Okay, so again let us continue with our discussion on algebraic geometry, so you know let me recall the setup in which we are working, K is an algebraically closed field for example you can think of K to be complex numbers if that is convenient for you and then the whole idea is you look at KN with the Zariski topology on the one side which is called as affine N space over K okay and this is the rather this is a geometric side and then on the other side you have the commutative algebraic side and that is supposed to be the ring of functions on that space and in this case of course it is a ring of polynomial functions on that space, so it is actually K of X1 through XN, so X1 through XN are N indeterminates they are N variables and if you think of this ring as a ring of functions on affine space because you give any polynomial you take any polynomial then you can think of it as a map from affine space to AN to A1 which is just A1 is just K okay, so these are functions with values in K okay and therefore this is the ring of polynomial functions they are functions on the space okay, so the geometric side has got to do with the affine space the commutative algebraic side has got to do with the functions see it is a supposed to be a statement of Felix Klein who said that you know the geometry of a space is supposed to be controlled by the functions on the that you allow on the space and this is so geometry actually kind of comes into play when you have a space and you define what the functions on your space are going to be okay, so in this case the space is affine space and the functions are the polynomial functions okay and I told you that so of course the just to recall what we have seen so far you know the given any subset S here you associate to S the common zero locus of S which is a set of all points in affine space which at which every polynomial in S vanishes okay and then there is also a map that goes in this direction given a subset T of affine space then you associate to T which is an ideal in the polynomial ring and this ideal of T is just all those functions which vanish at every point of T okay and then this correspondence goes on the one side the objects that are important here are the sub-objects which correspond to ideals and the objects that are important here are the so called algebraic sets which are the closed sets which has zero sets of this form okay and of course you should remember that if I changed or if I replaced S with the ideal generated by S you will still get the same zero set okay. So basically what you get is you get on this side if you take the set of radical ideals you get a bijection with the set of all closed subsets and this bijection is a it is an inclusion reversing correspondence that which is quite obvious to see because the larger the ideal is the common zeros of all the functions the ideal will grow smaller okay and the other important thing is that so if you look at radical ideals you get closed subsets on the other hand if you look at maximal ideals then under so you should remember so maybe I think let me write this below let me leave some space in between so there are maximal ideals so the maximal ideals are also they are also radical ideals because actually you should perhaps check as an exercise probably you have already done it in competitive a course in competitive algebra or algebra that if an ideal is prime then it is already radical and since a maximal ideal is prime a maximal ideal is also radical but of course there are radical ideals are not even prime okay but in any case maximal ideals the collection of maximal ideals is a subset of this okay and this under this bijective correspondence goes to the smallest possible closed subsets which are actually the points of course when I say points I am thinking of a point here as a singleton subset of an okay so in other words you can actually write an here and of course the notation for this is the maximal spectrum of the ring of functions okay so max spec of a ring competitive ring with one means the collection the set of all maximal ideals in the competitive ring so what you must understand is a point here is a maximal ideal of the ring okay and the fact is that given a maximal ideal you get a point and conversely okay so in particular for example if you take a maximal ideal of the maximal ideal will always look in this form it will be generated by Xi-lambda i for a tuple of a for an n tuple lambda 1 etc lambda n which will be a point of an and this is the correspondence because 0 set of this is this the ideal of this will be that okay so you should remember that in this direction the map is taking the ideal it is the I map and in this direction the map is a Z map which associates the 0 set the common 0 locals okay and in fact I told you that this is that this is this is this is this also uses in a way it is an avatar of the Nullson and Sats probably weaker or stronger probably weaker but let us look at that in the exercises but the point is this is a non-trivial statement okay what is trivial is if you give me an ideal like this it is that it is maximal is reasonably trivial to check but to conversely say that every maximal ideal is of this form which is true only when k is algebraically closed or at least when k is algebraically closed it is non-trivial okay and that uses Hilbert's Nullson and Sats so the what lies in between are the prime ideals the prime ideals the collection of prime ideals they correspond to what is called the spectrum of the commutative ring and you see the spectrum of the commutative ring is supposed to be the set of all of its prime ideals and therefore you think of a prime ideal here as a point in the spectrum so a point here is a maximal ideal and a point here is a prime ideal okay and of course this is contained as I told you prime ideals are radical maximal ideals of prime okay and what happens is that so what corresponds to prime ideals on this side are what are called as affine varieties in An so by this I mean the so the definition is these are all algebraic these are all algebraic sets these are all closed sets which are irreducible okay so prime ideals correspond to irreducible subsets which are closed okay and that is the theorem I stated in the previous lecture and I just stopped with that because the last lecture actually stopped with the definition of what an affine variety is okay an affine variety is an irreducible closed subset of affine space okay and why are they important they are important because we will see later that any variety I mean that any closed subset can be broken down into a finite union of affine varieties and the decomposition is unique if you make sure that there are no redundancies that is no affine variety in this is contained in no affine variety in the decomposition is contained in some other affine variety in the decomposition so every closed subset can be broken down into a finitely many affine varieties only in a unique way essentially unique way I say essentially because you can always you will have you can always permute the pieces in the union but that should not affect the union or the decomposition and so that is one important thing about affine varieties because they are like building blocks of algebraic sets every algebraic set is broken down into union of affine varieties and this is very very important because later on for example I told you probably in the first lecture that there is a more advanced or I should say sophisticated language of algebraic geometry which involves what are called as schemes. And schemes are some spaces with functions okay with rings of functions and in fact the what you have is not just rings of functions you have sheaves of rings which means that you have rings of functions on every open subset of the space so it is data not only with the space with a ring of functions on the whole space but it also comes with for every open set in the space you will have a ring of functions. So your this whole data is called a sheaf of rings and a scheme is something to like that which consists of space and sheaf of rings okay where the sheaf is a collection of rings for every open set in the space but the important point is the technical point is this scheme is supposed to be locally modelled in this way it is supposed to be made of affine pieces. So for all of algebraic geometry no matter how general algebraic geometry you do these affine pieces these are the building blocks these are the building blocks and that is the reason why these have to be first studied okay. So you should understand that affine varieties are important because they are the building blocks even in the most sophisticated form of the theory okay and the other important thing is that the other important thing is of course that these sets are topologically irreducible okay and you know irreducibility I have told you is a very strong form of connectivity so they will have nice properties with respect to maps. So for example in topology you will learn that the image of a connected set under a continuous map is again connected okay so if you have a topological space and you have a map a continuous map from the topological space into another topological space then the image of a connected set if in the source topological space will be a subset of the target topological space which will be connected okay and the same thing will happen for irreducible subsets okay. So irreducibility is a very nice thing to have on a subset okay so now let me try to prove this part so let me recall definition so this I am just recalling a topological space a subset y of a topological space so I am just abbreviating topological space to top x is called irreducible if y cannot be written as y1 union y2 with y1 y2 non-empty proper close subsets okay. If a topological space can be written as y1 union y2 where y1 and y2 are non-empty proper close subsets then we say that the topological space is reducible okay and the definition of irreducible is that it should not be reducible okay and what happens in the case of varieties as we will see what happens in the case of algebraic sets here namely close subsets of avian space you will see that you will get it you will be able to break it down into not just a union of 2 you will be able to break it down into union of finitely many subsets which are each which are themselves irreducible okay and they will be called the irreducible components okay but and that will be called the reducible decomposition of your given close set okay so that is where we are heading to fine. So the of course this definition as I told you implies that if y is irreducible then it is connected okay because connected is for it to be connected for it to be connected you should not be able to write it as a disjoint union of proper close subsets non-empty close subsets okay and that is certainly not possible if you cannot even write it as a union of non-empty proper close subsets okay. So irreducibility is a very strong form of connectedness I told you that irreducibility has not lots of nice properties one thing that comes is that if space is irreducible then if a subset is irreducible then its closure is also irreducible so its irreducibility is not going to be affected if you add the boundary which is what you do when you take the closure of a set and this is also true for connectedness if a set is connected then its closure is also connected and then but the other more important thing is that you see the more important thing about an irreducible space is that every open every non-empty open subset is dense and is itself irreducible that is another very important property okay which I hope you would have tried as an exercise otherwise you should try it it is a pretty easy exercise. So what it tells you is that if you take an irreducible space and take a non-empty subset then you can test on that subset all those properties which will be preserved when you take a closure okay that subset will because the closure of that subset will be the whole space and that is so you can test on any non-empty open subset and any non-empty open subset will be dense and that also tells you that if you take any two non-empty open subsets they will intersect okay they cannot be disjoint from each other. So these are the nice properties of irreducibility and later on it will come we will again look at it probably it is not so hard you can even check it off hand I think that the image of an irreducible set continues to be irreducible under a continuous map okay which is the same kind of statement that you get for a connected set okay. Now I go to this theorem which I stated last time so the theorem is the following if i in the polynomial ring in n variables over k is an ideal then z of i the zero set of i the points in an which are common zeros of all the polynomials in i is irreducible this is a subset of this topological space you see this topological space is just kn given the Zariski topology okay. So since it is a subset of a topological space this definition applies and you can put the condition that this subset is irreducible and the theorem says that this is irreducible if and only if the radical of i is a prime ideal. So what this means is that if I already started with the radical ideal I mean if I already started with the prime ideal I already started with the prime ideal then z of i will be irreducible and conversely if I already started with the radical ideal then saying z of i is irreducible is same as saying that the ideal itself is prime okay and that is essentially what gives you this correspondence in the middle okay that the affine varieties in an they correspond to prime ideals okay. So well so the proof proof is quite straight forward so let us do both ways so let us begin with let us assume radical of i is prime suppose radical of i is prime what do I have to prove I have to prove z of i is irreducible to show z of i is irreducible okay but actually you see z of i is the same as z of rad i okay this is something that I told you last time two ideals j1 and j2 have the same set of common zeros if and only if the radical of j1 is equal to the radical of j2 so it is the since i and rad i have the same radical namely which is rad i they both have the same zero set okay and but even otherwise this is quite trivial to see directly okay because rad i mind you is defined to be all those polynomials some positive integral power of which lies in i so it is like taking the radical of ideals is like expanding that ideal to include n throats of its members positive n throats of its members okay that is n throats for positive n right and so anyway so how do I check a set is irreducible so the logic is I have to check that it is not reducible I have to check that it is not reducible so I have to check that if it can be written in this form with y1 and y2 as close subsets and if I assume that y1 and y2 are both non-empty and also that y1 and y2 are both proper that should not happen that is what I have to check so what I will do is I will assume that it can be written as in this form with y1 and y2 non-empty but I will assume that I will assume further that y1 is proper close subset and I will try to prove that y2 is not proper namely that y2 is everything if I do that then I am done okay so that proves that it is not reducible in other words that it is irreducible so what I will do is suppose that z of i is y1 union y2 where y1 y2 are non-empty closed so close subsets so here I have to go back to the definition and stress on something which I have not written there with y1, y2 non-empty proper close subsets mind you of y okay that is something that I had not written but I did say that in my last lecture so let me stress it when I say what do you so I told you y is just a subset of a topological space what is the meaning of saying that a subset of y is closed in y that you have closed subset of y this is the language of induced topology a subset of y is said to be closed subset of y if it is gotten by intersecting y with a closed subset of the ambient space the larger space x in which y sits okay so when I write z of i is y1 union y2 where y1 y2 are non-empty closed subsets of z of i what you must understand is that since z of i is already closed in x it follows that y1 y2 are not just closed subset of z subsets of z of i but they are actually closed subset of x subsets of x itself because a closed subset of a closed subset will continue to be a closed subset okay see in other words if when I say y1 is a closed subset of z of i it means y1 is z of i intersected with a closed subset of x okay but then you see z of i itself is closed in x and so if I intersect another closed subset of x the intersection of finitely many closed subsets is again a closed subset for in the topology because the axiom of topology for axioms for a topology if you take the axioms for closed sets tells you that you take any finite number of closed sets and you take the intersection the result is again a closed set in the whole space. So what this tells you is that y1 which is supposed to be the intersection of z of i with a closed subset of x is itself a closed subset of x so let me stress that suppose z of i is y1 union y2 where y1 y2 are non-empty closed subsets of z of i with y1 a proper subset of z of i of course I will then try to prove that y2 is equal to z of i that in other words that y2 is not proper okay then note that y1 y2 are closed in the larger space x which is actually in our case an because by definition as I just told you y1 has to be z of i intersection with a closed subset of an but z of i is already closed in an and the intersection of two closed subsets of a topological space is again a closed subset so the reason is since z of i is already closed in a by definition because you know that is how the Zariski topology was defined the Zariski topology was defined just by taking for the closed sets subsets of the form z of i okay so but what does that mean saying that y1 y2 are closed in an a n means that y1 y2 are the 0 sets of some ideals okay so this implies so y1 is z of i1 y2 is z of i2 for i1 i2 ideals in kx1 etc etc okay this is what you get but then what is now y1 union so if you take z of rad i which is z of i which is y1 union y2 is actually z of i1 union z of i2 okay and you know but this is the same as z of i1 i2 because this is something that we this is how we proved that sets of the form z of i they form a topology by declaring such sets as such subsets as closed subsets in fact what we proved is if you took z of s1 union z of s2 union etc up to z of sm where si's are subsets not even ideals then the union is just z of the product s1 times s2 times etc sm we proved that okay. So this is z of i1 i2 okay and you see it is at this point that you know I will use so I will now use the null cell insert okay so what will happen is you see apply i if you apply i the null cell insert tells you that since i of z of j is rad j okay mind you this is this is a statement that involves the null cell insert and that is valid for any ideal j actually i of z of j always contains rad j it is very easy to see the non-trivial thing is to say that i of z of j is contained in rad j namely it is a statement that if f if a polynomial is in i of z of j namely if a polynomial vanishes on z of j then some power of the polynomial is in j you cannot have a polynomial some power of which is not in j to vanish on all the zeros of j that cannot happen okay. So this is this statement uses in null cell insert and if I apply this on both sides what I will get is I will get radical of rad i is equal to radical of i1 i2 okay I just want to say that this contains i1 i2 okay this is the way I have to go okay so you see radical of an ideal always contains the ideal okay because the radical is supposed to be all those elements some positive power of which is in the ideal and the first positive power of every element of the ideal is in the ideal so the ideal itself is contained in its radical always and see the fact is that this is rad i okay because taking rad more than once is not going to change anything and this is prime this is prime. So what you are getting is you are getting a prime ideal contains the product of ideals now you see we use this following fact from commutative algebra it is very simple fact that if a prime ideal contains a finite product of ideals then it has to contain at least one of them okay. So here is a lemma very simple lemma from commutative algebra if a prime ideal contains a finite product of ideals then it has to contain at least one of those ideals one of the ideals in the product of course you know of course all this is in this statement the background I am assuming is that you are working in a commutative ring with one and you are having finitely many ideals and if you have finitely many ideals J1, J2 through Jm then their product is J1 dot J2 dot etc Jm which consists of just finite sums of products of M tuples taken from the Cartesian product of all the Js okay and if a prime ideal contains the product J1, J2, Jm then it has to contain some Ji and this is very easy to see because it is just definition of prime ideal that if a prime ideal contains a product then it has to contain one of the finite product then it has to contain one of the factors of the product it is just a restatement of that if you try to work it out. So what this lemma will tell you is that rad i has to contain I1 or rad i has to contain I2 but then this now you apply Z okay you apply Z to take the zero locus and remember that when you apply Z the inclusion is reversed. So what you will get is Z of rad of I is contained in Z of I1 or Z of rad of I is contained in Z of I2 this is what you get and mind you but Z of rad of I is mind you is just same as Z of I and Z of I1 is Y1 and Z of rad of I again is Z of I here and this is Y2. So what you are saying is Z of I is contained in Y1 or Z of I is contained in Y2 okay but what we have already started with was that Y1 and Y2 are contained in Z of I so what this means is that either Z of I is equal to Y1 or Z of I is equal to Y2 but then you are assuming that Z of I is not equal to Y1 so what this will tell you is that Z of I has to be equal to Y2 and that tells you that you cannot reduce Z of I okay. So let me write that down this implies that Z of I is equal to Y1 or Z of I is equal to Y2 which implies that Z of I is equal to Y2 since Z of I is supposed to properly contain proper of sorry supposed to properly contain Y1 okay and this implies that Z of I is reduced okay. So we have started with rad I prime okay and we are able to reduce that Z of I is reduced now we will do the other way we will assume Z of I is irreducible and show that rad I is prime okay so conversely assume that Z of I is irreducible assume that Z of I is irreducible we will show rad I is prime okay. So how do you so this is again this is again translation you just have to check the condition for a prime ideal you have to take a product how do you check something is a prime ideal how do you check an ideal is a primary you take a product of two elements of the ring as belonging to the prime ideal and demonstrate that one of the two factors of the product is that ideal okay and so what we will do is so let F times G belong to rad I let the product be in rad I okay. So what this will tell you this will tell you that the ideal generated by FG F into G is a subset of rad I okay because if an element belongs to an ideal and the ideal generated by that element is also in that ideal because the ideal generated by an element is just simply multiples of that element by ring elements okay. So if I now you apply Z if you apply Z I will get Z of FG contains Z of rad I okay and but you know Z of FG is just Z of F union Z of G okay this is exactly the same statement that Z of I1 union Z of I2 Z of I1 I2 okay. So Z of FG is Z of F union Z of G okay and of course when I write Z of F for a single element F by that I mean Z of a single element F is the same as Z of the subset consisting of the single element F and this is also the same as Z of the ideal generated by F they are all one and the same okay. So what happens is that so you know so now you can so what is what this tells you is that you see Z of rad I has been written as ZF intersection Z of rad I union ZG intersection Z of rad I you you see Z of F is a close Z F this union contains this so you intersect this with the smaller subset you will get back the smaller subset. So if I intersect this with Z of rad I I should get Z of rad I and that is and intersection you as you know distributes over the union by simple set theory okay so you get this but what you must realize is that this is if I call this as Y1 and if I call this as Y2 what you will notice is that Y1 is a closed set it is a closed set because it is the intersection of two closed sets so it is a closed set similarly Y2 is a closed set closed set and you have written Z of rad I as union of two closed sets but mind you Z of rad I is the same as Z of I but what is the assumption on Z of I is the assumption on Z of I is that it is irreducible. So the moral of the story is that either one of these is empty okay and if both are non-empty then one of them has to be then one of them has to be the whole Z I itself okay so let us let us write that out let me write that out here since Z of I equal to Z of rad I is irreducible we have and Y1, Y2 are closed we have the following possibilities so I want to say Y1 is non-empty Y2 is non-empty okay so let us write down Y1 is if Y1 is empty this will tell you that that means so that will imply that Z of F intersection Z of rad I is empty well this intersection is supposed to be Z of F union Z of the ideal generated by F union rad I right this is what it is supposed to be by definition because what is how do you show that the how do you show that the closed sets form a topology how do you show algebraic sets form a topology. So what you if you recall you can recall that Z of if you take intersection over alpha or lambda in capital lambda some indexing set of Z of S lambda this is just Z of the ideal generated by the union of S lambdas okay if S lambdas are subsets of the polynomial ring okay and you take the Z S lambdas these are a collection of closed sets how do you show that the intersection of see how do you show that the intersection of an arbitrary collection of closed sets is closed it follows from this calculation okay so Z of F intersection Z of rad I will be Z of F union rad I ideal generated by F union rad I and so what that will imply is if I apply I to both sides if you apply by I to both sides and use again use this use this Z of I of Z of J is rad J so what I will get is if I apply I to both sides you will get radical of the ideal generated by F union rad I will be if I apply I to the null set okay then I will get the whole ring okay because what is I of subset it is all those polynomials which vanish on the subset okay I of a subset of affine space is all those polynomials in the polynomial ring which vanish on that subset but if that subset is empty there is nothing to test every polynomial will satisfy this condition therefore I of null set will be just the whole polynomial ring okay and you know if this happens I mean I essentially have to show that if this happens I am done otherwise I have to proceed further okay so what does this mean this means that this means that F the ideal generated by F union rad I is itself the polynomial ring see so this is again a fact I am using that you know if an ideal if the radical of an ideal contains a unit then the ideal itself contains a unit because saying that the radical of saying that the radical of an ideal contains a unit say 1 tells you that there is some power of this which is equal to 1 there is some power of this the ideal generated by this union which is equal to 1 but then that if the some power of an element is equal to 1 then that element itself is unit okay that means that the ideal generated by this itself is the whole polynomial ring okay and what this will tell you is that so you know there is some so what this will tell you is the following that so there exists g1 etc gm in rad I such that sigma fg plus fg plus sigma over i i equal to 1 to m fi gi is 1 this is what it tells you okay I mean an element in the ideal generated by the union like this will look like this you will have to pick actually you have to pick finitely many elements from the subset and then take ring linear combinations of that and such a ring linear combination is equal to 1 because 1 is there on the right side and now what I want to say is that from this we will have to say that so you know if this happens that is if y1 is empty it should more or less follow that if y1 is say that again yes if y1 is empty yes z of z of rad i is equal to the remaining thing on that oh I see you will just get z of rad i is equal to z of i g intersection z of rad i and then you will get therefore z of rad i contain in z of g okay so if you apply i on both sides you will get g belongs to rad i right you will get g belongs to rad i okay okay okay so probably so this is not this is not this is not required you are right okay so let me get rid of this good let me get back to the easier part of the argument but I would strongly encourage you to think about that whatever I wrote down okay so and the fact is that also will lead to something okay that will also lead to what you want but you have to so the point is you have to keep translating back to the geometric side if you are on the geometric side you should translate to the ideal side if you are on the ideal side you should translate to the geometric side by applying this i and z appropriately okay so as you as one of you has rightly pointed out what you should do what one does is that if y1 is empty then I mean z of it is obvious that z of rad i is just z of g intersection z of rad i and so which means that z of rad i is contained in z of g right and because the right side is contained in z of g and now you apply i you will get i of z of rad i which is just rad of rad i which is rad i containing i of z of g will be just radical of the ideal generated by g okay and to which g belongs okay so if y1 is empty you get g is in rad i okay alright so you assume y1 is not empty if y2 similarly if y2 is empty then you will get f in rad i with mind you f times g is in rad i so you have to prove either f is in rad i or you have to prove g is in rad i okay and y1 equal to empty directly gives you g is in rad i similarly y2 is empty implies f is in rad i okay and you have done so assume both are not true okay so assume both y1 and y2 are non-empty okay so you will have to thrash out all the possibilities okay. Suppose suppose y1 is proper then irreducibility of your right z of rad i implies that if this is proper then that cannot be proper if y1 is proper then y2 cannot be proper so y2 has to be everything okay and in that case you see g belongs to rad i alright so you are done essentially so let me write that down suppose y1 is proper then irreducibility of z of rad i implies that z of rad i is y2 okay and this implies that again the same argument literally z of rad i is equal to z of g intersection z of rad i and this will imply that g I guess this will imply g is in rad i okay so if y1 is proper you will get g is in rad i similarly if y2 is proper okay you will get f is in rad i okay similarly y2 proper will imply f is in rad i and that complies with proof okay. So it is very clear that an ideal here is prime if and only if the radical of an ideal here is prime if and only if the corresponding 0 locus here is irreducible okay that proves it okay so I will stop here and then we will continue in the next part with some examples


This is the presentation of our paper rotational analysis in a differential linear perspective. Practical distinguishes for wrong-reduced ferrets, zudu, and aardets. I am Yunwen and this is joint work with Siwei Sun and Chaoli. Differential and linear commonalities are the most general method to analyze block size, and they have many variants, including the boomerang and the differential linear attack. These combined variants could be stronger than pure differential or linear ones, and they are especially effective in the trails covering a smaller number of rounds have a high probability, but the probability drops exponentially with the increased rounds. In this talk, we will focus on differential linear analysis. With the vectorial Boolean function, giving an input difference delta and an output schema, we can build a differential linear distinguisher by a linear approximation on the output difference. And the probability of the distinguisher is defined by the number of right inputs. Accordingly, we have the correlation as 2 times p minus 1, and the bias is half of the correlation. Now the problem is how to estimate the probability given delta and gamma. Often, we split the cipher into two parts. In some cases, three parts are with an intermediate layer connecting E0 and E1. In E0, we find a differential with probability p, and in E1, we have a linear approximation with bias. Then the total bias of the distinguisher is 2 times p times epsilon squared. This is a rough estimation, and for a better evaluation, many new approaches are proposed. For instance, attacks on air-exciber Chesky were proposed in Eurocrypt 2015 and Crypto2020 using differential linear analysis with the partitioning technique. In 2017, theoretical formula was proved for evaluating the differential linear bias which requires to enumerate the Boole intermediate mask space. In 2019, the differential linear connectivity table was proposed to better evaluate the middle layer between E0 and E1. In 2015, it was noticed that the theoretical and experimental differential linear bias have a huge gap due to dependence, and this problem was answered in a very recent Crypto2021 paper from an algebraic viewpoint. So our motivation for this paper is two parts. How to extend the framework of differential linear analysis, and can we improve the accuracy in the bias evaluation? Before further discussion, I will first explain rotational analysis. We will later show its connection with differential linear. Rotational analysis was proposed in 2010 for air-excibers based on a rotational property of modulation. Then Marwiski applied a rotational analysis on kitchen. Following works on rotational analysis were proposed for the applications of the technique, and in several papers on rotational exhaust analysis it was shown that the rotational analysis can be regarded as a generalization of differential analysis, where our rotational difference replaced the XOR difference. The definition of rotational XOR difference is to add a rotation on one operand. When the rotational amount is zero, then it generates two ordinary XOR difference. Having the rotational difference in mind, what we do next is to generalize the differential linear criminal analysis by replacing the XOR difference to rotational difference. So we proposed rotational differential linear criminal analysis where giving a pair of rotationally related inputs X and X prime. We studied the linear approximation with mass gamma on their output difference through the side. As this formula here, and accordingly we defined the bias of such distinguisher by the probability minus half. Now we can see that rotational differential linear is a generalization of differential linear because when the rotational amount is zero, the distinguisher becomes a differential linear distinction. We will come back to this observation later. So can we borrow the differential linear bias computation here? We tried the first approach. In fact, we can borrow it, the previous idea on differential linear of separating a cyber into two or three parts. And we find good rotational differentials and linear approximations to concatenate. So here is the detailed direction on computing the bias. Assume that we have a good rotational differential on the first half of the cyber and a good linear approximation on the second half. It can be shown that similar formula can be achieved. But instead of the bias of the linear parts squared, we have two linear biases here and where the masks are rotated. But using this formula can be inaccurate in some cybers because we didn't consider the connectivity effect. So we further proof link between rotational differential and linear approximations and extend the previous formula on differential linear to rotational differential linear as well. The details proof can be found in the paper on the print here. This theoretical formula gives a unified view on rotational differential linear and differential linear. In practice, it will still require us to enumerate all intermediate linear masks. And unlike differential probability, the correlations are signed. So estimation or subspace of linear masks doesn't reflect the real correlation over the whole space considering the transformation. So we decide to take it another approach, and it is based on the previous work by Maurice Key on rotational analysis with application to K-check F. Given a three dimensional state A, X, Y, Z, where the nonlinear operation is on the X axis, we rotate the states on the D axis to get a rotational pair. So in other words, the inputs rotational difference is zero. The K-check F permutation without constants is invariant under such a notation, and aim is to find out which position has a high probability that's output pair differs on that bit. There are three rules to compute the probability that's output difference being zero, giving the probability that's the input difference on certain bits are zero. So three operations, the end operation X or N not. For instance here, the probabilities for the input difference being one is B and Q for the end operation. And then after the end operation, the output difference is one with the probability of P plus Q minus P times Q divided by two. So with these three rules, we can predict the probability of each output bits being n equal, round by round. We first observed that the rotational distinguisher on K-check F was a special case of rotational differential linear, where the output mask is one bit. Then our second observation is that the probability of the output bits being n equal through a Boolean function can be predicted by the following formula, where in the submission, the first term shows the difference transition probability. And the second term here gives the initial probability distribution of the input difference. Then our third observation is on the effect that the constants have on the rotational pairs. Here I show an example to show the effects when there are consecutive nonzero bits in the constants, like here 011 constants. We actually introduce a new rotational difference, which is the constant C x4 with C rotated. We call it delta C with difference. If the bits of that rotational difference is nonzero, then we should flip the rotational differential linear probability of the specific bits on the state. And this is called the adjusted zero. So, given an input rotational difference delta, the initial probability is fully determined, and we can evaluate the round function by regarding its circuit with Boolean operations. So we can compute the rotational differential linear probability, round by round, and to find out the position of the output bits that is the most biased. Here we show an example on Zudu permutation. Zudu is like with permutation designed by the kitchen team arranged in three times four cube with 32 bits on each side. One round of Zudu has five steps, where except for the step four, all the others are linear. Notice that the constant addition is before the linear layer. We can control the input rotational difference such that the difference before step four in the first round is zero. And that helps us to extend the distinguisher one round for three. So this is the input difference, rotational difference, which is. So with this input rotational difference, the rotational amount is set to be one bit left. And we get a full round rotational differential linear distinguisher with correlation one and output mask is one non zero bits on the 16th bit of the cell one zero. Next, we are going one step further to extend rotational differential linear analysis to areas. First, we'd like to mention that what works for rotational differential linear also works for differential just with the rotational amount being zero. So in this talk, we will speak about differential linear on Eric's and the full discussion can be found in our paper. To get the probability of propagation rule for addition modular to the end, we found that's the dependency in the carry function cannot be ignored. So if we see we apply the three and rules here and two extra rules here, then it will give us bias estimation. So what we did is the following using the observation to from our previous. We didn't use a cable carry room that takes the dependency into full constellation. So giving the probability of the input difference being zero, you can predict the carry difference being zero using this expression. Then it follows. That's the modular addition rule for differential linear propagation can also be right here as this. Let's see an example. Given the input differences being seven and seven to eight beats modular addition, and the probabilities that's the output difference beats are no zero can be computed as this table shoes. And our experiment confirms the results. And this is particularly efficient for modular additions with 64 bits or even more, where a direct computation of the differential linear bias would be computationally invisible. Another interesting thing that we observed is that the differential linear probability seems to have rotational behavior when the input differences rotate. For instance, when we rotate the difference zero one to zero two, the resulting probability is also shifted here. And this can be used to explain an experiment experimental results on the rotational property property in the differential linear distinguishes of seep hash. And we give a theoretical evaluation of the found differentially new distinguishes there. Then we apply rotational differential linear analysis and the new method of evaluating the probability to several cryptographic permutations. Here I showed the application of RZ for some more details. RZ is a 64 bits area space permutation presented in crypto 2020. It has two branches so each with 32 bits. The structure has only modular addition rotation and XOR because our previous propagation rules show the modular addition for differential linear. It contains quadratic and higher degree terms, and also the size of this permutation is not too large. So we can actually use the quadratic constraints programming in Groovy to search for a good input difference, where our objective is to minimize overall probability for all one bits output masks. And we observed that the input differences with low or high hemorrhage tend to give better rotational differential linear distinguishes in RZ and we also carry out experiments to compare the probability with the theoretical evaluation. For instance, here we have input difference, this one in a differential linear setting and the results are depicted in the following figure. The X axis is the position of output difference from 0 to 63 and the Y axis is the probability. The statistics show basically the same pattern for theoretical evaluation and experiments. And then overall of all our applications can be found here in this table. We found a certain round of rotation differential linear distinguisher for the permutation threads, four round for azuto and four round for RZ. We tested the experimental probability to verify the distinguishes whenever possible. The distinguishes show an advantage over traditional differential or linear distinguishes either in the number of current rounds or the probability. To conclude, in this paper we proposed rotational differential linear analysis as a generalization of differential linear and the theoretical analysis on rotational differential linear is given. Then a new measure for computing the probability of rotational differential linear distinguisher is presented, which is efficient by evaluating round by round. Finally, our technique is applied to three permutations where practical distinguishes are obtained. Thank you for your attention.


Next question is from janky garage jim. Can you provide some tips for dips? Oh Yes, you know, you know, it's it's funny when you list the top when you ask the average lifter to list the top best upper body muscle building Exercises the ones that are the most functional that build the most muscle give you the most bang for your buck For some reason body weight dips Oftentimes doesn't appear in that top lift and or list and I think it's just because they kind of fell out of favor because the reality is bodyweight dips done properly is Easily one of the best exercises you could do for your shoulders triceps and even your chest. It's a fun It's like a pull-up for your back except this is more incredibly difficult for a lot of people They're very difficult and so you I mean just being good You just pointed to the back of the pull-up like just being good at pull-ups and dips would build an incredible upper by itself Absolutely those two things if you got really good at pulling your body weight up and dipping really well enough to where you got so Good that you can do 15 reps of both those no problems. So you have to add weight You have an amazing body you her body. You have a great upper body guaranteed so here's some tips for dips now if you have the strength to do a bodyweight dip Start at the top grip the handles really tight and maintain good tension Lean slightly forward do not drop down below the point you lose control and tension This is the big mistake I see with people dips is they challenge the depth past the point that they have stability or proper stability so they go all the way down and they tend to relax at the bottom and Then press their way up. That's a recipe for shoulder injury. It's a recipe for For failure so always have full control throughout the whole thing now If you can't do a body weight dip which a lot of people can't because it's a it's a full body weight exercise One easy way to get better at them is to use a resistance band put it around the bars both bars Step on it or put your knee on it and now you have assistance and you can do body weight dips with assistance This is how I train my son when I have him do I put the band around the two handles He puts his foot in there and then he the band helps them so he can get better at the motion of dips and then of course as you get Stronger, I'll take the band off. So I have to give Justin some cool credit here because you know, there's we and we don't talk a lot about this on the air But you know when we all got together and started working together there there are a handful of things I think each of us probably Really learned from the other guys even with all the experience that we had and maybe we were familiar with Certain things, but the other guy was way more into it and so it really kind of opened your eyes for things and Justin was like Justin's like hardcore about tension like he was like the stick mobility thing You know those have been listening for a long time knew the the whole invention thing that he did like and so he was all about like all these tension moves and Something that I used to notice when I would do dips cold like if I came in cold I went over the dip sometimes my shoulder would bother my shoulder a little bit about my elbow a little bit like And it would feel I have this kind of uncomfortable feeling when I did it and something I thought you know what this is an area where I bet if I did like a tension exercise before I actually Do the perform the movement I could probably gain some value here And so what it looked like is this a lot of places you go to a dip the dip bar and the dip bars Elevated you got to step up to it and then you get on it And then you come down and then you drop down into this like new range of motion area And that's where people either hurt themselves or their shoulder feels clicky and weird So instead of that like I and I'll either use a bench to slide up or I'll find one or in our case We can we can manually lower eyes I want to be able to stand in it like I can always bend my knee and do a dip So I want to be able to stand it and then I actually start in the most deep position I want to go so I actually bend my knees let let the dip and I want to I want to position my elbows Right where I want them to be I want my chest and my shoulders right where I want to be And then I create tension and I create tension in that position for about five seconds And then I push out of it from that that position when I when I started doing that I completely eliminated all that weird Clicking in my shoulder or a little bit of pain that I would feel sometimes of being cold and dropping into the dip So going into it in the deepest range of motion where I'm supporting myself with my leg So it's not like I'm putting my body weight there And then actually creating tension in that position for a good five to ten seconds and then coming out of it to start Man, I made a big difference on my dip. Yeah, that's a great. Yeah, that's a great tip I also think too like The hand position itself like I know with some clients of mine They would feel like an immediate sort of impingement in the shoulder based off of like how narrow or how wide Their their hand grip was just to kind of manipulating that on your own just initially to Feel where you're most comfortable and a lot of it times it emulates and matches like where you would grab the bar for an overhead press And so for me, I would experiment with that Definitely adding the tension is is a panty dropper. I'm gonna go ahead and throw that out It's a big move to you know, to put you into the right Direction and then challenge it too with instability. Yeah, and one other tip I like this one too is to you can practice negatives on dips pretty well So what you would do is you bring a bench up to the bar you'd stand in the dip Apparatus so you're kind of already at the top bend your knees hold that tension and then slowly lower yourself Until you touch the floor with your feet and then get out Rest for about five seconds and then try that again if you're not strong enough to do a dip You probably are strong enough to lower yourself with control and so that's one way to slowly build up your strength


Why most people fail at lucid dreaming? All right, so I wanted to talk in this video about why most people fail at lucid dreaming. There's one thing in particular that means it's pretty much the reason why 95% of people just can't lucid dream. Because if they were doing this thing, then they would naturally lucid dream and they'd be having lucid dreams every night. So there is just one particular thing which most people just aren't doing, okay? And that thing is just general self-awareness. Now what I mean by that is awareness of what you're doing throughout the day and being critical of the things you're doing. A huge percentage of people, probably more than 95% actually, just aren't aware of what they're doing. They're in these routines where they just sort of go maybe to work or they go about their daily life relying on these autopilot routines without really thinking and being aware of what they're experiencing. A great example of this is most people can't remember their commutes to work. Because they're in autopilot. Ask yourself the question. Can you honestly remember every part of the journey on your last commute to work? What did you see? What people did you see, you know, what were they wearing? Can you honestly remember every detail of that commute? For most people they can't simply because they're just not paying attention and you know this critical awareness is much more important than just remembering what people were doing on your commute. It's about being present in the moment and actually experiencing things instead of just being submerged in your thoughts, right?  So you get into the real world as an experience what you're actually doing now. Don't just be caught up in your thoughts and internally conversing with yourself thinking, you know, what am I going to do today at work? Because you miss everything. You miss all of the details around you. You miss all of the experiences that are already happening. And as a result, because you're not aware of these things, then you lose you lose that self-awareness and you don't lose a dream. This is pretty much why 95% of people fail a lucid dream before they've even started because lucid dreaming fundamentally is about awareness. It's about realization and you can't really have that awareness if you're not, right, aware. It sounds simple, doesn't it? But if you're not aware of yourself during the day, if you're not paying attention to what you're doing and mentally, you know, focusing on what you're actually doing, what you're saying, where you're going, then, of course, you're not going to lose a dream because you're not paying attention.  Now, a lot of what we experience in a waking life will translate to the dream. I've said this in other videos and I'll keep saying it for future videos as well. What you experience in the day will be mirrored in the lucid dream. So in Naiman's term, say if you always dream about making a co- if you say if you always make a coffee in the morning and you always go about it in exactly the same way, then the chances are that when you dream about making a coffee, you'll make it in the same way. The same can be said of your routine, of your commute to work, of the way you talk, and everything really. Everything that you do in waking life is sort of mirrored in a lucid dream. And you do pretty much the same stuff in the dream. And so a way that you can become lucid a lot more regularly is just to pay attention. Just pay attention to what you're doing, be critical, ask yourself questions and don't just walk around every day without really thinking. I know that it's easy to do this, especially if you have a boring commute or if you're maybe it's a long one hour long or more train journey or something, whatever it is. It can be easy to just fall into autopilot and just play a game on your phone and just not really pay attention.  But also you don't really, you're not really paying attention and you miss things that maybe otherwise would have been great opportunities. This is why, one of the main reasons why the law of attraction is so powerful, because it's not some spiritual strange law. It's just a very simple fact, right? If you focus on, I don't know how much you know about a law of attraction.  It's just like if you buy something like, say if you buy a car and it's a certain color, say if you buy a blue Audi, for example, suddenly from that point on, you're gonna then notice all of the other blue Audi's driving around and it's not that there are more of them. It's just that because you've now focused on that because you've brought one, you're now paying attention and you subconsciously or consciously notice the other Audi's. They were always there. You just only noticed them because you've been focusing on it. And it's exactly the same thing with law of attraction. If you want to earn more money, if you want to, whatever it is you want, if you focus on it, you'll notice the opportunities. And so that's another great benefit of being self aware, you know, regarding lucid dreaming is that you'll also be aware of opportunities. You'll also be more focused and in tune with what's happening now. So you'll be able to spot those opportunities. You'll be able to overhear those conversations that maybe you would otherwise ignore that you can actually jump in and say, Oh, I know about this. And then right there, that's like a that's a business relationship form or that's a friendship. Or, you know, that's an opportunity that you would have otherwise missed if you weren't paying attention. So this is it guys. This is why 95% of people don't lucid dream. It's also why 95% of people don't achieve the goals they want in life because they're just not paying enough attention. So get back in the the zone, I guess you could say, get back into your body and your awareness and just pay attention to what you're doing. Don't just go through life as a zombie because you'll miss so much.


Today, a Centers for Disease Control Report stated that suicide was the second leading cause of death for 10 to 24-year-olds after a category of accidental deaths that includes motorcycle, excuse me, motor vehicle crashes, falls, drowning, and overdoses. With us today, we have a panel, and we'll be speaking about policies, procedures that we can help ring the bell and reduce teenage suicide. And I'm pleased to share with you that I'm Tina Patterson, I'm in Germantown, Maryland. I am sitting in for Chuck Crumpton, who is on well-deserved vacation today. With us, we have an outstanding panel, and I'm pleased to, I feel kind of like Julie Andrews, a few of my favorite people are here with me today, and we're going to have a very exciting conversation. I do want to share with you at the onset of this conversation that we are not mental health care practitioners, and if you do need assistance, we'll talk at the end of the session about resources that are available to you. But if you are in distress at this moment, please call 911. With us, our first guest is retired judge Sandra Sims from Hawaii, and she is not only a retired judge, but she is author of Tales from the Bench, Essays on Life and Justice. Also with us this evening is Louise Ng, who is a partner at Dentons in the Honolulu Office. Louise believes the better you know the community, the more you can help your client. Also with us this evening is Rebecca Rattler, a world-renowned mediator and arbitrator at Jams, and she is a global thought leader, conflict resolution and DEI consultant, risk management professional, and adjunct law professor at Howard University, and recently takes on the title of being best-selling author. We also have with us this evening a special guest, and I like to think of her as my sister from another mister, and that is Lorraine Delaporta, who hails from Cranston, Rhode Island, so we are both, it's amazing, Lorraine and I never met each other, I originally grew up in Rhode Island, but she's from Cranston, and she is the president of the Resolution Collaborative. So welcome, and as I mentioned, tonight's topic is teen suicide.  Anyone? I would hope not. I would hope not. This is such a critical issue, as you given those statistics, I would think that we have to really, really be more and more and more concerned about what is happening to our young people. We've come out of this, I don't say we've come out of this COVID, but we went and we have come out of an era in which it's had drastic ramifications for all of us, not just here in the United States, but all around the world, coming out of the isolation. We're not able to attend schools, and when they did, they were under some very restrictive conditions. Families were stressed at home. Employers and employees were stressed in their work environments, and in the midst of this, there were parents and aunts and grandparents raising children. So this can't be politicized. There was an article I just was looking at that was in the Pacific Business News talking about our children growing up in an age of anxiety, which not just the children, but we are adults as well in this age of anxiety, this sort of, I don't want to call it post-COVID, because we're not really out of it yet. We're still having to deal with its ramifications, but that's my thought. I think it's important to note that COVID is still among us. It's not a pandemic, but it's endemic. I have a friend who's an infectious disease doctor. When he said endemic, I had to look that up. I didn't know what it meant, but it's like the flu. It will be among us now ongoing and forward. And as you eloquently stated, Judge Sims, we still do not know what the effects of COVID are. As a society, we have not recovered from the thick of things globally. And the same confusion and instability that's being experienced by adults is being experienced by children for one of the reasons you mentioned, lack of socialization for really a couple of years, and that really affects children and academic. And I see even in higher level learning the effects that being away, being isolated, the change in how curriculum was presented, I see the effects that that has even on students, even at a law school, at a level such as law school as an example. And so we're still trying to figure out the next normal. We just, you know, there was instability caused by overlapping pandemics. So there's COVID and then I've been published to say overlapping pandemics are COVID, civil unrest, weather events, the economic downturn, which then caused other issues. Food instability, job instability, a lot of issues surfaced and the age of anxiety that you speak of is prevalent everywhere and is being felt by young and old. I think, Tina, if I can chime in, in terms of fear of this becoming politicized, one of the things I worry about is with politics is access to services for young people, access to good health care, access to mental health services, you know, for young women, access to reproductive health. All of those things, I think, have run the risk of being politicized and that keeps me up a little bit at night. I had to think about that question, Tina.  We're seeing that in states where there is not only, not only is there the age of anxiety, all those, all these crisis things, but states that are trying to suppress the rights of LGBTQ kids, for instance, denying gender-affirming care, denying the teaching of history that fully recognizes, you know, the bad parts of our history that we should be learning from. So yes, I think that there's definitely that. I'm hopeful that there seems to be more attention paid to mental health. There's obviously not enough because every time we have a mass shooting, it's often by somebody who has had mental health concerns, but nothing seems to be done about the gun control issue or increasing mental health services in those states that espouse the use of guns so widely. Louise, it's interesting you should mention guns. I'm going to share a factoid with you that I found in the Centers for Disease Control report, and that is guns were used in 54% of suicides in 2021. I don't know if that number will shift and spiral down, but it is noticeable that, again, to see such a high percentage. One of the things that experts across the board have been saying, and we have data that goes back as early as back to the 1980s, one thing that experts continue to say is that preventing suicide starts with an open and honest conversation. And I'm curious, based on your experience and your interface with youth and adolescents, what do you suggest in terms of where the conversation begins? Well, I think we have to be good listeners for one. It seems and also maybe just be aware. It just seems like from what the anecdotal stories I've heard, it's kids who feel alone and isolated and seem to have nowhere to turn. And we as adults are really oftentimes just focused on living our own lives or trying to worry about the care of our kids and maybe not putting ourselves in their shoes very much. One of the good things that happened during the pandemic is that the Bar Association did have a presentation by a woman who has an expert who has worked in the suicide prevention area and they have a video. I wonder if it's out there, but she did talk about red flags and things to watch out for. Do you remember the name of it, Louise? No, I'm going to have to look. But Liam Deely, who is the head of our substance abuse program for lawyers and judges, it's his mother, actually, Mrs. Deely, who worked in that area, who gave a presentation and I think maybe also had an associate helper. I was watching the view the other day. And one of the topics was how involved should parents be? They had a conversation in the video. How involved should parents be in their watching how their children socialize? Should parents be checking cell phones? Should parents be reading a diary, for instance? And Sunny was saying that she still suffers some trauma from feeling betrayed because I forget who it was. Somebody in her family read her diary, who was very personal. I forget it might have been one of a sibling, I forget. But then there was this robust discussion around whether parents should be reading a diary. And I believe it was Joy Behar who said, well, I think it depends.  I mean, how are you going to know depends on the age, depends on the relationship, depends on the child's personality style, your observations. And I thought those were really good points because while, you know, depends again on the age of the child, how do they need to be, you know, until they have, they deserve or they've earned privacy or, you know, there are so many things going on now in our society and we have to keep, you know, tabs on. I hadn't really decided whether I thought parents should read a diary or not. But I agreed with Joy and then Whoopi weighed in and was saying that she agreed that you've got to know the child. And sometimes you think you do know the child, but there are things going on with social media. Everybody's overexposed, even, you know, seasoned adults. And so with that, you know, just given the realities in our society, it's important to know what's going on with your child because some people have been surprised by their younger and older children's suicides. Yeah. Yeah. And another piece of that, those are good points to bring up, Rebecca, but another piece of it, though, when we're looking at the suicide rates among teams, we're looking at kids who are not necessarily in situations where they often have access to a parent that is even listening or concerned about what's in their diary. Great point. I think those are the I'm not to dismiss the others, but I think when we look at these rates, those are the ones that we as a society really they've got they don't have that, you know, unless they're organizations or like mental health or whatever that are advocating for those kids. They're out there. They're just kind of out there and they're more probably more susceptible to all the other places. Some of them are like one of the things I used to notice when I was on the state council mental health. We talked about we often talked about kids that run away, you know, the whole our laws about truancy and and and all those things having to address why kids run away. And oftentimes when they're running away, they're running away from something, particularly the girls, there is something that has happened. And it is that group that you've got to be because it's not to diminish anyone else, but it's that group just recently there was a report that came out about the detention center, one of the juvenile detention centers here in Honolulu. That you're beginning to address that issue. Like what is it that's going on rather than just saying, you know, you ran away or you didn't go to school today or whatever. Let's find out what's taking place. Let's move back to what you were saying, at least about the listening. You didn't listening is a way to begin because otherwise knows what's going on in their lives and they're in their in their minds. And the media, social media is a is a is a plus and a minus. I mean, it's right. It's good and it's not good sometimes. But I it's it's a really tough issue, Tina. Why do you pick this? This is really hard because, you know, it's we have to be concerned about it and we have to do what we can to not make it so it is politicized. I tried to, you know, sugarcoated, but it is. I mean, it goes back to the issues, particularly with regard to gun control. Can you imagine the kids who have grown up doing shooting drills in school? I mean, good Lord, when you starting like at six years old, particularly the kids that came out of Sandy Hook, they're like, college age now, they've not gotten over that trauma. There's no way. What are they carrying with that? Lots of trauma. So, so, Tina, I took in preparation for tonight, I took a look. So CDC has great information about what are some risk factors. And I looked at the list of risk factors. And one thing that I found incredibly stunning was all of them seem to have some tie to unresolved conflict either at school in the family, in the community. So this is where I kind of thought about, you know, those of us that do conflict resolution, what can we do to help build some capacity for, you know, places people can go to get some skills and build some resilience and some capacity to engage these conflicts. So conflicts on social media, the bullying, you know, the unresolved issues in the family that cause teens to run away. All of it seems to tie back to they don't have a way to resolve those conflicts and to gain that resilience. Yeah, a way to process it. I was reading, Tina was kind enough to send us some articles just in case we had time to look. And really quickly, I was I was looking at one of the articles and a point that was made, which I think is important, is that a lot of times. So minority children, black children, specifically, have a higher rate of and growing. So, yeah, the statistic is from 2018 to 2021, the racial group that has seen the largest increase in suicides among people, 10 to 24 years old was black individuals with an increase of 37 percent from 2000 to 2020. So the these are shocking. These are shocking rates. And one of the things that was mentioned as a I don't know if it's a cause, but it's a reality is that sometimes mental health issues are not diagnosed and they are called behavioral issues versus having the child assessed for, yeah, for a mental health diagnosis. And so you have children who are chemically imbalanced. And then, as Judge Sims mentioned, you may have a child who was an orphan, lost of a parent. There are lots of lots of reasons that trauma exists here in Atlanta. There is Covenant House. It is a home for it's a place for homeless teenagers. You're familiar. And my initial in my initial acquaintance with Covenant House, I asked myself as we were doing community service here with the Insurance Association that I was leading here. They were our pick twice in the year for community service. And my thought was. Ignorantly, why would why would teenagers be? I had a teenager. So why would teenagers be homeless? Well, there are lots of reasons.  So runaways, rejections, rejection because of the LGBT community. So many different reasons. This really is a topic. Different adverse childhood experiences. And I guess A.C.E. is a term that's being used in connection with these causes for for teenage suicide. You asked a good question. Sorry. I'll wait, Louise. You're going to say something. Oh, no, I was just going to say, you know, just as Sandra said, and it's really an issue that should not be politicized because, you know, everybody has kids. We all are concerned about them. You think part of the problem is, you know, mental health issues or just talking about them are soaked under the rug. Also that, you know, we need to educate ourselves more about what to look for in terms of red flags. And it's not necessarily the person who's got an extreme mental health problem. Yeah, yeah, I think some self education is needed for everybody. I don't know. I think that's that's that's critical.  Oh, there's a mental health issue. We have to address the mental health issue. I always tend to back away because I want to say we have to really educate ourselves about mental health is a lot of people have mental health issues. They don't all shoot people. That's true.  They're just having issues that need to be addressed that lead to other things such as particularly with our, you know, the teen girls, the severe, you know, depression. It's when kids are depressed, that's what's going to get you into this place of contemplating things like suicide because of all the pressures that are around it. So I think we really do have to educate ourselves about what is mental health and mental health treatment. I'm kind of happy to see and I don't want to use the term happy, but at least that even on social media, there is now this emphasis on having access or finding, you know, people who can address some of your mental health, you know, just somebody to talk to. Yes. And sometimes that's just that may be it. Someone to thank you.  And one of the ladies was saying. She said, just doing this, just sitting here. We went and looked at our exhibit and had lunch. And she said, just doing this, this means a lot. I am ready, you know, just to, just to connect with people. We were kind of in an open air place and just to listen to people's experiences and stories and laugh. And she said, I needed this. And sometimes maybe that's kind of what we need ourselves so that we can prepare to listen to what the kids are saying because we got all our own heads all messed up with stuff, too. So that just struck me. And she just was so just grateful for the opportunity just to be out and connect what they call connecting with people. Not so much always, you know, hanging out at your house or anything, but just connecting. Feeling connected. Yeah. That was what she emphasized. And I thought, oh, this really is the thing that we just need that sometimes. So, yeah, I know I'm on that board of mental health. And one of the things that mental health has been doing is doing some trainings in this area for youth suicide and training for people to learn about the recognized signs for, you know, for bullying and bullying prevention.  So you're not just, as my daughter is saying, you know, instead of approaching someone from their perspective about what's wrong with you, you know, why don't you do right rather than trying to find out what what happened? How can I help you? What is it, you know? Chandra, you raised a question and I'll be transparent. What made me bring this topic forward? And it's actually, I have to give credit to what credit is due. Professor Renelia Randall had spoken to me back in late March, early April, and she shared a statistic. And it's from a report that was published in April of 2023 by Professor Janelle Goodwill with the University of Chicago. And the study is included in the Journal of Racial and Ethnic Health Disparities. And this was literally talking about that as Rebecca stated, this spike in suicides, especially among black or African American children or youth. And one of the key findings was the feeling of hopelessness about the future as being one of the primary reasons black young adults consider suicide. And that was, it was hard for me to hear that because we have so much going on and people blame all the internet and social media and all these other factors. But if there is a feeling of hopelessness, there's a center, there's a route to that hopelessness. And how do we as adults recognize that message is having a detrimental effect on our children? I think what Louise talked about, there's an intersectionality. So let's talk about, or I may be think about you have a African American youth who happens to also identify or be consider themselves a part of the LBGQ plus community. Yeah. So now there's this intersection and that feeling of hopelessness where you've got legislation and you don't have the wraparound services that are available. That's what maybe you want to talk about it, not necessarily as a mental health practitioner, but as leaders of change in our communities where we can say, you know, I'm on this board or my jurisdiction, our school board has said, we're going to offer mental health services. And we want to communicate to our youth, if you need help, if you're feeling isolated, if you are feeling suicidal and you don't know what quite to call it, but you feel like life isn't worth living, talk to a trusted adult. Doesn't have to be mom or dad, but it should be a trust that an adult that you can trust that's going to listen to you without judgment and get you the help that you need. And I will say this, and I think I could say this for all five of us, if you are doing something to hurt yourself, we have an obligation to intervene and get you the help that you need. And any trusted adult is going to do that. They're not telling on you to get you in trouble, but they're telling on you because they want you to live. And that's what started this conversation for me. I thought, you know, this is something we need to talk about. I, Professor Randall, unfortunately, wasn't available this evening.  You interface with youth. And that's why I wanted to bring this up. We have less than a minute remaining. So I'm going to go round Robin and give you an opportunity to share your final thoughts on this topic of teen suicide, policies, procedures. What should we be doing? What can we be doing? And we'll start with LaRaine. Perfect. So thank you for convening this. This is an important conversation. My parting thoughts are that we need to build an infrastructure in schools and in communities to engage conflicts and things that are troubling youth. All right, FAB3, I'm usually part of the FAB4. So I'll leave it up to you. I'm so glad that LaRaine is here this evening and I didn't want her to go last, which is what she would try to do. So FAB3, I'm going to leave it up to you. Final words. 98. Oh, sorry. So does everyone know what 98? Yes. You can text the message to 988. It's time to say more about 988, please. That is the line that's sort of come to replace to address mental health and suicide issues. So you don't have to call up 911. If you're in that situation or you know if someone or we as those sick and being those trusted adults, we can down 988. And it connects the national, it's a national number. So it is. I was thinking listen and lean back to what Louise was saying, we need to be listening and we need to allow people who are in crisis to lean on the resources available. That's great. And yeah, I think it takes a village too. It's not just family. As LaRaine says, it's schools too. That's usually where the kids have the most interaction with adults and I think that just means again, training and awareness of what we should be looking out for and also making people feel, I think adults too, we're not alone in this struggle. So often you hear that people have a great feeling of relief when they're able to talk to other people and share stories, you know, that they're not the only ones feeling inadequate or hopeless that we all can have, you know, hopefully we can all overcome and help others overcome. Yeah, yeah. Louise, you summed it up nicely, helping people overcome. On behalf of Think Tech Hawaii and it's time for a responsible change episode, I wanna thank you for being here, LaRaine, Rebecca, Louise, Sandra. In two weeks, you'll have Chuck Crumpton who will be back from vacation. It's been a pleasure to work with you this evening and we'll see you in two weeks, hopefully. Aloha. Aloha. Aloha. Thank you so much for watching Think Tech Hawaii. If you like what we do, please click the like and subscribe button on YouTube. You can also follow us on Facebook, Instagram, and LinkedIn. Check out our website, thinktechawaii.com. Mahalo.