---
abstract: 'We review the notion of a classical random cipher and its advantages. We sharpen the usual description of random ciphers to a particular mathematical characterization suggested by the salient feature responsible for their increased security. We describe a concrete system known as $\alpha\eta$ and show that it is equivalent to a random cipher in which the required randomization is effected by coherent-state quantum noise. We describe the currently known security features of $\alpha\eta$ and similar systems, including lower bounds on the unicity distances against ciphertext-only and known-plaintext attacks. We show how $\alpha\eta$ used in conjunction with any standard stream cipher such as AES (Advanced Encryption Standard) provides an additional, qualitatively different layer of security from physical encryption against known-plaintext attacks on the key. We refute some claims in the literature that $\alpha\eta$ is equivalent to a non-random stream cipher.'
author:
- |
    Ranjith Nair[^1], Horace P. Yuen, Eric Corndorf, Takami Eguchi, and Prem Kumar\
    Center for Photonic Communication and Computing\
    Department of Electrical Engineering and Computer Science\
    Northwestern University, Evanston, IL 60208
title: Quantum Noise Randomized Ciphers
---

PACS: 03.67.Hk, 42.50.Ar

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

The possibility of achieving greater secrecy by introducing additional randomness into the plaintext of a cipher before encryption was known, according to [@massey88], already to Gauss, in the form of the so-called ‘homophonic substitution’. Such a procedure is an example of a *random cipher* [@massey88; @yuen05qph]. The advantage of a random cipher not present in standard nonrandom ciphers is that it can provide information-theoretic security of the key against statistical attacks, and possibly known-plaintext attacks (See Appendix A and also [@yuen05qph]). A somewhat detailed description of these possibilities is one of the goals of this paper. In spite of the potential advantages of random ciphers, a large obstacle in their deployment is the bandwidth expansion, or more accurately data rate reduction, that is needed to operate all previous random ciphers. Also, it is not currently possible to generate true random numbers at speeds high enough for random ciphers to operate at sufficiently high data rates ($\sim$ Mbps is the current upper limit for random number generation). The quantum noise in optical coherent-state signals may be utilized for this purpose, and quantum optical effects seem to be the only technologically feasible way to generate $>$ Gbps true random numbers. A particular quantum noise-based random cipher, called $\alpha\eta$, that also does not entail data rate reduction, has already been proposed and implemented [@barbosa03; @yuen03] at Northwestern University. In a previous preprint [@yuen05qph], $\alpha\eta$ was discussed concomitantly with that of the closely related key generation system called $\alpha\eta$-KG. Since the features of $\alpha\eta$ direct encryption are subtle and complex enough, we take the approach in this paper of discussing just the $\alpha\eta$ encryption system in its own right, and analyze quantitatively its random cipher feature. Doing so will hopefully also avert many possible confusions with $\alpha\eta$-Key Generation, such as those in [@nishioka04; @nishioka05]. In particular, we will set up in detail the proper framework to understand and analyze the security issues involved. Note that the present paper can be understood independently of ref. [@yuen05qph], the relevant terminology and results from which are summarized in Section 2.1 and Appendix A of this paper.

Following our discussion of random ciphers in general and the $\alpha\eta$ cryptosystem, we show that $\alpha\eta$ security is equivalent to that of a corresponding classical random cipher. We show how quantum noise allows some degree of randomization in $\alpha\eta$ without sacrificing data rate, and quantify the randomization by two different parameters corresponding to ciphertext-only and known-plaintext attacks. We also show how $\alpha\eta$ can be operated on top of a standard cipher like AES to provide additional, qualitatively different, security based on quantum noise against known-plaintext attacks on the key. However, information-theoretically, ciphertext-only attack on the key is possible with the original $\alpha\eta$. We will indicate what additional techniques can alleviate this problem, without going into any detailed analysis to be presented at a later time. Generally, only search-complexity based security will be quantitatively described in this paper. Finally, we rebut the claims in [@nishioka04; @nishioka05] that $\alpha\eta$ security is equivalent to that of a standard stream cipher and that $\alpha\eta$ is nonrandom.

The plan of this paper is as follows: In Section 2, we provide the necessary review of standard cryptography. In addition, we define the random cipher concept quantitatively and point out the available results on random cipher security. This sets the stage for our definitions in Section 3 that characterize a *quantum cipher* and a *quantum random cipher*, which are both ciphers in which the ciphertext is in the form of a quantum state. In Section 4, we describe the $\alpha\eta$ system in detail, show its quantum random cipher characteristics, and highlight its advantages. In Section 5, we respond to the criticisms on $\alpha\eta$ made by Nishioka et al [@nishioka04; @nishioka05] in a further elaboration of the quantitative random cipher character of $\alpha\eta$.

Standard Cryptography and Random Ciphers
========================================

Standard Symmetric-Key Cryptography
-----------------------------------

We review the basics of symmetric-key data encryption. Further details can be found in, e.g., [@massey88; @stinson]. Throughout the paper, random variables will be denoted by *upper-case* letters such as $K,X_1$ etc. It is sometimes necessary to consider explicitly sequences of random variables $(X_1,X_2,
\ldots, X_n)$. We will denote such *vector* random variables by a *boldface* upper-case letter $\mathbf{X}_n$ and, whenever necessary, indicate the length of the vector ($n$ in this case) as a subscript. Confusion with the $n$-th component $X_n$ of $\mathbf{X}_n$ should not arise as the latter is a boldface vector. Particular values taken by these random variables will be denoted by similar *lower-case* alphabets. Thus, particular values taken by the key random variable $K$ are denoted by $k, k'$ etc. Similarly, a particular value of $\mathbf{X}_n$ can be denoted $\mathbf{x}_n$. The plaintext alphabet will be denoted $\mathcal{X}$, the set of possible key values $\mathcal{K}$ and the ciphertext alphabet $\mathcal{Y}$. Thus, for example, the sequences $\mathbf{x}_n \in \mathcal{X}^n$. In most nonrandom ciphers, $\mathcal{X}$ is simply the set $\{0,1\}$ and $\mathcal{Y}=\mathcal{X}$.

With the above notations, the $n$-symbol long *plaintext* (i.e., the message sequence that needs to be encrypted) is denoted by the random vector $\mathbf{X}_n$, the *ciphertext* (i.e., the output of the encryption mechanism) is denoted by $\mathbf{Y}_n$ and the secret key used for encryption is denoted by $K$. In this paper, we will often call the legitimate sender of the message ‘Alice’, the legitimate receiver ‘Bob’, and the attacker (or eavesdropper) ‘Eve’. Note that although the secret key is typically a sequence of bits, we do not use vector notation for it since the bits constituting the key will not need to be singled out separately in our considerations in this paper. In standard cryptography, one usually deals with *nonrandom ciphers*. These are ciphers for which the ciphertext is a function of only the plaintext and key. In other words, there is an encyption function $E_{k}(\cdot)$ such that: $$\label{encryption}
\mathbf{y}_n=E_{k}(\mathbf{x}_n).$$ There is a corresponding decryption function $D_{k}(\cdot)$ such that: $$\label{decryption}
\mathbf{x}_n=D_{k}(\mathbf{y}_n).$$ In such a case, the $X_i$ and $Y_i, i=1,\ldots, n$ are usually taken to be from the same alphabet. In contrast, a *random cipher* makes use of an additional random variable $R$ called the *private randomizer* [@massey88], generated by Alice while encrypting the plaintext and known only to her, if at all. Thus the ciphertext is determined as follows: $$\label{randomcipher}
\mathbf{y}_n=E_{k}(\mathbf{x}_n, r).$$ Because of the additional randomness in the ciphertext, it typically happens that the ciphertext alphabet $\mathcal{Y}$ needs to be larger than the plaintext alphabet $\mathcal{X}$ (or else, $\mathbf{Y}$ is a longer sequence than $\mathbf{X}$, as in homophonic substitution). It may even be a continuous infinite alphabet, e.g. an analog voltage value. However, we still require, as in [@massey88], that Bob be able to decrypt with just the ciphertext and key (i.e., without knowing $R$), so that there exists a function $D_{k}(\cdot)$ such that Eq.(\[decryption\]) holds. We note that random ciphers are called ‘privately randomized ciphers’ in Ref. [@massey88] – we will however use the shorter term ‘random cipher’ (Note that ‘random cipher’ is used in a completely different sense by Shannon [@shannon49]).

We note that the presence or absence of the private randomizer $R$ may be indicated using the conditional Shannon entropy (We assume a basic familiarity with Shannon entropy and conditional entropy. See any information theory textbook, e.g., [@cover91].). For nonrandom ciphers, we have from Eq.(\[encryption\]) that $$\label{nonrandom}
H(\mathbf{Y}_n|K \mathbf{X}_n)=0.$$ On the other hand, a *random cipher* satisfies $$\label{random}
H(\mathbf{Y}_n|K\mathbf{X}_n) \neq 0,$$ due to the randomness supplied by the private randomizer $R$. The decryption condition Eqs.(\[decryption\]) for both random and nonrandom ciphers has the entropic characterization: $$\label{decrypt} H(\mathbf{X}_n|K\mathbf{Y}_n)=0.$$ Note that this characterization of a random cipher is problematic when the ciphertext alphabet is continuous, as could be the case with $\alpha\eta$, because then the Shannon entropy is not defined. It may be argued that the finite precision of measurement forces the ciphertext alphabet to be discrete. Indeed, in Sec. 2.2, we define a parameter $\Lambda$ that characterizes the “degree of randomness” of a random cipher. In any case, the definition makes sense, similar to Eq. (\[random\]), only when the ciphertext alphabet is finite, or at most discrete.

In the cryptography literature, the characterization of a general random cipher is limited to that given by Eqs. (\[randomcipher\]) and (\[random\]). See, e.g., [@massey88]. In the next section, we will see that the purposes of cryptographic security suggest a sharper quantitative definition of a random cipher involving a pertinent security parameter $\Gamma$. This new definition, unlike (\[random\]), will be meaningful irrespective of whether the ciphertext alphabet is discrete or continuous. Before we discuss the above new definition of random ciphers, we conclude this section with some important cryptographic terminology.

By *standard cryptography*, we shall mean that Eve and Bob both observe the same ciphertext random variable, i.e., $\mathbf{Y}^{\rm E}_{n}=\mathbf{Y}_{n}^{\rm B}=\mathbf{Y}_{n}$. Thus, standard cryptography includes usual mathematical private-key (and also public-key) cryptography but excludes quantum cryptography and classical-noise cryptography [@maurer93]. For a standard cipher, random or nonrandom, one can readily prove from the above definitions the following result known as the *Shannon limit* [@massey88; @shannon49]: $$\label{shannonlimit} H(\mathbf{X}_n|\mathbf{Y}_n) \leq H(K).$$ This result may be thought of as saying that no matter how long the plaintext sequence is, the attacker’s uncertainty on it *given the ciphertext* cannot be greater than that of the key. This condition is of crucial importance in both direct encryption and key generation, as brought out in refs. [@yuen03; @yuen05qph; @yuen05pla; @pra05; @yuan05], but was missed in previous criticisms of $\alpha\eta$ [@nishioka04; @nishioka05; @loko05].

By *information-theoretic security* (or *IT security*) on the data, we mean that Eve cannot, even with unlimited computational power, pin down uniquely the plaintext from the ciphertext, i.e., $$\label{ITsecurity}
H(\mathbf{X}_n|\mathbf{Y}_n)\neq 0.$$ The level of such security may be quantified by $H(\mathbf{X}_n|\mathbf{Y}_n)$. Shannon has defined *perfect security* [@shannon49] to mean that the plaintext is statistically independent of the ciphertext, i.e., $$H(\mathbf{X}_n|\mathbf{Y}_n)=H(\mathbf{X}_n).$$ With the advent of quantum cryptography, the term ‘unconditional security’ has come to be used, unfortunately in many possible senses. By *unconditional security*, we shall mean near-perfect information-theoretic security against all attacks consistent with the known laws of quantum physics.

Incidentally, note that the Shannon limit Eq. (\[shannonlimit\]) immediately shows that perfect security can be attained only if $H(\mathbf{X}_n)\leq H(K)$, so that, in general, the key needs to be as long as the plaintext.

Random Ciphers – Quantitative Definition
----------------------------------------

As mentioned in the previous section, the characterization of a general random cipher merely using Eq. (\[randomcipher\]) or (\[random\]) is perhaps not well-motivated. The reason for studying random ciphers is in fact the belief that they enhance the security of the cipher against various attacks. By bringing into focus the intuitive mechanism by which a random cipher may provide greater security than a nonrandom counterpart against known-plaintext attacks, we will propose one possible quantitative characterization of a general random cipher (or more exactly, a general random *stream* cipher. See below.). For a description of known-plaintext and other attacks on ciphers, together with the known results on their security, we refer the reader to Appendix A.

We now discuss the intuitive mechanism of security enhancement in a random cipher. To this end, a schematic depiction of encryption and decryption with a random cipher is given in Fig. 1. For a binary alphabet $\mathcal{X}=\{0,1\}$, let $\mathcal{X}^n=\{a_1,\ldots, a_N\}$ be the set of $N=2^n$ possible plaintext $n$-sequences. Let $k$ be a particular key value. One can view the key $k$ as dividing the ciphertext space $\mathcal{Y}^n$ into $N$ parts, denoted by the $\mathcal{A}_{a_j}^k, j \in \{1, \ldots, N\},$ in the figure. Encryption of plaintext $a_j$ proceeds by first determining the relevant region $\mathcal{A}_{a_j}^k$ and randomly selecting (this is the function of the private randomizer) as ciphertext some $y\in\mathcal{A}_{a_j}^k$. The decryption condition Eq.(\[decryption\]) is satisfied by virtue of the regions $\mathcal{A}_{a_j}^k$ being disjoint for a given $k$. Also shown in Fig. 1 is the situation where a different key value $k'$ is used in the system. The associated partition of $\mathcal{Y}^n$ consists of the sets $\mathcal{A'}_{a_j}^{k}$ that are shown with shaded boundaries in Fig. 1. The *important point* here is that the respective partitions of the ciphertext space for the key values $k$ and $k'$ should be sufficiently ‘intermixed’. More precisely, for any given plaintext $a_j$, and any observed ciphertext $\mathbf{y}_n$, we require that there exist sufficiently many key values $k$ (and hence a sufficiently large probability of the set of possible keys corresponding to a given plaintext and observed ciphertext) for which $\mathbf{y}_n \in
\mathcal{A}_{a_j}^k$. In other words, a given plaintext-ciphertext pair can be connected by many possible keys. This is the intuitive basis why random ciphers offer better quantitative security (as measured either by Eve’s information on the key or her complexity in finding it; see Sec. 4.2-4.4 for a discussion of $\alpha\eta$ security) than nonrandom ciphers against known-plaintext attacks.

\[htbp\]

While the above arguments hold for any type or random cipher whatsoever, we will restrict our scope to the so-called *stream ciphers*. Most ciphers in current use (which are all nonrandom), such as AES, are stream ciphers [@stinson]. In a nonrandom stream cipher, the key $K$ is first expanded using a deterministic function into a much longer sequence $(Z_1,\ldots,
Z_n)$ called the *keystream* or *running key*. The defining property of a *stream cipher* is that the $i$-th ciphertext symbol $y_i$ be a function of just the $i$-th keystream symbol $z_i$ and the earlier and current plaintext symbols $x_1,\ldots,x_i$: $$\label{streamcipher}
y_i=E^i(x_1, \ldots, x_i;z_i).$$ It follows that decryption of the first $i$ symbols of plaintext is possible from the first $i$ symbols of ciphertext and the running key. A *synchronous* stream cipher is one for which $$\label{syncstreamcipher} y_i=E^i(x_i;z_i).$$ Thus, the $i$-th ciphertext symbol depends only on the $i$-th plaintext symbol and the $i$-th keystream symbol, i.e., the cipher is memoryless. For our discussion of random ciphers, we will restrict ourselves for concreteness to the case of *random stream ciphers*, that are defined by: $$\label{randomstreamcipher}
y_i=E^i(x_1, \ldots, x_i; z_i; r_i).$$ Here, the $\{R_i\}$ are randomizers that may be assumed to be independent random variables (this is the case in $\alpha\eta$), but this is not necessary. In the rest of the paper, a *random cipher* will always mean a *random stream cipher*.

For a nonrandom stream cipher given by Eq. (\[streamcipher\]), it is usually the case that given the plaintext vector $\mathbf{x}_i$ of length $i$ and ciphertext symbol $y_i$, the value of the keystream $z_i$ is uniquely determined. This is typically the case also in a random stream cipher *when the value $r$ taken by the randomizer $R_i$ is known*. In the absence of such knowledge, however, the different possible values taken by $R_i$ will in general allow many different values of the keystream for the given plaintext vector and ciphertext symbol. The more such possibilities exist, the less information is obtained about the keystream and the more ‘secure’ the cipher is. Our quantitative definition of random cipher given below introduces a parameter $\Gamma$ that provides one way of quantifying the different knowledge of the keystream obtained in the above two scenarios by the number of additional possible keystreams for a given pair of input data and corresponding ciphertext symbols.\
\
**Definition** (*$\Gamma$- Random Cipher*) **:**\
A $\Gamma$-Random Cipher is a random stream cipher of the form of Eq. (7) for which the following condition holds:\
For every plaintext sequence, $\mathbf{x}_i$, for every $i$, for every ciphertext symbol $y_i$ obtainable by encryption of $\mathbf{x}_i$, and for every value $r$ of $R_i$, $$\label{Gammarandomcipherdef}
|\{z_i | y_i=E^i(x_1, \ldots, x_i; z_i; r') \; \textup{ for some}
\; r'\}| - |\{z_i | y_i=E^i(x_1, \ldots, x_i; z_i; r)\}| \geq
\Gamma.$$\
The bars $|\cdot |$ indicate size of the enclosed set. For a nonrandom stream cipher, the keystream $z_i$ is uniquely fixed by the plaintext vector $\mathbf{x}_i$ and the ciphertext symbol $y_i$. Therefore, if the randomizer in (\[Gammarandomcipherdef\]) is ignored so that it applies to a nonrandom cipher, a nonrandom cipher would have $\Gamma = 0$. Note that the sets whose sizes appear in the above equation, both for random ciphers and their nonrandom reductions, are constructed only on the basis of the $i$-th ciphertext symbol $y_i$, and not on the basis of the entire ciphertext sequence. Thus, the definition of $\Gamma$ only gives the *number of possible keys per symbol of ciphertext* under known-plaintext attack, while the number of possible keys based on the entire ciphertext sequence (that is illustrated schematically by the overlap sets in Fig. 1) may be significantly less. In this sense, our definition has a restricted symbol by symbol scope but is easy to calculate with, similar to the independent particle approximation in many-body physics. It does not by itself determine the precise security of the cipher, but rather is the starting point of precise analysis, which is a difficult task just as correlations in interacting many-body systems are always difficult to deal with in a rigorous quantitative manner.

It is possible to satisfy the random cipher condition (\[random\]) with $\Gamma =0$. This happens, e.g., when (\[Gammarandomcipherdef\]) holds for some ciphertext symbols with $\Gamma >0$ but some others with $\Gamma=0$, so the overall condition (\[Gammarandomcipherdef\]) is only satisfied for $\Gamma=0$. A different measure of randomization $\Lambda$, bearing directly on (\[random\]), may be introduced which has the property that $\Lambda=0$ is equivalent to a nonrandom cipher. For the case where the ciphertext alphabet is finite and for given $\mathbf{x}_i,z_i$ and $r$, let $$\label{lambdarandomcipherdef}
\Lambda=|\{ y_i | y_i = E^i(x_1, \cdots , x_i;z_i;r') \; \textup{ for some}
 \; r' \}| - |\{ y_i | y_i = E^i (x_1, \cdots , x_i;z_i;r)
\}|.$$ Thus, condition (\[random\]) is equivalent to $\Lambda >0$ for some $\mathbf{x}_i,z_i$ and $r$. It follows that $\Lambda=0$ for all $(\mathbf{x}_i,z_i)$ is equivalent to the cipher being nonrandom. $\Lambda+1$ is the number of possible output signal symbols corresponding to a given input symbol and running key value. Thus, the parameter $\Lambda$ measures directly the degree of per symbol ciphertext randomization, while $\Gamma$ measures the per symbol key redundancy. It is possible that a $\Gamma=0$ random cipher is still useful due to the additional loads on Eve to record and store more information from her observation. On the other hand, for the *typical* case where $z_i$ is in one-to-one correspondence with $y_i$ for given $\mathbf{x}_i$ and $r$, $\Gamma >0$ implies $\Lambda>0$ for every $\mathbf{x}_i$ and $z_i$, which in turn implies that a cipher with $\Gamma
> 0$ is random in the sense of (\[random\]). A simple application of the $\Gamma$ and $\Lambda$ characterizations to $\alpha\eta$ leads to information-theoretic lower bounds on the unicity distances $n_0$ and $n_1$ for CTA and KPA, as discussed in Sec. 4.3. The following simple example also serves to illustrate the above definitions: **Example** (Random cipher) Let $\mathcal{X}=\{0,1\}$, $\mathcal{K}=\{k_0,k_1,k_2,k_3,k_4\}$ and $\mathcal{Y}=\{a,b,c,d,e\}$. Fig. 2 lists the possible ciphertexts for each plaintext and key pair.

[|c|c|c|]{} $x$ & $k$ & $y$\
$0$ & $k_0$ & $a,b$\
$1$&$k_0$ & $c,d,e$\
$0$ & $k_1$ & $c,d$\
$1$ & $k_1$ & $e,a,b$\
$0$ & $k_2$ & $e,a$\
$1$ & $k_2$ & $b,c,d$\
$0$ & $k_3$ & $b,c$\
$1$ & $k_3$ & $d,e,a$\
$0$ & $k_4$ & $d,e$\
$1$ & $k_4$ & $a,b,c$\

For this cipher, one can easily verify that at least 2 key values connect every possible plaintext-ciphertext pair. In addition, every plaintext-key pair can lead to at least two different ciphertexts. In terms of the definitions given above, this cipher has $\Gamma=1$ and $\Lambda=1$.

Quantum Random Ciphers
======================

The known and possible advantages of a random classical cipher over a nonrandom one were discussed in the previous section. While it is possible to implement a random cipher classically using random numbers generated on Alice’s side, this is not currently practical at high ($\sim$ Gbps) rates. As will become clear in the sequel, the quantum encryption protocol $\alpha\eta$ (Various implementations are described in [@barbosa03; @corndorf03; @pra05; @ptl05; @hirota05] - The protocol in [@hirota05] is a variation on the original $\alpha\eta$ of [@barbosa03]) effectively implements a random cipher from Eve’s point of view for a given choice of her measurement, the difference from a classically random cipher being that it uses coherent-state quantum noise to perform the needed randomization. Before we describe $\alpha\eta$, we define some concepts that capture the relevant features of a quantum random cipher. As emphasized in Section 2.2, we will confine our attention to *stream* ciphers. First, we straightforwardly extend the usual stream cipher to one where the ciphertext is a quantum state. Our motivation for this definition is that, from the point of view of the legitimate users Alice and Bob, $\alpha\eta$ is a quantum stream cipher with negligible $\lambda$ in the sense given below:\
\
**Definition** *($\lambda$-Quantum Stream Cipher (QSC)*)**:**\
A quantum stream cipher is a cipher for which the following two conditions are satisfied:

A.  The encryption map $e_k(\cdot)$ takes the $n$-symbol plaintext sequence $\mathbf{x}_n$ to a quantum state $n$-sequence $\mathbf{\rho}$ in the $n$-fold tensor product form:

    $$\label{quantcipher}
    \mathbf{\rho}= e_k(\mathbf{x}_n) = \rho_{1}(x_1;z_1) \otimes
    \ldots \otimes \rho_{n}(x_1, \ldots ,x_n;z_n),$$

    and

B.  [ Given the key $k$, there exists a measurement on the encrypted state sequence, that recovers each plaintext symbol $x_i$ with probability $P_{dec} > 1 - \lambda$.]{}

Here, as in Section 2.2, $(Z_1, \ldots, Z_n)$ is the keystream generated from the seed key $K$. A few comments will help clarify the definition. First, note that the tensor product form of the state in condition A retains for a quantum cipher the property of a classical cipher that one can generate the components in the $n$-sequence of states that constitute the output of a cipher one after the other in a time sequence. Note also that, analogous to a classical stream cipher, the $i$-th tensor component of $\rho$ depends on just $z_i$ and $(x_1,\ldots,x_i)$. Condition B is the generalized counterpart of the decryption condition Eq.(\[decryption\]) for a classical cipher – we now allow a small enough decryption error probability. Thus, the per-symbol error probability is bounded above by $\lambda < 1$.

We now want to bring the concept of classical *random* cipher defined in the previous section into the quantum setting. Our motivation in doing so is to show that, for an attacker making the same measurement on a mode-by-mode basis without knowledge of the key, $\alpha\eta$ reduces to an equivalent $\Gamma$-Random Cipher with significantly large $\Gamma$. Since the output of a quantum cipher is a quantum state and not a random variable, we will need to specify a POVM $\{\Pi_{\mathbf{y}_n}\}$ whose measurement result $\mathbf{Y}_n$ supplies the classical ciphertext. Note that in this quantum situation different choices of measurement may result in radically different kinds of ciphertext. Note also that the user’s and the attacker’s measurements may be different. Our definition of a quantum random stream cipher below will apply relative to a chosen ciphertext $\mathbf{Y}_n$ defined by its associated POVM. We will also assume that, from the eavesdropper’s viewpoint, the same measurement is made on each of the $n$ components of the cipher output. In other words, the POVM defining the ciphertext $\mathbf{Y}_n$ is a tensor product of identical POVMs $\{\pi_y\}$.\
\
**Definition** (*$(\Gamma,\lambda,\lambda',\{\pi_y\})$- Quantum Random Stream Cipher (QRC)*)**:**\
An $(\Gamma,\lambda,\lambda',\{\pi_y\})$ - quantum random stream cipher is a $\lambda$-quantum stream cipher such that for the ciphertext given by the result of the product POVM $\{\Pi_{\mathbf{y}_n}= \bigotimes_{i=1}^{i=n} \pi_{y_i}\}$,

A.  one has an $\Gamma$-random stream cipher satisfying Eq.(\[Gammarandomcipherdef\]), and

B.  the probability of error per symbol $P_{dec}'$ using the key *after* measurement is $P_{dec}' > 1 -\lambda'$.

Several comments are given to explain this definition:

1.  [ While condition QRC-B above appears similar to the condition QSC-B for a quantum stream cipher, there is a crucial difference. In the latter, the decryption probability $P_{dec}$ takes into account the possibility that the *quantum measurement* (as well as classical post-processing) made on the cipher state can depend on the key, i.e. it refers to Bob’s rather than Eve’s error probability. In QRC-B, we are considering the probability of error involved for Eve when she decrypts using a quantum measurement independent of the key followed by classical post-processing that is , in general, “collective” and depends on the key. Thus, the parameter $\lambda'$ is related to the symbol error probability under this latter restriction while the parameter $\lambda$ in QSC-B is tied to the symbol error probability for a quantum measurement allowed to depend on the key. We see that there are two measurements implicit in our definition of a QRC - one made by the user with the help of the key, and the other given by $\{\pi_y\}$ made by the attacker without the key. See also Item 3 below. As we shall see, $\alpha\eta$ satisfies QRC-B with negligible $\lambda'$ under a heterodyne or phase measurement attack by Eve.]{}

2.  [ $\Gamma$ in QRC-A, as in Eq.(\[Gammarandomcipherdef\]), is a measure of the ’degree of intermixing’ of the regions of ciphertext space corresponding to different key values on a symbol-by-symbol basis. If $\{\pi_y\}$ describes a discrete measurement, a $\Lambda$ corrresponding to Eq.(\[lambdarandomcipherdef\]) can also be introduced.]{}

3.  [ Our stipulation that the same POVM be measured on each of the components of the cipher output is tantamount to restricting the attacker to identical measurements on each tensor component followed by collective processing. We will call such an attack a *collective attack* in this paper (also in [@yuen05qph]). This definition is different from the usual collective attack in quantum cryptography [@gisin02]: in the latter, following the application of identical probes to each qubit/qumode, a joint quantum measurement on all the probes is allowed. In our case, there is no probe for Eve to set as we conceptually allow her a full copy of the quantum state. Doing so, we can upper bound her performance. (This is an important feature of our so-called KCQ approach to encryption and key generation. See [@yuen03] for discussion.) Thus, allowing a joint measurement, as also nonidentical measurements on each output component, will be called a joint attack.]{}

4.  [In analogy with the classical random cipher definition Eq. (\[Gammarandomcipherdef\]), one may wonder why the private randomizers $R_i$ used in that definition are missing from that of the quantum random cipher. Indeed, one may randomize the quantum state $\rho_i(x_1, \ldots
    ,x_i;z_i)$ to $\rho_i(x_1, \ldots ,x_i;z_i;r_i)$ using a private random variable with probability distribution $p_{r_i}$. However, since the value of $R_i$ remains unknown to both user and attacker (Indeed, the user should not need to know $R_i$ in order to decrypt or even to encrypt in the case of $\alpha\eta$), one sees that all probability distributions of Bob’s or Eve’s measurements in this situation are given by the state $\rho'_i(x_1, \ldots
    ,x_i;z_i)=\sum_{r_i}{p_{r_i}\rho_i(x_1, \ldots ,x_i;z_i;r_i)}$, in which there is no explicit dependence on $r_i$. In particular, we mention here that exactly such quantum state randomization, called Deliberate Signal Randomization (DSR), has been proposed in the context of $\alpha\eta$ in [@yuen03] for the purposes of enhancing the information-theoretic security of $\alpha\eta$.]{}

5.  [It is important to observe that the definitions given above both for classical and quantum random ciphers are not arbitrary ones, but rather the mathematical characterizations of very typical situations involving randomization in classical and quantum cryptosystems.]{}

We present an example of a QRC in the next section: the $\alpha\eta$ cryptosystem.

The $\alpha\eta$ cryptosystem
=============================

Operation
---------

We now describe the $\alpha\eta$ system and its operation as a quantum cipher:

(1) Alice and Bob share a secret key $\mathbf{K}_s$.

(2) Using a *key expansion function* $ENC(\centerdot)$, e.g., a linear feedback shift register or AES in stream cipher mode, the seed key $\mathbf{K}_s$ is expanded into a running key sequence that is chopped into $n$ blocks: $\mathbf{K}_{Mn}=ENC(\mathbf{K}_s)=(K_1, \ldots , K_{mn})$. Here, $m=\log_2(M)$, so that $Z_i \equiv (K_{(i-1)m +1}, \ldots,
    K_{im})$ can take $M$ values. The $Z_i$ constitute the *keystream*.

(3) The encrypted state $e_{\mathbf{K}_s}(\mathbf{X}_n)$ of Eq.(\[quantcipher\])is defined as follows. For each bit $X_i$ of the plaintext sequence $\mathbf{X}_n = (X_1, \ldots, X_n)$, Alice transmits the *coherent state* $$\label{state}
    |\psi(X_i,Z_i)\rangle=|\alpha e^{i\theta(X_i,Z_i)}\rangle.$$ Here, $\alpha \in \mathbb{R}$ and $\theta(X_i,Z_i)$ takes values in the set $\{0,\pi/M,\ldots,(2M-1)\pi/M\}$. The function $\theta$ taking the data bit and keystream symbol to the actual angle on the coherent state circle is called the *mapper*. In this paper, we choose $\theta(X_i,Z_i)=[Z_i/M+(X_i\oplus
    Pol(Z_i))]\pi$. $Pol(Z_i)= 0$ or $1$ according to whether $Z_i$ is even or odd. This distribution of possible states is shown in Fig. 2. Thus $K_i$ can be thought of as choosing a ‘basis’ with the states representing bits $0$ and $1$ as its end points. In general, one has the freedom to vary the mapper in various ways for practical reasons. See, e.g, [@pra05].

(4) In order to decrypt, Bob runs an identical ENC function on his copy of the seed key. For each $i$, knowing $Z_i$, he makes a quantum measurement to discriminate just the two states $|\psi(0,Z_i)\rangle$ and $|\psi(1,Z_i)\rangle$.

\[htbp\]

To decrypt in step (4) above, Bob, in general would need a phase reference. This is effectively provided by the use of Differential Phase Shift Keyed (DPSK) signals in the implementations of $\alpha\eta$. See [@corndorf03; @pra05; @ptl05] for details. Doing so does not compromise security as we still assume that Eve has a perfect copy of the transmitted state.

If the line transmittance between Alice and Bob is $\eta$, Bob receives a coherent state with energy $\eta S$ instead of $S
\equiv |\alpha|^2$. The optimal quantum measurement [@helstrom76] for Bob has error probability $$\label{pB}
P^B_e \sim \frac{1}{4} \exp(-4\eta S).$$ It is thus apparent that $\alpha\eta$ is a $\lambda$-quantum cipher in the sense of Section 3 with $\lambda \sim \frac{1}{4}
\exp(-4\eta S)$. For the $S \sim 4 \times 10^4$ of [@pra05], over a distance of 80 km at a loss of 0.2 dB/km, we have $\eta S
\sim 10^3$ photons. For this mesoscopic level, $\lambda$ is $\sim
\exp(-1000)$, which is completely negligible compared, say, to the standard acceptable BER limit of $10^{-9}$, which arises from device imperfections, for an uncoded optical on-off keyed line.

Let us briefly indicate how this system may provide data security by considering an *individual attack* on each data bit $X_i$ by Eve. Under such an attack, one only looks at the per-bit error probability ignoring correlations between the bits. Under this assumption, Eve, not knowing $Z_i$, is faced with the problem of distinguishing the density operators $\rho^0$ and $\rho^1$ where $$\label{rho}
\rho^b=\sum_{Z_i}\frac{1}{M}|\psi(b,Z_i)\rangle\langle\psi(b,Z_i)|.$$ For a fixed signal energy $S$, Eve’s optimal error probability is numerically seen to go asymptotically to $1/2$ as the number of bases $M \rightarrow \infty$ (See Fig. 1 of [@barbosa03]). The intuitive reason for this is that increasing $M$ more closely interleaves the states on the circle representing bit 0 and bit 1, making them less distinguishable. Therefore, at least under such individual attacks on each component qumode [^2] of the cipher output, $\alpha\eta$ offers any desired level of security determined by the relative values of $S$ and $M$. While we are not concerned in this paper with key generation, it may be observed that unambiguous state determination (USD) attacks on $\alpha\eta$ are totally ineffective due to the large number of $2M$ states involved.

In our security analysis, Eve is always assumed to be at the transmitter so that $\eta=1$ for her. Without knowing the key, however, her performance on the data is still poor as described in the above paragraph. Her attacks on the key are described in the following. We have assumed that the users can utilize the signal energy $\eta S$ to maintain a proper bit error rate without channel coding, despite possible interference from Eve. This does not place a stringent requirement on $\eta$ itself as one can typically go around 80 km in fiber before the signal needs to be amplified. In case Eve’s interference is too strong and causes error, it would be detected in a message authentication code which always goes with encryption. There is clearly no need to do separate intrusion detection in this direct encryption case, but it turns out there is also no need in the key generation regime [@yuen05qph; @yuen03] which we do not discuss in this paper.

$\alpha\eta$ as a Random Cipher
-------------------------------

We showed in the previous subsection that $\alpha\eta$ may be operated in a regime of $S$, $\eta$ and $M$ where it is a $\lambda$-quantum cipher for $\lambda \sim 0$. We now show, that from Eve’s point of view, under both a heterodyne and phase measurement attack, $\alpha\eta$ appears effectively as a quantum *random* cipher according to the characterization of Section 3. Note that the randomization in $\alpha\eta$ can also be effected in principle by using an additional classical random number generator. This is not required in $\alpha\eta$ as high-speed randomization is automatically provided by the coherent-state quantum noise.

To see the quantum random cipher characteristic of $\alpha\eta$, consider employing the following two measurements for obtaining $\{\pi_y\}$ in the quantum random cipher definition:

1)  [ (Heterodyne measurement) $\pi_y = \frac{1}{\pi} |y\rangle\langle y|, y \in
    \mathbb{C}.$]{}

2)  [ (Canonical Phase measurement) $\pi_{\theta} = \frac {1} {2\pi} \sum_{n,n'=0}^{\infty} e^{i(n-n')\theta} |n\rangle \langle n'|, \theta \in [0,2\pi).$ ]{}

To show that the conditions for a QRC are satisfied, let us first consider QRC-B. It may be shown [@yuen03] that the error probabilities $\lambda'$ involved are respectively $\sim
\frac{1}{2}e^{-S}$ and $\sim \frac{1}{2}e^{-2S}$ for the heterodyne and phase measurements.

Turning to QRC-A, let us estimate the value of $\Gamma$ under heterodyne and phase measurement. For a signal energy $S$, the heterodyne measurement is Gaussian distributed around the transmitted amplitude with a standard deviation of $1/2$ for each quadrature while the phase measurement has an approximately Lorentzian distribution around the transmitted phase with standard deviation $\sim 1/{\sqrt{S}}$. If we assume that, given a certain transmitted amplitude/phase, the possible ciphertext values are uniformly distributed within a standard deviation on either side and ciphertext values outside this range are not reached (this will be called the *wedge approximation*), we get the following estimates $N_{het}$ and $N_{phase}$ for the number of keystream values $z_i$ covered by the quantum noise under heterodyne and phase measurements: $$\label{numberestimate}
N_{het}=2N_{phase} = M/(\pi \sqrt{S}).$$ If the value of the randomizer $R$ is fixed (corresponding to rotation by a given angle within the wedge), $Z_i$ is fixed by the plaintext and ciphertext. Thus we have according to Eq. (\[Gammarandomcipherdef\]) that $$\label{Gammahet}
\Gamma_{het}=N_{het} -1 \cong M/(\pi \sqrt{S}),$$ and that $$\label{Gammaphase}
\Gamma_{phase} \cong \Gamma_{het}/2 \cong M/(2 \pi \sqrt{S}).$$ As expected, the $\Gamma$’s of both measurements increase as the number of bases $M$ increases, and decrease with increasing signal energy $S$ that corresponds to decreasing quantum noise. For example, using the experimental parameters in [@pra05] of $S \sim 4 \times 10^4$ photons and $M \sim 2
\times 10^3$ has $\Gamma_{het} \sim 3$. The $\Lambda$ (cf. Eq. (\[lambdarandomcipherdef\]) characteristics of $\alpha\eta$ will be considered in Sec. 5.2 in connection with the Nishioka group attack. The relevance of these parameters for security is considered in detail in the next subsection and in Sec. 5.2.

$\alpha\eta$: Information-theoretic and Complexity-Theoretic Security
---------------------------------------------------------------------

Before discussing $\alpha\eta$ security, we comment that $\alpha\eta$ direct encryption is often compared to BB84 key generation followed by the use of the generated key in either one-time pad or a standard cipher like AES. This is not an appropriate comparison because $\alpha\eta$ already assumes that the users share a key. Perhaps the source of the confusion is that both $\alpha\eta$ and BB84 involve the use of quantum states. In any case, the appropriate comparison would be between $\alpha\eta$ and a standard cipher like one-time pad or AES - we do make such a comparison in the following.

We will consider in turn the information-theoretic (IT) and *complexity-theoretic (CT)* security of $\alpha\eta$. In standard cryptography, no rigorous result is known about the quantitative security level of any cipher, save the one-time pad. Since $\alpha\eta$ includes a classical stream cipher ENC (See Fig. 1), we may in general expect a similarly murky state of affairs regarding its quantitative security. However, it will turn out that, under known-plaintext attacks, one can claim *additional* security from the physical coherent-state noise for a suitably modified $\alpha\eta$ with any cipher ENC, as compared to ENC alone.

### Information-theoretic (IT) Security: Qualitative discussion

Considering first IT security, we discuss in turn qualitatively the cases of ciphertext-only, known-plaintext, and statistical attacks on the data as well as the key. Subsequently, for the former two cases, we give lower bounds for the unicity distances $n_0$ and $n_1$ (See Appendix A for definitions).

As mentioned in Appendix A, for a nondegenerate ENC box cipher, one can protect the key completely and attain data security up to the Shannon limit under CTA. If the same ENC box is used in $\alpha\eta$ one may consider, as in Sec. 4.1, an attack in which Eve attacks each data bit using only the measurement result from the corresponding qumode. Although under such an assumption IT security obtains as $M/\sqrt{S} \rightarrow \infty$, this attack is too restrictive since Eve does gain information on the key from each qumode measurement that could be useful in learning about other data bits as well. Such attacks utilizing key correlations across data bits may be launched against standard stream ciphers. Under the wedge approximation of Sec. 4.2, Eve is able to narrow her choice of basis down to $\Gamma$ possible values. Even if $\Gamma$ is large, the key security (and hence data security) is not as good as that of the ENC box alone for which case the keystream bits are *completely* random to Eve. However, one can still derive a unicity distance lower bound (See below). This defect of $\alpha\eta$ may be removed by the use of Deliberate Signal Randomization (DSR) introduced in [@yuen03]. However, the concrete analysis of systems using various forms of DSR are still under progress. But see [@yuen06pla].

Let us now consider the case of known-plaintext attacks on the key. As discussed in Appendix A, most nonrandom ciphers have a nondegeneracy distance $n_d$ at which the key is fixed under a known-plaintext attack. We also mentioned that for random ciphers, such a distance may not exist, so that it is unknown whether or not they possess IT security against KPAs. Since $\alpha\eta$ is random, the same remark applies to it. However, a finite unicity distance $n_1$ may exist for $\alpha\eta$ and other random ciphers beyond which the key is fixed in a KPA. While rigorous analysis is difficult and is so far limited to the unicity distance bound given below, we believe that such is the case for the original $\alpha\eta$ with no modification, so that it has no IT security for large enough $n$.

The statistical attacks fall between the above two extremes. Thus, there may exist a crossover point where $\alpha\eta$ security becomes better than that of the ENC box alone as one moves from CTA towards KPA. However, no quantitative results, e.g., the unicity distance under STA, are known. To summarize, we believe that under all cryptographic attacks, $\alpha\eta$ has no IT security for large enough $n$, i.e., $\lim _{n \rightarrow \infty}
H(K|\mathbf{Y}_n^E) = 0$. However, the use of $\alpha\eta$ should extend the unicity distance beyond that of the cipher ENC used in it for some statistical attacks and for known-plaintext attacks.

### Information-Theoretic (IT) Security: Unicity Distance Lower Bounds

Nonrigorous estimates of the unicity distance $n_1$ against KPA for standard stream ciphers are often made via a capacity argument in the so-called “correlation attacks” (See, e.g., [@chepyzhov00]). The bound $$\label{capacitybound} n \geq |K|/C,$$ where $C$ is the capacity of Eve’s effective channel, follows from the converse to the coding theorem [@cover91]. The application of (\[capacitybound\]) to correlation attacks is nonrigorous because the assumption of independent noise in each bit is not valid. In the case of $\alpha\eta$, *rigorous* lower bounds on $n_0$ and $n_1$ can be obtained from (\[capacitybound\]) because of the independent qumode to qumode coherent-state noise. Under the wedge approximation to the noise distribution for evaluating Eve’s capacity in (\[capacitybound\]), it may be shown [@eguchi06] that for uniform data, the CTA unicity distance $$\label{n0bound}
n_0 \geq \frac{|K|}{\log_2(\frac{M}{\Lambda +1})},$$ and for KPA, $$\label{n1bound}
n_1 \geq \frac{|K|}{\log_2(\frac{M}{\Gamma +1})}.$$In terms of the experimental parameters of [@pra05], this gives $n_0 \geq 550, n_1 \geq 490$. While these are much bigger than $n_0 \sim 120$ bits for English, no precise practical conclusion can be drawn, both because they are just lower bounds and because the actual complexity of key determination as a function of $n$ is not yet known. For the numbers above, the cryptosystem would be secure if the optimal complexity is exponential in $n$.

### Complexity-theoretic (CT) Security

Apart from IT security, the issue of complexity-theoretic (CT) security is of great practical importance. Indeed, in [@yuen05qph], we have argued that large enough search complexity security is as good as information-theoretic security in reality. For standard ciphers, we have seen that there is no IT security beyond the nondegeneracy distance. Thus, standard ciphers rely for their security under KPA basically on the complexity of algorithms to find the key. We now compare the situation with that of $\alpha\eta$. For any attack, the mere fact that $H(K|\mathbf{Y}_n^E)=0$ (for CTA and STA) or $H(K|\mathbf{Y}_n^E
\mathbf{X}_n)=0$ (for KPA) does not mean that the unique key can be readily obtained from $\mathbf{Y}_n^E$ (and $\mathbf{X}_n$ in the case of KPA). For most ciphers, one needs to run an algorithm to obtain it. At worst, this algorithm can be a *brute force search* - one decrypts $\mathbf{Y}_n^E$ with all the $2^{|K|}$ possible keys until a valid plaintext is obtained. This search can easily be made prohibitive by choosing $|K|$ large enough – $|K|\sim 4000$ used in experimental $\alpha\eta$ [@pra05] is already way beyond conceivable search capability. A better procedure that we call an *assisted brute force search* can exploit partial knowledge of the possible running key values for each bit as follows. Since each basis is specified by $m=\log_2(M)$ bits of the running key, and the seed key is revealed by a $|K|$-bit sequence of the running key for an ENC box of Fig. 3 that is an LFSR with known connection polynomial, we obtain an *assisted brute-force search complexity* of $$\label{searchcomplexity}
\mathcal{C}=\Gamma^{|K|/m}.$$ For $|K|=4400$ used in [@pra05], $\mathcal{C} \sim 2^{630}$ which is far beyond any conceivable search capability. While it is not known what Eve’s *optimal* search complexity is, the advantage here is that this degree of randomization is achieved automatically by the coherent-state quantum noise at the $\sim$ Gbps rate of operation of the system. Note also that it is not hard to increase $M$ while maintaining the same data rate because the number of bits needed to select a basis on the circle scales logarithmically with $M$.

In practice, heuristic algorithms based on the structure of the ENC cipher are used to speed up the search. The rigorous quantitative performance of these algorithms is unknown for standard ciphers. However, one may view $\alpha\eta$ as an “enhancer” of security by providing an additional ‘physical encryption’ on top of the standard ‘mathematical encryption’ provided by the ENC box as follows.

For the ENC of Fig. 3 used as a standard cipher, so that $$\label{ENC}
Y_i=X_i\oplus K_i, \ K_i=ENC(\mathbf{K}_s),$$ let the unicity distance for KPA be $n_1$ . Let us assume that there exists an algorithm ALG($Y_{n_1},X_{n_1})$) whose output is the seed key $K_s$ and that ALG has complexity $C$ when used with inputs of length $n_1$. In order to compare this complexity with that of $\alpha\eta$, we assume that the same ENC is used in an $\alpha\eta$ system. However, since $m$ bits of the keystream output of ENC are used to choose the basis for one data bit in $\alpha\eta$, we first ’match’ the data stream and keystream in $\alpha\eta$ as follows.

We *expand* the ENC output keystream by applying $m$ deterministic $m$-bit to $m$-bit functions $\{f_j\}_{j=1}^m$ to each keystream symbol $Z_i$ to get a new keystream $\mathbf{Z'}$ as follows: $$\label{kprimeprime} Z' = (f_1(Z_1), \cdots,
f_m(Z_1), f_1(Z_2), \cdots, f_m(Z_2), \cdots).$$ We then use $Z'$ instead of $Z$ to choose the basis for each data bit.

The above modification results in the $i$-th $m$-block of ciphertext $Y_{(i-1)m} \cdots Y_{im}$ being dependent only on $K_{(i-1)m} \cdots K_{im}$ and $X_{(i-1)m} \cdots X_{im}$ for *both* ENC and $\alpha\eta$ with ENC. Under a KPA on ENC alone, using a known plaintext of length $n_1$, $K_1\ldots
K_{n_1}$ is known *exactly*. For ENC augmented with $\alpha\eta$ in the described manner, it *may* happen that because of the randomization of $Z'_1 \cdots Z'_{n_1}$, $K_1\ldots
K_{n_1}$ is not fixed by $\mathbf{Y}_{n_1}$ and $\mathbf{X}_{n_1}$. In the latter case, we have IT security above that of ENC alone, even though such security may be lost for large enough $n$, as mentioned in the previous subsection.

Let us assume that, at the nondegeneracy distance $n_1$ of ENC, $\alpha\eta$ with ENC does *not* have IT security, so that $H(K|\mathbf{X}_{n_1}\mathbf{Y}_{n_1})=0$. Assume also that $n_1=mk$. Even in such a case, it appears harder to implement the algorithm ALG that finds the key. As discussed in Section 2.2, the reason is that the randomization of the ciphertext $Y_i$, for each $i$, leaves each $Z_i$ undetermined immediately after the measurement, even though, by our present assumption, only one possible seed key $K$ can lead to the observed measurement results. If the number of possibilities for each $Z_i$ is $l$, Eve may need to run the algorithm ALG $l^{k}$ times resulting in a complexity of $l^{n_1/m}C$ versus $C$ for ENC alone. Of course, there may exist a clever algorithm that enables her to do much better. All we claim here is that $\alpha\eta$ provides an additional but unquantified layer of security over that of the ENC box against KPA, both in the IT and CT senses. Thus, $\alpha\eta$ can be run on top of any standard cipher in use at present, e.g. AES (Advanced Encryption Standard), and provides an additional, qualitatively different layer of physical encryption security over AES under a known-plaintext attack.

An interesting point is that, if the above level of CT security against known-plaintext attack is sufficiently high for some data length $n$, there is at least as much security against CTA for the same $n$. However, this comparison may not be practically meaningful as a CTA can typically be launched for the entire sequence of data while usually only a much smaller segment of known-plaintext is available to the attacker. Typically, this would imply the attacks can be parallelized. On the other hand, the situation is practically favorable with AES used in the ENC - see ref. [@yuen06pla], where the immunity of $\alpha\eta$ against fast correlation attacks with and without DSR are also treated.

Overview of $\alpha\eta$ Features
---------------------------------

We summarize the main known advantages and rigorous security claims regarding $\alpha\eta$ compared to previous ciphers:

(1) For known-plaintext attacks on the key, $\alpha\eta$ using an LFSR has an additional brute force search complexity given by $\Gamma^{|K|/m}$. When reconfigured as in Sec. 4.3.3, it also has at least as much IT security as the ENC box alone for the same length $n$ of data.

(2) It may, when supplemented with further techniques [@yuen03], have information-theoretic security against known-plaintext attacks that is not possible with nonrandom ciphers, and would also have maximal information-theoretic security against ciphertext-only attacks.

(3) With added Deliberate Signal Randomization (DSR) [@yuen03], it is expected to have improved information-theoretic security on the data far exceeding the Shannon limit.

(4) It has high-speed private true randomization (from quantum noise that even Alice does not know), which is not possible otherwise with current or foreseeable technology.

(5) It suffers no reduction in data rate compared to other known random ciphers, because Bob needs to resolve only two and not $M$ possibilities (i.e, one data bit is transmitted per qumode).

(6) It provides physical encryption, different from usual mathematical encryption, that forces the attacker to attack the optical line rather than simply the electronic bit output.

Nishioka et al’s criticisms of $\alpha\eta$
===========================================

In this section, we discuss the criticisms made by Nishioka et al [@nishioka04; @nishioka05] and respond to them. This section has some overlap with [@nair05] (that was not published), but contains new material.

Claims in Nishioka *et al* [@nishioka05]
----------------------------------------

Nishioka *et al* claim that $\alpha\eta$ can be reduced to a classical non-random stream cipher under the attack that we now review. For each transmission $i$, Eve makes a heterodyne measurement on the state and collapses the outcomes to one of $2M$ possible values. Thus, the outcome $j \in \{0, \cdots, 2M-1\}$ is obtained if the heterodyne result falls in the wedge for which the phase $\theta \in [\theta_j-\pi/2M, \theta_j+\pi/2M]$, where $\theta_j= \pi j/M$. Further, for $q \in \{0, \cdots, M-1\}$ representing the $M$ possible values of each $Z_i$, Nishioka *et al* construct a function $F_j(q)$ with the property that, for each $i$, and the corresponding running key value $Z_i$ actually used, $$\label{nishiokadecryption}
F_{j^{(i)}}(Z_i)=r_i$$ with probability very close to 1. In fact, for the parameters $S=100$ and $M=200$, they calculate the probability that Eq.(\[decryption\]) fails to hold to be $10^{-44}$, which value they demonstrate to be negligible for any practical purpose.

The authors of [@nishioka05] further claim that the above function $F_{j^{(i)}}(q)$ can always be represented as the XOR of two bit functions $G_{j^{(i)}}(q)$ and $l_{j^{(i)}}$, where $l_{j^{(i)}}$ depends *only* on the measurement result. Thus, they make the claim that the equation $$\label{reduction}
l_{j^{(i)}}=r_i \oplus G_{j^{(i)}}(Z_i)$$ holds with probability effectively equal to 1. They then observe that a classical additive stream cipher [@stinson] (which is non-random by definition) satisfies $$\label{streamcipher}
l_i=r_i \oplus \tilde{k_i},$$ where $r_i$, $l_i$, and $\tilde{k_i}$ are respectively the $i$th plaintext bit, ciphertext bit and running key bit. Here, $\tilde{k_i}$ is obtained by using a seed key in a pseudo-random-number generator to generate a longer running key. The authors of [@nishioka05] then argue that since $l_{j^{(i)}}$ in Eq.(\[reduction\]), like the $l_i$ in Eq.(\[streamcipher\]), depends just on the measurement result, the validity of Eq.(\[reduction\]) proves that the security of Y-00 is equivalent to that of a classical stream cipher. In particular, they claim that by interpreting $l_{j^{(i)}}$ as the ciphertext, Y-00 is not a random cipher, i.e., it does not satisfy Eq.(\[random\]) of the next section.

We analyze and respond to these claims and other statements in [@nishioka05] in the following section.

Reply to claims in [@nishioka05]
--------------------------------

To begin with, we believe that Eq. (\[decryption\]) (Eq. (14) in [@nishioka05]) is correct with the probability given by them. This content of this equation is simply that Eve is able to decrypt the transmitted bit from her measurement data $J_N$ and the key $\mathbf{K}_s$. In other words, it merely asserts that Eq.(\[decryption\]) holds for $\mathbf{Y}_N=J_N$. As such, it does not contradict, and is even *necessary*, for the claim that $\alpha\eta$ is a random cipher for Eve. In fact, we already claimed in [@yuen03] and [@yuen05pla] that such a condition holds. In this regard, note also that the statement in Section 4.1 of [@nishioka05] that “informational secure key generation is impossible when ( Eq.(\[decryption\]) of this paper) holds” is irrelevant, since direct encryption rather than key generation is being considered here. Furthermore, we have already pointed out [@yuen05qph; @yuen03; @yuen05pla] that the Shannon limit prevents key generation with the experimental parameters used so far, a point missed in [@nishioka04; @nishioka05; @loko05]. See also [@note1].

We also agree with the claim of Nishioka *et al* that it is possible to find functions $l_{j^{(i)}}$ and $G_{j^{(i)}}(q)$, the former depending only of the measurement result $j^{(i)}$, such that Eq.(\[reduction\]) holds, again with probability effectively equal to one. The *error* in [@nishioka05] is to use this equation to claim, in analogy with Eq. (\[streamcipher\]), that $\alpha\eta$ is reducible to a classical nonrandom stream cipher.

To understand the error in their argument, note that, for Eq. (\[streamcipher\]) to represent an additive stream cipher, the $l_i$ in that equation should be a function *only* of the measurement result, and $\tilde{k_i}$ should be a function *only* of the running key. While the former requirement is true also for the $l_{j^{(i)}}$ in Eq. (\[reduction\]), the latter is certainly *false* for the function $G_{j^{(i)}}(Z_i)$ in Eq. (\[reduction\]), since it depends *both* on the measurement result $j^{(i)}$ and the running key $Z_i$. Indeed, it can be seen that the definition of the function $F_{j^{(i)}}(Z_i)$, and thus, $G_{j^{(i)}}(q)$ depends on the sets $C_{j^{(i)}}^+$ and $C_{j^{(i)}}^-$ defined in Eq. (12) of [@nishioka05]. The identity of these sets in turn depends on the relative angle between the basis $q$ and Eve’s estimated basis $\tilde{j^{(i)}}= j^{(i)} \bmod M.$ Thus, it is clearly the case that $G_{j^{(i)}}(Z_i)$ must depend both on $j^{(i)}$ and $Z_i$, a fact also revealed by the inclusion of the subscript $j^{(i)}$ by the authors of [@nishioka05] in the notation for $G$.

Notwithstanding the failure of Eq. (\[reduction\]) to conform to the requirements of a stream cipher representation Eq. (\[streamcipher\]), Nishioka *et al* reiterate that Y-00 is nonrandom because $$\label{l}
H(L_N|R_N, \mathbf{K}_s) =0$$ holds, where $\mathbf{L}_N=(l_{j^{(1)}}, \ldots, l_{j^{(N)}})$. This equation follows from Eq. (\[reduction\]) and so by considering $\mathbf{L}_N\equiv \mathbf{Y}_N$ to be the ciphertext, the Eq.(\[random\]) is not satisfied, thus supposedly making Y-00 nonrandom. The choice of $\mathbf{L}_N$ as the ciphertext is supported by the statement in [@nishioka05] that “It is a matter of preference what we should refer to as “ciphertext”.” This is indeed true, especially considering that there are different possible quantum measurements that may be made on the quantum state in Eve’s possession, each giving rise to a different ciphertext. This point is also highlighted by our definition of a qauntum random cipher. However, if one wants to claim equivalence to a non-random cipher for some particular choice of ciphertext $\mathbf{Y}_N$, one must show that Eq. (10) is violated *and* that Eq. (11) is satisfied using the chosen ciphertext in *both* equations. In other words, no equivalence to any kind of cipher is shown unless one can also decrypt *with the chosen ciphertext* and key alone. However, one may readily see that, taking $\mathbf{Y}_N=\mathbf{L}_N$, Eq. (\[decrypt\]) is not satisfied, i.e., $H(\mathbf{R}_N|\mathbf{L}_N,\mathbf{K}_s)\neq 0$. The reason is that, as we noted from our analysis above of the function $G_{j^{(i)}}(q)$, decrypting $r_i$ requires knowledge of certain ranges in which the angle between the basis chosen by the running key and the estimated basis $\tilde{j^{(i)}}$ falls. To convey this information *for every possible* $j^{(i)}$, one needs at least $\log_2(2M)$ bits. It follows that the single bit $l_{j^{(i)}}$ is insufficient for the purpose of decryption, and so Eq. (\[decrypt\]) cannot be satisfied for $\mathbf{Y}_N=\mathbf{L}_N$. Therefore, we conclude, that in the interpretation of $\mathbf{L}_N$ as the ciphertext, decryption is not possible even if Eve has the key $\mathbf{K}_s$. Indeed, it is $\mathbf{J}_N$ that can be regarded as a possible ciphertext, since Eq. (\[decrypt\]) is satisfied for $\mathbf{Y}_N=\mathbf{J}_N$. However, with this choice of ciphertext, Y-00 necessarily becomes a *random* cipher, because $H(\mathbf{J}_N|\mathbf{R}_N,\mathbf{K}_s) \neq 0$, a fact admitted by Nishioka *et al* in [@nishioka05].

We hope that the discussion above makes it clear that the ‘reduction’ of $\alpha\eta$ in [@nishioka05] to a non-random cipher is false, and that in fact, no such reduction can be made under the heterodyne attack considered in [@nishioka05]. Indeed, as detailed in previous sections, the representation of ciphertext by $\mathbf{Y}_N=\mathbf{J}_N$ does reduce it to a *random* cipher under the heterodyne attack. Its quantitative random cipher characteristics, namely $\Gamma$ of Eq. (\[Gammarandomcipherdef\]) and $\Lambda$ of Eq. (\[lambdarandomcipherdef\]), are as follows, for various definitions of “ciphertext” adopted.

If the full continuous observation on the circle is taken as the ciphertext, then (\[Gammahet\]) shows that $\Gamma \sim 3$ for typical experimental parameters. If the ciphertext alphabet is digitized and taken to be the $2M$ arc segments around the $2M$ states on the circle, then $\alpha\eta$ has, for any $(\mathbf{x}_i,z_i,r)$, $\Lambda+1 =2(\Gamma+1)$ where $\Gamma$ is given by (\[Gammahet\]). If one attempts to ‘de-randomize’ the ciphertext by clubbing together the possibilities, $\Gamma$ would increase while $\Lambda$ would decrease. In the nonrandom limit where a fixed half-circle observation is taken to represent each bit value, which is the nonrandom reduction discussed in [@yuen05pla], $\Gamma$ would increase from that of Eq. (\[Gammahet\]) to $M$, making attacks on the key completely impossible. On the other hand, while $\Lambda =0$ for a binary ciphertext alphabet, the $2M$-outcome ciphertext would lead, from Eq. (\[Gammahet\]), to an error probability per ciphertext bit for Eve [@yuen05pla]: $$\label{wedgeerror}
P_b^E \sim 2/{\pi \sqrt{S}}.$$ Eq. (\[wedgeerror\]) is obtained in the wedge approximation on a per qumode basis for Eve, under the assumption that the state is uniformly distributed on the circle which is satisfied for uniform data and an LFSR for the ENC box of Fig. 3. It leads to $0.1-1\%$ error rate for Eve on the ciphertext (not data [@note2]) for the experimental parameters of [@barbosa03; @pra05]. As a consequence, the data security will far exceed the Shannon limit (\[shannonlimit\]) because she would make many errors even when the correct key is given to her for decryption. For any other ciphertext alphabet division of the circle, it is clear that $\Lambda >0$ for any $z_i$ and $\mathbf{x}_n$ from the same randomization for states near the ciphertext alphabet boundaries on the circle.

In sum, there can be no nonrandom reduction of $\alpha\eta$. If the ciphertext alphabet is chosen to make $\alpha\eta$ nonrandom, then known-plaintext attack on the key is impossible and the ciphertext itself would be obtained with significant noise.

We conclude this section by responding to some other statements made in [@nishioka05].

In Section 3.3, Nishioka *et al* claim that “The value of $l_{j^{(i)}}$ does not have to be the same as that of $l_{j^{(i')}}$ when $i \neq i'$, even if $j^{(i)}=j^{(i')}$ holds.” This statement is in direct contradiction to their previous statement in the same subsection that “$l_{j^{(i)}}$ depends only on the measurement value $j^{(i)}$”.

In the same subsection, Nishioka *et al* claim that “In ([@nishioka04]), we showed another concrete construction of $l_{j^{(i)}}$ ...”. We could find no explicit construction of $l_{j^{(i)}}$ in that paper. We were led to the choice of $l_i$ described in [@yuen05pla] by the attempt to make the stream cipher representation Eq. (\[streamcipher\]) valid. In fact, such a representation is claimed by Nishioka *et al* in their Case 2 of [@nishioka04]. It turned out, however, that decryption using that $l_i$ suffered a $0.1-1$% error depending on the value of $S$ used as noted above. See [@yuen05pla] for further details. While it was later claimed that they have a different reduction in mind [@nishioka05], the reduction in [@yuen05pla] is the only one that makes $\alpha\eta$ nonrandom (but in noise). In any case, as we have shown above, no construction of a single-bit from the heterodyne or phase measurement results can satisfy Eq.(\[decryption\]) with the extremely low probability given in [@nishioka05].

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

We would like to thank Greg Kanter, Chuang Liang, and Koichi Yamazaki for useful discussions. This work was supported by DARPA under grant F30602-01-2-0528 and by AFOSR under grant FA9550-06-1-0452.

Appendix A – Security under Statistical and Known-Plaintext Attacks {#appendix-a-security-under-statistical-and-known-plaintext-attacks .unnumbered}
===================================================================

In this appendix, we summarize some relevant terminology and results from ref. [@yuen05qph] on the key security of a random cipher. We first present an overview of the various possible cryptographic attacks possible on a cipher and some early results on the subject. We also present our result on the security of a nonrandom cipher under known-plaintext attacks. In the process, we define the important term ‘unicity distance’ coined by Shannon and broaden it to include the notion of ‘unicity distance under known-plaintext attack’ for both random and nonrandom ciphers. We also define the important concept of ‘nondegeneracy’ for both random and nonrandom ciphers that is needed to make the concept of unicity distance meaningful. Finally, we discuss how random ciphers may enhance security against known-plaintext attacks.

The following terminology in regard to cryptographic attacks has bee used in this paper, as in [@yuen05qph]. This terminology is not standard, however. In the cryptography literature, what we call statistical attacks are sometimes referred to as ciphertext-only attacks (See, e.g., [@stinson], Ch. 2) but are also often lumped together with known-plaintext attacks.

By a *ciphertext-only attack (CTA)*, we refer to the case where the probability distribution $p(\mathbf{X}_n)$ is completely uniform, i.e., $p(\mathbf{X}_n) =2^{-n}$ to Eve, so that her attack cannot exploit input frequencies or correlations and must be based only on the ciphertext in her possession. By a *statistical attack (STA)*, we refer to the case where the probability distribution $p(\mathbf{X}_n)$ is nonuniform, so that Eve may in principle exploit input frequencies or correlations to launch a better attack. Such an attack is typical when the plaintext is in a language such as English. It is also the attack that obtains when the $\{X_i\}$ are independent and identically distributed (i.i.d.) but each $p(X_i)$ is nonuniform. By a *known-plaintext attack (KPA)* we mean the case where Eve knows *exactly* some length $m$ of plaintext $\mathbf{x}_m$. Finally, by a *chosen-plaintext attack (CPA)*, we mean a KPA where the data $\mathbf{x}_m$ is chosen by Eve.

In standard cryptography, one typically does not worry about ciphertext-only attack on nonrandom ciphers. The reason is that, under CTA, Eq. (\[shannonlimit\]) is satisfied with equality for large $n$ for the designed key length $ |K|=H(K)$ under a certain ‘nondegeneracy’ condition [@jkm] that is readily satisfied. Thus, in practice, the data security is assumed to be sufficient if $H(K)$ is chosen large enough by adjusting the key length. In this paper, we would essentially make the same assumption and, with few exceptions, do not discuss data security per se. However, it follows from (\[shannonlimit\]) that no meaningful lower bound on $H(\mathbf{X}_n|\mathbf{Y}_n)$ exists for $n \gg |K|$. A new fundamental treatment of data security in symmetric-key ciphers has to be developed separately. Under CTA, it is also the case for nonrandom nondegenerate ciphers that [@jkm] $$\label{keysecurity}
 H(K|\mathbf{Y}_n)=H(K),$$ i.e., the key is *statistically independent* of the ciphertext. Thus, no attack better than pure guessing can be launched on the key.

The above two results do not hold for statistical and known-plaintext attacks. Eve can indeed launch an attack on the key and use her resulting information on the key to get at future and past data. In fact, it is such attacks that are the focus of concern for standard ciphers such as the Advanced Encryption Standard (AES). For STAs, Shannon [@shannon49] characterized the security by the so-called unicity distance. The *unicity distance* $n_0$ of a cipher is the smallest input data length for which $H(K|\mathbf{Y}_{n_{0}}) = 0$. In other words, if a plaintext sequence of length $n_0$ is encrypted by the cipher, the ciphertext contains enough information to fix the key (and hence, the plaintext) uniquely – the cipher has no information-theoretic security. For nonrandom ciphers defined by Eq. (\[nonrandom\]), Shannon, in [@shannon49], derived in terms of the data entropy an estimate on $n_0$ that is independent of the cipher. This estimate is actually *not* a rigorous bound. Indeed, it can be shown that one of the inequalities used in the derivation goes in the wrong direction. Even so, the estimate works well empirically for English language plaintexts, for which $n_0 \sim
25$ characters are found to be sufficient to break many ciphers.

We now consider, in some detail, security against known-plaintext attacks. Here, a natural quantity to consider is $H(K|\mathbf{X}_n\mathbf{Y}_n)$, since it provides a measure of key uncertainty when both plaintext and ciphertext are known to the attacker. Before we state the main result, we define the notion of nondegeneracy distance. The reader can readily convince himself that a finite unicity distance exists only if, for some $n$, there is no *redundant key use* in the cryptosystem, i.e., no plaintext sequence $\mathbf{x}_n$ is mapped to the same ciphertext $\mathbf{y}_n$ by more than one key value. With redundant key use, one cannot pin down the key but it seems that this may not enhance the system security either, and so is merely wasteful. The exact possibilities will be analyzed elsewhere. For now, we call a cipher *nondegenerate* in this paper if it has no redundant key use for some finite $n$ or for $n \rightarrow \infty$. Under the condition $$\label{nondegenerate}
\lim_{n \rightarrow \infty} H(\mathbf{Y}_n|\mathbf{X}_n) =
H(K),$$ which is similar but not identical to the definition of a ‘nondegenerate’ cipher given in [@jkm], one may show that, when Eq. (\[nonrandom\]) also holds, one has $$\label{broken} \lim_{n \rightarrow \infty}
H(K|\mathbf{X}_n,\mathbf{Y}_n) = 0,$$ so that the system is asymptotically broken under a known-plaintext attack. More generally, for a nonrandom cipher, we define a *nondegeneracy distance* $n_d$ to be the smallest $n$ such that $$\label{nondegdist}
 H(\mathbf{Y}_{n}|\mathbf{X}_{n})=H(K)$$ holds, with $n_d =\infty$ if (\[nondegenerate\]) holds and there is no finite $n$ satisfying (\[nondegdist\]). Thus, a nonrandom cipher is nondegenerate in our sense if it has a nondegeneracy distance, finite or infinite. In general, of course, the cipher may be *degenerate*, i.e., it has no nondegeneracy distance. We can readily show (see Appendix A of [@yuen05qph]) that, under known-plaintext attack, a nonrandom nondegenerate cipher is broken at data length $n=n_d$, in the sense that $$\label{kpabroken}H(K|\mathbf{X}_{n_d}\mathbf{Y}_{n_d})=0.$$

More generally, for both random and nonrandom ciphers, we define the *unicity distance under known-plaintext attacks*, denoted by $n_1$, to be the smallest integer such that $$\label{unicitydistKPA}
H(K|\mathbf{X}_{n_1}\mathbf{Y}_{n_1}) = 0.$$ If no such integer exists, the unicity distance under KPA is taken to be infinite if $\lim_{n \rightarrow \infty}
H(K|\mathbf{X}_n\mathbf{Y}_n) = 0$. Thus, $n_1$ is the minimum length of data needed to break the cipher for *any* possible known-plaintext $\mathbf{X}_n$. For a nonrandom cipher, it is equal to the nondegeneracy distance.

Many ciphers including the one-time pad and LFSRs (linear feedback shift registers [@stinson]) have finite $n_d$. Similar to the case of $n_d$ for nonrandom ciphers, $n_1$ for a random cipher may not always exist. For our definition of $n_1$ to make sense for random ciphers, we will impose a ‘nondegeneracy’ restriction on random ciphers: A *random cipher* is said to be *nondegenerate* if and only if *each* nonrandom cipher resulting from an assignment $\mathbf{R}=\mathbf{r}$ of the randomizer is nondegenerate. Then we say it has *information-theoretic security against known-plaintext attacks* if $$\label{ITsecurityKPA}
\inf_n H(K|\mathbf{X}_n,\mathbf{Y}_n) \neq 0,$$ i.e., if $H(K|\mathbf{X}_n,\mathbf{Y}_n)$ cannot be made arbitrarily small whatever $n$ is. In other words, $n_1$ does not exist. The actual level of the information-theoretic security is quantified by the left side of (\[ITsecurityKPA\]). One major motivation to study random ciphers is the *possibility* that they possess such information-theoretic security. Some discussion on this point is also available in Appendix A of [@yuen05qph].

Even in the absence of information-theoretic security, nondegenerate random ciphers can be expected (see the discussion in Section 2.2) to have larger unicity distance $n_1$ under KPA compared to the case where the randomization is turned off. This would, as assumed in cryptography practice, increase the complexity of attacking the key significantly. If Eq. (\[kpabroken\]) holds when $\mathbf{X}_{n}$ is replaced by a specific $\mathbf{x}_{n}$, $n$ defines the unicity distance corresponding to $\mathbf{x}_{n}$. The overall unicity distance under KPA may be defined by $$\label{overallunicitydistance}
\bar{n}_1= \min_{H(K|\bf{X}_n = \bf{x}_n, Y_{n})=0} n \textup{ for
some } \bf{x}_n.$$

The above result has not been given in the literature, perhaps because $H(K|\mathbf{X}_n\mathbf{Y}_n)$ has not been used previously to characterize known-plaintext attacks. Nevertheless, it is assumed to be true in cryptography practice that $K$ would be pinned down for sufficiently long $n$ in a nonrandom ‘nondegenerate’ cipher.

We now discuss the advantages that a random cipher provides as compared to nonrandom ciphers. For the case of STA on the key when the plaintext $\mathbf{X}_n$ has nonuniform but i.i.d. statistics, the so-called *homophonic substitution* method provides complete information-theoretic security, i.e. $H(K|\mathbf{Y}_n)=H(K)$ [@jkm]. The original form of homophonic substitution involves assigning to each plaintext symbol a number of possible *sequences* of length $l$ proportional to its a priori probability in such a way that all possible $l$-sequences are covered. Then, for every input symbol, if one of its assigned $l$-sequences is generated at random, the net effect is to generate $l$-sequences of plaintext with i.i.d. uniform statistics. These sequences may be passed through a non-degenerate cipher without revealing information on the key as per Eq. (\[keysecurity\]). To put it another way, a statistical attack has been converted to a ciphertext-only attack. A generalized homophonic substitution that allows each symbol to be coded into sequences of variable length is discussed in [@jkm], for which it is shown that sometimes data compression instead of data expansion results.

Unfortunately, this reduction of a STA to a CTA does not work for known-plaintext attacks. However, we emphasize that there is *no result* on random ciphers analogous to Eq. (\[kpabroken\] ) with $n_d$ replaced by any definite $n$ depending on the cipher, since under randomization, Eq. (\[nonrandom\]), and usually (\[nondegdist\]) also, does not hold for any $n$. Indeed, an inspection of the defining equation Eq. (\[Gammarandomcipherdef\]) for a random cipher (or Fig. 1) suggests how a random cipher may provide greater security against KPAs. For a given plaintext-ciphertext sequence pair, Eq.(\[Gammarandomcipherdef\]) suggests that one has some residual uncertainty on the value of the keystream $(Z_1,\ldots,
Z_n)$, which does not exist for a corresponding nonrandom cipher. On the other hand, Eq.(\[Gammarandomcipherdef\]) refers only to the per-symbol uncertainty of the key stream calculated without regard to the ciphertext observed for the other symbols in the sequence. When such correlations are taken into account, the uncertainty on the keystream may be drastically reduced and we can give no general quantitative assertions of information-theoretic security. Note, however, that due to the randomization, the unicity distance $n_1$ of a random cipher under known-plaintext attacks can be expected to be bigger than that of any of its nonrandom reductions. Thus, the complexity-based security would be greater.

In fact, the general problem of attacking a random cipher has received limited attention because they are *not used in practice* due to the associated reduction in effective bandwidth or data rate as is evident in homophonic substitution, due to the need for high speed random number generation, and also due to the uncertainty on the actual input statistics needed for, e.g., homophonic substitution randomization. Thus, the rigorous quantitative security of symmetric-key random ciphers against known-plaintext attacks is not known theoretically or empirically, although in principle random ciphers have actual and potential advantages just discussed.

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The point that the throughput for $\alpha\eta$ key generation using the optimal quantum receiver principle is low, which we already remarked in [@yuen03; @yuen05pla], was also made in [@nishioka04; @nishioka05]. We would like to point out here that the throughput can be increased to the level of the original data rate by DSR. This involves moving the state to the basis boundary with Bob utilizing a different matching quantum measurement. For the case of key generation with uniform data, the output state is still uniformly distributed on the circle on a per-qumode basis to Eve. However, Bob’s optimal performance has not been characterized with DSR. A detailed discussion of $\alpha\eta$ key generation will be presented later. We may observe that the use of DSR in direct encrypion would necessitate the use of an error-correcting mechanism in contrast with the original $\alpha\eta$, and would break the Shannon limit (12). The significance and practicality of such variations of $\alpha\eta$ have been briefly mentioned in [@yuen03], and will be treated elsewhere.

Note that for the purpose of bounding Eve’s information in key generation by granting her the key after her measurement, she would make at least $0.1-1\%$ errors as asserted in [@yuen05pla].

[^1]: Email: nair@eecs.northwestern.edu

[^2]: When referring to an optical field mode, we use the term *qumode* (for ’quantum mode’, in analogy to ’qubit’).
