just think about it. Just a few paragraphs before, to get Joab's attention, he burns his field down. I mean, he was not appointed to a leadership role, even though his father was the king. I mean, he was a prince and he had no responsibilities given to him. Let's just tell you something right there. This man was a prince and he was this spoiled brat that just burned somebody's field down to get attention. He had no qualifications and he was not appointed to any role like this. So, you know, the lesson here, the takeaway is, you know, be careful not to assume roles that aren't yours. You know, stay in your lane. You know, with Absalom, the severity of what happened to Absalom was very serious. So especially, you know, refer people to leadership. Don't sit in the gates is the bottom line. What Absalom was doing was behind David's back. It ended in a short coup, but ultimately, it caused his death. It ended in his life was taken because of it. So, you know, in general, you need to make sure, you know, what Absalom should have done if he was sincere or what the guy that was listening to Absalom in verse number three when he said, but there is no man, deputy of the king, to hear thee. That guy should have said, well, let's go talk to the king about it. Let's go talk to David. But Absalom wouldn't have wanted that because he was scheming behind the king's back. So you need to be careful about that. It will always come back around on you.  Number one, you need to recognize that if you're listening to that, you're part of it. Period. And number two, those types of people that are constantly trashing people behind other people's back that, you know, the guy that's not there is the one he's trashing. When you're not there, he's trashing you. That's how those people are. And, you know, if he hasn't done it yet, he will eventually. So look, don't listen to it. Don't do it. You know, let's go talk to the king about it. It should be your reaction. You'll find those people in churches too. Those people will come in churches. They'll come in and they will want things done a certain way and they'll go behind everybody's back and say, I don't think we should do it this way. And I don't think we should do it.  I mean, we don't have it here, but I mean, we're going to see it here. I mean, I'm just warning you again. Okay. This is one thing that's going to happen here too. All right. So don't just don't be part of it. Don't do it. Let's go talk to the king. That's the bottom line. All right. This guy, I mean, I told you guys about a guy I met a couple nights ago. This guy did, I just had it happen to me actually. I'm sitting here and this guy comes up to us and we're at dinner with my whole family and he starts asking about, you know, why we're in Fresno. So we told him, you know, we started a church here and we're leading a church and he's like, oh, are you the pastor? I'm like, no, I'm not the pastor. And, and he says, right away, he picks up on that. He picks up on that I'm not the pastor of the church. And then he says to me, he says, well, what's the difference between the pastor? It's Holy Spirit and your Holy Spirit. I said nothing, but I'm not the pastor. You know, 1st Timothy, 1st Timothy, chapter three, verses one through seven has some very specific qualifications of the pastor. I am not the pastor of this church. Period. And, and he starts trying to cast shade on the pastor of this church and using, you know, what pride that I might, maybe I should be the pastor hot dog guy. You know, wow, you make good hot dogs and I'm the greatest. Thank you. I'm coming back here every day, but no, but you have to recognize those things. It's very clear when people are doing that. And that's one of the things that I picked up on him with him. And I was like, this is not a good guy, right? Because you got to pick up on these things when people start doing this.  I'm sure that's exactly what he is as he tells me that, you know, his hot dog stand is holy ground.  It's a shame because the hot dogs were good, you know? Anyway, just be careful with these things. Okay, people will try to cast shade on leadership especially. And you need to recognize that you need to pick up on it. All right? Because if it goes too far, it'll be very bad for you. Right? Like, it costs Absalom his life. Okay? All right. Reason number four that Christians lie. Go back to 1 Samuel chapter 15. Go back to 1 Samuel chapter 15. I mean, it's an interesting Friday night though. You go out for hot dogs and some guy tries to get you to join a cult.  So, I mean, it's an interesting story. It's one thing.  I'm like, don't touch me, bro. We're going to have a problem here. All right. What are we talking about? Okay. Manipulation.  Another reason Christians lie is to manipulate people. Look at 1 Samuel chapter 15 and verse number 20. And Saul said unto Samuel, Yay, I have obeyed the voice of the Lord. He lied, and have gone the way which the Lord sent me, and have brought ag-egg, the king of Amalek, and have utterly destroyed the amicolites. But the people took of the spoil and the sheep and oxen, the chief of the things which should have been utterly destroyed. So, we already saw that part. But then look what he does here in the last part of verse number 21. The people do it and to sacrifice unto the Lord thy God in Gilgal. He spiritualizes it. And you will see this all the time from Christians.  Saul attempted to spiritualize his disobedience, which, by the way, just made God matter. It just angered the Lord even more. But you know what? You'll see people do this all the time. They'll have a wrong decision in front of them, and they will spiritualize it. I can't do what God clearly commands me to do. Enter spiritual reason here. People do it all the time. Christians do it. Look, you'll never hear people say, I don't want to go to work because I'm lazy. You'll never hear that from Christians. I don't want to get in a good church because I love money. You'll never hear people say it that way, will you? I'm not going to sell out for the Lord in my life because I just love this worldly life that I'm living. You'll never hear people say it that way. Never. Instead, you will watch people spiritualize the wrong decisions in their life. They'll spiritualize it.   And look, Pastor mentioned this in his sermon on Sunday morning when he was here. But look, it will usually take people that do this years to realize that they have made the wrong decision if they realize it at all. That's why I wish people could just listen to preaching or wish the younger people could just listen to preaching and just not make those mistakes and not have to wake up in 10 years and be like, whoa, why didn't I just do that? I could have not gone through all this mess that I'm in right now. I could have not fallen into all these pits.  So look, it's unfortunate, but people will do it and you'll see it happen. Turn to Revelation chapter 21. So look, there's a lot of reasons that people justify lying. You know, we use lying actually out soul winning as a sin, example of a sin that everybody does.  Oh yeah, everybody's lied. But make sure we're using it in the right context is what I want to get at tonight as we as we close this evening. Look at Revelation 21.8. Everybody knows this verse, but the fearful and unbelieving and the abominable and murderers and hormongers and sorcerers and idolaters and all liars shall have their part in the lake with burneth with fire and brimstone, which is the second death. Look, it's a sin that we all commit, but the reason that it's listed with all the big sins is because it is a big sin. See? So make sure you're not presenting it to people when you're out soul winning like, oh, it's just nothing. We all lie. It's no big deal. No, it's something we all do, but it's a big sin. It's an abomination to the Lord. So, I mean, make sure you put it in the right context. It's a good example in Revelation 21.8 because everybody does it. Not everybody's a murderer. Not everybody's a hormonger. You know, but everybody's a liar, but it doesn't mean it's not a big sin. So don't let us get, you know, used to it because we say it so much. It's a big deal. It's a big deal. And here's the thing. It has big consequences. Big sins, big consequences. Not all sin is equal, folks. For kids, it should be a huge deal because guess what, folks? Lots of little kids in this room. It's easy to tell when your five-year-old's lying. It's easy because they're five. They're not as smart as you. They're not as experienced as you. It's easy, but guess what? They get to be 12, 13, 14 years old. Not so easy anymore because now they have experience. Now you've got a child who's got 10 years experience in lying.  That's like five years at a 40-hour-week job. If you raise a kid that is used to lying in 10 years, they're going to be an expert at lying. Lying needs to be one of those things that when your kids are small, you come down hard on them. And I mean little things. It could be little. Did you take the cookie? No, I didn't. And you know they took the cookie and you know they did it. That needs to be punished hard because they lied about it. It's not that you don't have any more cookies. It's not that you can't make more cookies. It's that they lied. They need to know that lying is serious because it starts out small but it will get serious and it'll get very serious when they get older. So like kids, kids, kids and adults, listen, remember this, turn to Luke chapter 8. Here's the whole thing when you think about lying. Here's the whole thing. Turn to Luke chapter 8 and verse number 17. All the kids and even the adults have to know is this. In Luke chapter 8 and verse number 17, here's your answer when you think about lying. The Bible says in Luke 8.17, for nothing is secret that shall not be made manifest. Neither anything hid that should not be known and come abroad. It's going to be found out. If you're saved, you're not getting away with it. So kids and adults the same, you just picture yourself getting caught because that's what's going to happen. Picture it coming out. When you want to blame somebody else for a mistake, you picture your integrity and your name being destroyed. When you think about blaming somebody else for your mistake, when you act differently around others, you picture your testimony being destroyed. Just picture it because that's what's going to happen. God promises you that. You're not getting away with anything.   In the case of Absalom, Absalom's case is the worst case when he's actually casting shade against leadership, against spiritual leadership, all those types of things. Look, if I would turn against the pastor of this church, it pains me to even say something like that.   You have to think about the consequences of these things. When you try to manipulate others into thinking that you're doing the right thing, just picture your life being destroyed because not only is that poor character, but you're actually, think about it, when you're trying to spiritualize the wrong decision, not only is it poor character to do that, but you're actually making the wrong decision and going down the wrong road. Two things. So you're destroying yourself in two ways.  So just look, you're not going to get away with it. Stay above board. Stay above board in your life. Operate that way and let the chips fall where they may. Look at that situation with my job. Nothing came of it. Nothing came of it. It would have been very easy for me to try to make excuses and maybe try to pass blame on somebody else. I got this file from so-and-so. Nobody would have bought it, but I owned the whole thing and all that really came of it is my boss brought it up for like the next three years in, you know, like reviews and things like, remember that time you messed that project up? I'm like, yeah, I remember that. That was it though. I mean, my boss just kind of brought it up to me and it's like, whatever, I'll take it. But stay above board. Let the chips fall where they may. Own everything right away. And look, just let God handle it from there. God will handle it. You do what you're supposed to do and God will handle it. That's the answer to so many things. Right? This church, we're just going to do what we're supposed to do and God will handle everything else. We're just going to do the work that we're supposed to do and God will handle everything else. I'm just going to do what I'm supposed to do and God will work everything else out. When it comes to that temptation of should I lie, should I not tell the truth here, just tell the truth. Just make the decision in your mind because look, it can happen quickly. You can be tempted to lie quickly. People can put you on the spot. You just stay above board with everything all the time and it will always work out. God will handle it for you from there. Okay? Let's bow our heads and have a word of prayer. Dear Heavenly Father, we thank you for the Word of God and all the lessons that we can learn. Lord, from the Bible, I thank you for today. I thank you for the wonderful time out soul winning. I thank you for the souls that were saved, Lord. Lord, help us to have enough faith to know that if we just follow the right path and stay above board and just tell the truth in our lives, Lord, that you'll just take care of the rest. I mean, it's a matter of faith. Lord, give us that faith. Give us that faith that we'll know that we know that if we do what's right, you'll handle it from there. Lord, we love you. We ask you to make sure that everybody gets home safe tonight. We thank you for today. We thank you for this church. In Jesus' name, we pray. Amen.