---
abstract: 'The potential weakness of the Y-00 direct encryption protocol when the encryption box ENC is not chosen properly is demonstrated in a fast correlation attack by S. Donnet et al in Phys. Lett. A 356 (2006) 406-410. In this paper, we show how this weakness can be eliminated with a proper design of ENC. In particular, we present a Y-00 configuration that is more secure than AES under known-plaintext attack. It is also shown that under any ciphertext-only attack, full information-theoretic security on the Y-00 seed key is obtained for any ENC when proper deliberate signal randomization is employed.'
author:
- |
    Horace P. Yuen [^1] and Ranjith Nair\
    Center for Photonic Communication and Computing\
    Department of Electrical and Computer Engineering\
    Department of Physics and Astronomy\
    Northwestern University, Evanston, IL 60208
title: 'On the Security of Y-00 under Fast Correlation and Other Attacks on the Key'
---

Introduction
============

The quantum-noise based direct encryption protocol Y-00, called $\alpha\eta$ in our earlier papers \[1-6\], was repeatedly misrepresented in previous criticisms, but that situation has apparently changed with our recent papers \[7-9\]. For the first time, a meaningful attack on Y-00 type protocols beyond exhaustive search has been developed in [@donnet]. A fast correlation attack (FCA) was presented that was shown to succeed by simulations for moderate signal levels when the ENC box in Y-00 is a LFSR (linear feedback shift register) of a few taps and length up to 32. Even though such Y-00 is already insecure against what we call assisted brute-force search [@nair06] due to the small seed key size $|K| \leq 32$, such FCA is of interest as it brings forth the whole issue of Y-00 seed key security against similar and other attacks.

The attack in [@donnet] is geared toward only the experiment reported in [@optlett03]. We have emphasized all along [@yuen04; @ptl05; @spie05] that the use of LFSR in the reported experiments was just for proof of principle demonstration, that the ENC box must be chosen appropriately in a final design, and that other techniques need to be deployed for proper security. To quote from [@spie05], “Similar to encryption based on nonlinearly combining the LFSR’s, Eve can launch a correlation attack using the following strategy: $\ldots$ many of the LFSR’s could be trivially attacked.” Thus, we were aware of the possible weakness of some ENC and in particular of FCA type attacks. Indeed, Hirota and Kurosawa [@hirota06] have already desribed a counter-measure to FCA via a “keyed mapper”, the incorporation of which in ASK-signal Y-00 [@hirota05] has been developed and is being tested. Generally speaking, it is important to study LFSR-based Y-00 despite its possible weakness, because LFSR is a practically convenient choice in various applications similar to the situation in standard cryptography.

In this paper, we first briefly describe general attacks on the Y-00 seed key as a problem of decoding in real noise – a viewpoint which includes all FCA’s. For both ciphertext-only attacks (CTA) and known-plaintext attacks (KPA), we show that Y-00 may be considered as a classical stream cipher, the ENC box, with real physical noise added on top. We comment on the possible defenses involving just a properly chosen LFSR, or an added keyed mapper, or with a keyed connection polynomial for the LFSR. We describe an AES-based Y-00 that is more secure against KPA than AES (Advanced Encryption Standard) alone, in the sense that if it is broken then AES is also broken but not the other way around. The practical security advantage of such AES-based Y-00 will be indicated. Finally, for CTA, we show that Deliberate Signal Randomization (DSR) introduced in [@yuen04] provides full information-theoretic security on the Y-00 seed key for any ENC. We hope that these results would establish beyond doubt that Y-00 is an important cryptosystem to consider in theory and in practice.

Attacks on Y-00 seed key
========================

Consider the original quantum-noise randomized cipher Y-00 [@prl; @yuen04] as depicted in Fig. 1. Alice encodes each data bit into a $2M-$ary phase-shifted coherent state in a qumode of energy $\alpha^2_0$. A seed key $K$ of bit length $|K|$ is used to drive a conventional stream cipher ENC to produce a running key $K'$ that is used to determine, for each qumode carrying the bit, which pair of antipodal coherent states, referred to as a basis, is to be used as a binary phase-shift keying (BPSK) signal set for Bob. With a synchronous ENC at the receiver, Bob discriminates the BPSK signals for each qumode by an appropriate receiver. With a differential (DPSK) implementation [@prl; @yuen04; @optlett03; @ptl05; @pra05; @spie05], there is no need to phase lock between Alice and Bob as is true in ordinary communications.

\[htbp\]

The optimum quantum receiver performance for both Bob and Eve is the same as in the non-differential case in principle, the differential implementation being a practical convenience. Even with a full copy of the quantum state granted to Eve in our KCQ approach of performance analysis [@yuen04; @pla05; @yuen05qph; @nair06], security on the data is nearly perfect when the seed key induced correlation is neglected [@prl]. Generally, it is a horrendous problem with yet no solution for meaningfully quantifying the data security of a symmetric-key cipher. In current practice, it is assumed that CTA on the data is not a problem if $|K|$ is “large”, and attention is focused on KPA on the key.

For conventional or standard [@yuen05qph; @nair06] ciphers, the key is usually completely protected from CTA for uniformly random data. This is, however, not the case for the bare Y-00 [@yuen04; @pla05; @yuen05qph; @nair06]. In this paper, we address both CTA and KPA on the Y-00 seed key, the (classical) ciphertext being obtained from some quantum measurement on the qumodes assumed to be in Eve’s possession. It is seen from Fig. 1 that a CTA or KPA on the Y-00 seed key is equivalent to the corresponding attack on the standard stream cipher ENC with its output stream observed in noise resulting from the coherent state randomization of the signal phase. Thus, it is equivalent to a CTA or KPA on the ENC alone as a stream cipher but with noise on top. The connection between the running key bits $K'$ and the basis, called the “*mapper*” [@pra05; @spie05; @hirota06], a crucial component of Y-00, and the noise effect on $K'$ are described in [@spie05; @donnet]. In a FCA on a conventional stream cipher composed of, say, a nonlinear combination of the outputs of a bank of $m$ LFSR’s, one focuses on one LFSR $L_i$ at a time and looks for correlation between the final stream cipher output $K'$ and the output $k'_i$ of $L_i$. Thus, even though the complete cipher is nonrandom, $K'$ constitutes a noisy observation of $k'_i$ from which a good estimate of $k'_i$ may perhaps be obtained. Such a divide-and-conquer strategy can be repeated to yield all the keys $k_i$ for each $L_i$. For Y-00, there is real noise from the coherent states, but a similar FCA can be launched if there is a significant correlation between $K'$ and the observed $2M$-ary signal, as obtained, say, by heterodyning.

In general, attack on the Y-00 seed key is *exactly* a decoding problem on a memoryless channel for both CTA and KPA. This can be seen by regarding the seed key as information bits and the observed sequence of $2M$-ary signals translated by the mapper to $K'$ as the codeword, with independent coherent-state noise for each qumode so that the memoryless channel alphabet has size $\log_2 2M$ in a CTA and $\log_2 M$ in a KPA. Note that this code from ENC, as in the case of AES, could be nonlinear with no useful linear approximation, making linear decoding not a viable attack. It is not known whether information-theoretic security may be obtained in Y-00 for a properly designed ENC, i.e., whether a (decoding) algorithm may be found that would succeed in determining the seed key with some nonvanishing probability [@yuen05qph; @nair06]. And there is the further question, if such an algorithm exists, of its complexity as the general syndrome decoding of even a linear code is exponential. In contrast, for KPA on standard “nondegenerate” nonrandom ciphers, the key is actually uniquely determined at a bit length $n_1=n_d$, the nondegeneracy distance [@yuen05qph; @nair06] which is often not very long. Thus, such cipher has no information-theoretic security against KPA, although there is still the problem of attack complexity in finding $K$ that may allow complexity-based security which can be practically as good as information-theoretic security [@yuen05qph]. The key point in this connection is that randomization introduces real noise that is otherwise absent in a nonrandom cipher, signifying its role in adding security to KPA.

For standard stream ciphers built upon LFSR’s, the class of FCA described above is powerful enough to break them for sufficiently long observed length $N$ of the output. However, the complexity of all known FCA algorithms is exponential in either the memory needed or the number $t$ of tap coefficients in the LFSR \[13\]. Thus, practically there are LFSR-based stream ciphers that are not broken by any known attack when the LFSR length $|K|$ and $t$ are sufficiently large. Shorter LFSR’s or ones with long $|K|$ and with few taps are more convenient and cheaper to use in practice, but are vulnerable to computationally intensive but feasible attacks. If such LFSR is used in the ENC in Y-00, the cipher becomes vulnerable even for moderate signal level if long enough $N$ is employed when that does not lead to an undue increase in memory required. For the $|K|=32$ single LFSR case reported in [@donnet], only $N=1500$ is needed in a CTA to undermine the system at the signal level $\alpha^2_0 \sim 1.5\times 10^4$, roughly the numbers used in [@optlett03]. The convolutional-code based algorithm chosen in [@donnet] is not suited to attacking long $|K|$ LFSR with a few taps, and thus would not be able to break the $|K|=4400$ and $t=3$ LFSR used in our system in [@pra05; @corndorf04]. However, a different FCA would no doubt be able to break that system, such as those designed for small $t$.

Defenses Against Fast Correlation Attacks
=========================================

We have already observed that one may use practical LFSR that resists known FCA in the ENC of Y-00. There are many other ways to defeat such and even more general attacks on the Y-00 seed key, as we will discuss in the rest of this paper.

First, a properly designed deterministic mapper that determines the $2M-$ary signal from the running key $K'$ would spread the noise into the different bit positions of $K'(m)$, increasing the minimum complexity of attacks. The mapper may also be keyed, e.g., the mapper function may be chosen for each qumode from the running key $K_m'$ from another ${ENC}_m$ with another seed key $K_m$. This results in a product cipher of ENC in noise and $ENC_m$, for which no obvious modification of the FCA can be made that does not involve exponential search over $K_m$. In particular, one cannot plot Fig. 3 in [@donnet] which is the basic starting point of their attack. This defense has already been proposed [@hirota06], although there is “correlation immunity” for such ciphers only under an approximation.

Secondly, the connection polynomial in the LFSR can be keyed, i.e., chosen randomly from an exponential number of possibilities. The known FCA’s on LFSR all require knowledge of the LFSR connection polynomial. In a future paper, we will present information-theoretic analysis of the effect of a keyed connection polynomial. Such ciphers can clearly be implemented in software, and to a considerable extent in hardware with field programmable logic, thus retaining much of the convenience of LFSR in practical applications. We do not believe information-theoretic security can be obtained this way, but it may greatly increase the complexity of at least FCA type attacks, thus providing useful practical security in some situations.

Thirdly, we now give an ENC design for Y-00 that leads to exponential complexity for CTA according to current knowledge, and more security that AES for KPA generally. Consider the ENC of Fig. 2 where a bank of $m$ parallel AES in a stream cipher mode is used to provide the $m=\log_2 M$ bits running key segment $K'(m)$ which determines, through the mapper, the basis of a qumode. Typically in our previous experimental demonstrations, $m \sim 10$ and $|K|$ is in the thousands. Thus each $K_i$ may be readily chosen to be of 256 bits. Under heterodyne or any other quantum measurement by Eve, the result is a noisy version of $K'(m)$ with independent coherent-state induced randomization for each qumode. According to the present state of knowledge, no KPA on AES is better than exhaustive (exponential) search [@stinson]. Even in a divide-and-conquer type attack as in FCA, so that a single AES is to be considered, one needs to deal with the KPA problem of artifical noise from such strategy with the addition of *real* coherent-state noise, in a CTA on the Y-00 seed key. Let $N_1$ be the length of the qumode sequence used for the attack, so that Eve may parallelize $N/{N_1}$ attacks simultaneously from the total length $N$. It is clear that even without noise, the attack complexity remains exponential for any realistic $N \leq 2^{80}$ and any $N_1$. In a KPA, the comparison is to be made with the same $N_1$ for no parallelization. Thus, Y-00 is equivalent to AES in a stream cipher mode with output observed in noise, thus harder than AES alone which does not have the decoding in noise problem. In particular, it is easily seen that if the Y-00 in the configuration of Fig. 2 can be broken, then each $AES_i$ itself can be broken.

\[htbp\]

[ ![image](newfigs1.eps){width="4.5in" height="2in"}]{}

The question arises as to what constitutes a fair comparison between a given stream cipher ENC versus Y-00 on top of ENC. A different configuration was given for ENC in [@nair06], where a single classical stream cipher (say AES) is used without parallelization but is adjusted to give the same clock rate for encrypting each data bit. The present scheme appears simpler in principle and more secure in practice when AES is used in ENC, because the functionality of multiple AES in parallel cannot be replaced by a single AES. However, with such parallelization for maintaining the same clock rate as AES (or ENC alone), the question arises as to whether the added security from Y-00 can be obtained from, say, nonlinearly combining the parallel AES’s. This question cannot be answered until security is precisely defined and quantified. However, it may be observed in this connection that there is no known attack developed for AES observed *in noise*, and the intrinsic nonlinearity of AES renders all known decoding attacks inapplicable.

The major qualitative advantage of Y-00 [@yuen05qph; @nair06] compared to a standard nonrandom cipher is that the quantum noise automatically provides high speed true randomization not available otherwise, thus giving it a different kind of protection from nonrandom ciphers. Furthermore, one has to attack such physics-based cryptosystem at the communication line with physical (measurement) equipment, which is not available to everyone at every place, whereas one only needs to sit at a computer terminal to attack conventional ciphers. In this connection, it may be mentioned that the high rate heterodyne attack needed on Y-00 is currently not quite technologically feasible, though it may be in the not-too-far future.

Y-00 can be employed to realize these benefits not available otherwise. However, if it is intrinsically less secure than conventional ciphers, its utility would be in serious doubt. The configuration of Fig. 2 shows this is not the case – it can in fact be more secure that ENC or $AES_i$ by itself. There is also no known attack applicable to AES in noise.

Deliberate Signal Randomization
===============================

In contrast to a nondegenerate nonrandom classical cipher for which the key is completely protected in the information-theoretic sense against CTA when the data is uniformly random [@yuen05qph; @nair06], there is little distinction between CTA and KPA for the bare Y-00. Only a factor of 2 in the per qumode alphabet size is obtained in KPA versus CTA as indicated above, and expounded in [@nair06]. The question arises as to whether full information-theoretic security against CTA can be restored by modifying the bare Y-00. The authors of [@donnet] appear to be pessimistic on the possibility of achieving this. To quote: “While randomization methods might increase the security level, it remains to be seen if they will provide perfect secrecy.” In the following, we show how this is possible with Deliberate Signal Randomization (DSR) independently of the mechanism of running key generation.

The reason why the seed key cannot be attacked in CTA is clear for an additive stream cipher with uniformly random data. The “channel” between the seed key and the output observation has zero capacity due to the data which acts as random noise. In particular, it is clear that no FCA can be launched. The coherent-state noise in Y-00 is not big enough for high signal level to produce a similar effect. However, further randomization may in principle be produced to achieve this end, both classically and quantum mechanically.

Since the coherent-state noise in Y-00 can in principle be replaced, in an equivalent classical system, by deliberate randomization of the classical signal from Alice as we have repeatedly emphasized [@yuen04; @pla05; @yuen05qph; @nair06], we first consider this classical situation. Let $\theta_s$ be the signal point on the circle of Fig. 1, $x$ the data bit, $k'$ the running key segment that determines the basis. Before deliberate or noise randomization, $\theta_s(x,k')$ is uniquely determined by $x$ and $k'$. From $\theta_s$ one randomizes it to $\theta_r$ according to a probability density $p(\theta_r|\theta_s)$. We use continuous $\theta$’s here but the argument is identical for discrete $\theta$’s. More generally, let $\theta$ be Eve’s observed signal point, so that $\theta=\theta_r$ in a classical noiseless system with deliberate randomization. Then, $$\label{pdf}
p(\theta|x,k')=\int
p(\theta|\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d} \theta_r.$$

In the classical noiseless case with just signal randomization, $p(\theta|\theta_r)=\delta(\theta-\theta_r)$, the BPSK signal may be correctly discriminated when the observed $\theta$ falls within the half-circle centred around $\theta_s$. Thus we pick $p(\theta_r|\theta_s)$ to be the uniform distribution on the half-circle with midpoint $\theta_s$. If $x$ is uniformly random, then from (1) $$\label{pdf2}
p(\theta|k')=\frac{1}{2} \sum_{x=0,1} p(\theta|x,k')$$ is the uniform distribution on the full circle independent of $k'$. This proves the observation of $\theta$ to Eve yields no information at all on $k'$. In other words, Eve’s channel on $k'$ has zero capacity from DSR and uniformly random data which acts as added noise unknown to her, similar to a nondegenerate nonrandom stream cipher.

For coherent-state noise described in the wedge approximation [@pla05; @nair06], whereupon a heterodyne or phase measurement the observed $\theta$ is taken to be uniformly distributed within a standard deviation around $\theta_r$ and zero outside, the same $k'-$independence for $p(\theta|k')$ obtains when $\theta_r$ is chosen in a discrete number of positions for given $\theta_s$ so that $p(\theta_r|\theta_s)$ fills out a uniform half-circle again. We have assumed an integral number of wedges would do this, which can be guaranteed by choice of the signal level $\alpha_0$. Going beyond the wedge approximation, one needs to determine the function $p(\theta_r|\theta_s)$ in (1) for a coherent state/fixed measurement $p(\theta|\theta_r)$ so that $p(\theta|x,k')$ is uniformly distributed in a half-circle, where $p(\theta|\theta_r)$ is obtained from Eve’s optimal individual qumode quantum measurement. In this case, there is the problem that the resulting error probability for Bob may be higher than the designed level even with knowledge of the seed key $K$. In principle, this problem can be handled in one of two different ways without affecting the data security as measured by the Shannon limit [@yuen05qph; @nair06].

First, one may increase $S$ and correspondingly $M$ while maintaining the same Y-00 random cipher characteristic $\Gamma =
M/{\pi\sqrt{S}}$ defined in [@nair06]. Doing so will make the tail of the probability distribution that causes Bob’s error arbitrarily small. Indeed, in the classical limit $S \rightarrow
\infty, M \rightarrow \infty, M/{\sqrt{S}} \rightarrow \pi\Gamma$, a constant, the error vanishes. A second way is to employ an error correcting code for Bob and randomize the entire codeword of $n$-bits in a correlated fashion in the signal space $\mathcal{C}^n$, where $\mathcal{C}$ is the coherent-state circle in $\mathbb{R}^2$. This is done by moving the $n$-bit codewords within mutually exclusive but jointly exhaustive regions that fill the entire signal space $\mathcal{C}^n$, similar to the filling of the circle $\mathcal{C}$ in the one-bit case. Detailed quantitative treatment of these will appear elsewhere.

Note that Y-00 is only a random cipher for a given quantum measurement, it is not a random quantum cipher. See [@nair06]. A convenient way to make it a quantum random cipher is to randomize the parameter $\theta_s$ to $\theta_r$ that determines the quantum state $\rho(\theta_r)$ to be transmitted. The resulting output state is then, analogous to (1), $$\label{state}
\rho(x,k')=\int
\rho(\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d}\theta_r.$$ It may be seen from (3) that by uniformly randomizing $\theta_s$ as above, for any state modulation $\rho(\theta_r)$, the output quantum state itself is independent of $k'$ upon averaging over $x$ as before. Thus, such quantum DSR would protect the key against CTA with the most general joint (quantum measurement) attack. In this case, there is generally a larger probability of error that Bob would decide on $x$ incorrectly as compared to no DSR, similar to the specific coherent state case under heterodyne attack. One of the above two approaches in the fixed measurement case can be similarly employed to bring the error down to any desired level.

It may be noted that the deployment of full DSR just described above is practically difficult at present if only because high speed random numbers are needed. On the other hand, it may be possible to delve into the qumode sequence to take advantage of the randomization inherent in such sequence for selected deliberate randomization while providing essentially the same overall result. Detailed treatment of concrete DSR on Y-00 will be given elsewhere.

Conclusion
==========

We have shown that Y-00 can be designed to be secure against fast correlation attacks including that of ref. [@donnet], and that it can be configured to be more secure than AES while retaining the same high speed and its advantage as a physics-based cipher. We also prove the full information-theoretic security of Y-00 with proper deliberate signal randomization against ciphertext-only attacks. Quantitative security against known-plaintext attacks, as in the case of conventional ciphers, is a difficult, open, and important area of research.

Acknowledgements
================

We would like to thank E. Corndorf, G. Kanter, P. Kumar, and C. Liang for useful discussions. This work has been supported by DARPA under grant F30602-01-2-0528 and AFOSR grant FA9550-06-1-0452.

[10]{} url \#1[`#1`]{}urlprefix

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[^1]: yuen@ece.northwestern.edu
