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Key Generation: Foundations and a New Quantum Approach

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Key Generation: Foundations and a New Quantum Approach JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS 1 Key Generation: Foundations and a New Quantum Approach Horace P. Yuen (Invited Paper) Abstract—The fundamental security and efficiency consider- noise randomized direct encryption protocol AlphaEta (αη) or ations for fresh key generation will be described. It is shown Y-00 [8], [9], [10], which is KCQ direct encryption, although that the attacker’s optimal probability of finding the generated their implementations are closely connected. key is an indispensable measure of security and that this probability limits the possibility of privacy amplification and the A fresh key is, by definition, statistically independent of amount of fresh key that can be generated. A new approach to other system information the attacker may possess – it has quantum cryptography to be called KCQ, keyed communication information theoretic security according to conventional termi- in quantum noise, is developed on the basis of quantum detection nology [11]. However, except in the limits of none or all infor- and communication theory for classical information transmission. mation on a bit sequence, it has never been made clear what KCQ key generation schemes with coherent states of considerable energy will be described. The possibility of fresh key generation operational or empirical meaning and significance the usual is demonstrated for binary and N-ary detection systems under quantitative measures of information theoretic security have heterodyne attacks. The security issues of these schemes will be in the context of cryptography. This is clearly an extremely discussed and compared with BB84. The emphasis throughout is serious foundational issue and it occurs in all key generation on concrete finite bit-length protocols. protocols, classical or quantum. The problem is compounded Index Terms—Quantum cryptography, Key Generation, by the use of a shared secret key during the process of key Information-theoretic Security generation which is necessary both in KCQ and in BB84 type QKD protocols where a “public authentic channel” has to be created. This paper will exhibit the inadequacies of the usual I. INTRODUCTION entropy or any single-number measure for appropriate security HIS paper studies the possible generation of a fresh guarantee. The security issues will be elaborated for realistic T key between two users via the process of advantage finite protocols. creation, which is derived from the different ciphertexts or The case of KCQ qubit key generation will first be presented signal observations by the user and an attacker. The quantum due to its close similarity to BB84, which illustrates the issues key distribution (QKD) protocol of BB84 and its variants [1], involved in a familiar context. The foundational issues of [2], [3] are the most well-known examples, although classical key generation in general will be discussed, especially those scenarios of key generation were available before [4], [5]. on proper measure of security as well as the meaning and There are various problems in utilizing BB84 type protocols in possibility of fresh key generation via a shared secret key. The concrete realistic applications, most of which can be traced to principles of KCQ key generation follow. The coherent-state the small microscopic signals involved and the need to carry αη key generation scheme is presented next as a different way out estimation of the intrusion level for such protocols. of utilizing the αη direct encryption scheme. A generalized This paper proposes a new approach to QKD via the scheme called CPPM will then be described and shown to have optimal quantum receiver principle for advantage creation: many desirable characteristics of a key generation protocol arXiv:0906.5241v1 [quant-ph] 29 Jun 2009 the structure of a quantum receiver that delivers the optimal under the universal heterodyne attack. Effects of loss and other performance depends on knowledge of the signal set [6], [7]. aspects of security will be discussed among a comparison with We call this new approach [8] KCQ (keyed communication BB84. Some concluding remarks will be given. The situation in quantum noise) key generation due to the explicit use of of theoretical security in concrete BB84 type protocols are a secret key in the generation process. This KCQ approach discussed in the appendices. We focus throughout on issues does not exist in a classical world in which a single universal important in the operation of realistic cryptosystems of any observation is optimal for all signal sets. The crucial point bit length. Note that the new schemes presented are far from of KCQ in contrast to BB84 type QKD protocols is that being fully developed and merely constitute the basis of further intrusion level estimation may be omitted as a consequence of development of this new approach of KCQ key generation. the optimal quantum receiver principle, which makes possible This paper deals with subtle issues of many facets which are among other advantages the use of strong signals. It is hoped often neither purely mathematical nor intuitive physical, but that KCQ would facilitate the adoption of physical key gen- of a different conceptual kind [12], [13] that arises from the eration methods in practical optical systems. Note that KCQ nature of cryptology, especially when the crypto mechanism key generation is in principle totally distinct from the quantum involves physical principles in addition to purely mathematical ones. As in some theoretical papers in cryptology, this paper Horace P. Yuen is with the Department of Electrical Engineering and Com- puter Science and the Department of Physics and Astronomy, Northwestern tends to be wordy and demands careful reading. It is hoped University, Evanston, IL, 60208 USA e-mail: [email protected] that the careful formulation described in this paper would JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS 2 facilitate further discussions and developments of this subtle, complicated subject of physical key generation. ρ k 1 X x II. KCQ QUBIT KEY GENERATION Mod Alice data Kr Consider two users A (Alice) and B (Bob) and an attacker E (I) Ks (II) ENC (Eve) in a standard 4-state single-photon BB84 cryptosystem, 0 1 key in which each data bit is represented by one of two possible Channel Ks bases of a qubit, say the vertical and horizontal states 1 , ENC | i key 3 and the diagonal states 2 , 4 . In standard BB84, the Bob Kr | i | i | i 0 X choice of basis is revealed after Bob makes his measurements, Demod and the mismatched ones are discarded. It has been suggested [14] that some advantages obtain when a secret key is used Fig. 1. The qb-KCQ scheme. Left – Two bases, I and II. Right – Overall encryption involves modulation with bases determined by a running key Kr for basis determination with usual intrusion level estimation generated from a seed key K via an encryption mechanism denoted by the and the resulting protocol is also secure against joint attacks box ENC. [15]. Clearly, no key can be generated after subtracting the basis determination secret key if a fresh key is used for each qubit. It was proposed in refs. [14], [15] that a long m- bit secret key is to be used in a longer n-qubit sequence Next, we show that the seed key has complete information- with repetition. However, even if such use does not affect theoretic security against ciphertext-only attacks. We use the average information that Eve may obtain, it gives rise upper case for random variables and lower case for the k to such an unfavorable distribution that security is seriously specific values they take. Let ρx be the quantum state compromised. This is because with a probability 1/2, Eve corresponding to data sequence x = x xn and running 1 ··· can guess correctly the basis of a whole block of n/m qubits key k1 kn. For attacking the seed key K or running key ··· k k by selecting the qubits where the same secret bit is used Kr, the quantum ciphertext reduces to ρ = Px pxρx where 3 repetitively. For a numerical illustration, let n = 10 and px is the apriori probability of x. In our KCQ approach, m = 102. Then with a probability 2−15 0.3 10−4, Eve we grant a full copy of the quantum state to Eve for the can successfully launch an opaque (intercept/resend)∼ × attack purpose of bounding her information [17]. By an optimal that gives full information at the dangerous 15% level [2] on measurement on the qubits, Eve’s probability of correctly the total bit sequence while yielding no error to the users. In identifying K may be obtained via ρk. Since each qubit general, the strong correlation from such repetitive use would is modulated by its own corresponding data bit, we have seriously affect the appropriate quantitative security level, and ρk = ρk1 ρkn . For uniform data commonly assumed for x x1 ⊗···⊗ xn the effect of such guessing attacks on some portion of the key generation, the Xi are independent, identically distributed data bit-sequence has not been accounted for with or without (i.i.d.) Bernoulli random variables with equal probabilities. ki privacy amplification included. Thus each ρ = Ii/2 after averaging over xi for any value k n This problem is alleviated when a seed key K is first passed of ki, with resulting ρ = Ni=1 Ii/2 completely independent through a pseudo-random number generator (PRNG) to yield of k. So Eve can obtain no information on Kr or K at all a running key Kr that is used for basis determination, as even if she possesses a full copy of the quantum signal. We indicated in Fig. 1. In practice, any standard cipher running summarize: in the stream-cipher mode [16] can be used as a PRNG.
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