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Quantum Unicity Distance [8440-32] Quantum Unicity Distance Chong Xiang and Li Yang* State Key Laboratory of Information Security, Institute of Information Engineering, CAS, Beijing 100195, China Graduate University of the Chinese Academy of Sciences, Beijing 100049, China ABSTRACT We attempt to develop Shannon’s concept of the unicity distance into the quantum context. Based on the defini- tion of information-theoretic security for quantum cryptography, we present a quantum probabilistic encryption algorithm with bounded information-theoretic security, and then work out its quantum unicity distances. The result shows that quantum unicity distance is much bigger than the unicity distance of classical cryptography. Keywords: unicity distance, quantum unicity distance, quantum private key encryption, quantum probabilistic encryption. 1. INTRODUCTION Shannon provided the concept of unicity distance in 19491 . He showed a way of calculating approximately how many ciphertexts were necessary to recovery the unique key for the classical encryption protocols, the length of ciphertexts needed was determined by the entropy of the key space and the redundance of the plaintext space. This parameter can be used to measure the number of times one key can be reused securely, while the length of ciphertexts encrypted by same key is not more than the unicity distance, this key can not be confirmed under the Known Ciphertext Attack(KCA). According to Deavours, the unicity distance of DES is worked out as 8.2 ASCII characters or 66 bits2, 3 . Mao and Wu4 used the unicity distance to analysis the security of hashing algorithms, Liu, Zhang and Li5 then develop it into multimedia hash scheme. The quantum key distribution has been proved to be unconditionally secure and applied in practice.6–10 Though it is seem that quantum one-time pad(which usually called private quantum channel11–13) can be achieved theoretically with the quantum key distribution, we still need to consider the repetition of key for the well- known low efficiency of the quantum key distribution. The quantum unicity distance represents the theoretical limitation of how many times a key can be reused in quantum context. The quantum unicity distance of a quantum cryptography protocol can be used to measure its efficiency and the security in practice. In the latter part of this paper we provide a definition of information-theoretic security for quantum cryp- tography based on the information-theoretic security of classical cryptography suggested by O. Goldrich. We present a quantum probabilistic encryption with bounded information-theoretic security, and then calculate its quantum unicity distance. This quantum unicity distance is proved to be much bigger than that of a classical encryption protocol introduced by Shannon. 2. PRELIMINARIES 2.1 Unicity distance Although Shannon has shown an expression of the unicity distance1 , people always use the following expression in order to express the unicity distance under multi-system (P, C, K, ε, D): H(K) n0 = . RL log2 |P | *Corresponding author. Email: [email protected] Quantum Optics II, edited by Thomas Durt, Victor N. Zadkov, Proc. of SPIE Vol. 8440, 84400T · © 2012 SPIE · CCC code: 0277-786X/12/$18 · doi: 10.1117/12.922555 Proc. of SPIE Vol. 8440 84400T-1 The unicity distance has great significance for one cryptography protocol as the amount of unicity distance reflects the security of the protocol in one aspect. It must be noticed that unicity distance is a average value, so people always need more ciphertexts to get the unique key. It can be seen that if the ciphertext that eavesdropper E owned is longer than the unicity distance, it can be decrypted into a meaningful plaintext only with the unique key, and be decrypted into meaningless message with any other bit-strings. We can get the following figure: Figure 1. While the length of the plaintext grows, the ciphertext sets encrypted from the same set of meaningful plaintexts with different keys must be almost mutually disjoint sets. Classical cryptography has a property that its decryption process is certain, which means every time we decrypt the same ciphertext with the same key will result the same plaintext; however, the quantum cryptography has a certain decryption result with the unique key but a uncertain decryption result with other bit-strings. It means that a quantum cryptography may not have unicity distance even if its ciphertext sets encrypted from the same set of meaningful plaintext with different keys are almost mutually disjoint sets. 2.2 Information-theoretic security In classical cryptography, the information-theoretic security is suggested by O. Goldreich14 as follows: Definition 2.1. An encryption is information-theoretically secure if for every circuit family {Cn}, every positive polynomial p(·), all sufficiently large n’s, and every x, y in plaintext space: n n 1 Pr[Cn(G(1 ),EG(1n)(x)) = 1] − Pr[Cn(G(1 ),EG(1n)(y)) = 1] < , (1) p(n) where G is a key generation algorithm. 3. DEFINITIONS 3.1 Structure of quantum encryption transformation Quantum encryption protocols can be divided into four different kinds: 1. both the plaintext and the key are classical bits; 2. the key is quantum state, and the plaintext is classical bit; 3. the plaintext is quantum state, and the key is classical bit; 4. both the plaintext and the key are quantum states. Meanwhile the ciphertext is always quantum state. Different kinds of protocols may lead to different definition of quantum unicity distance, here we just discuss the first type of them as the Figure 2. Proc. of SPIE Vol. 8440 84400T-2 Classical plaintext bit-string Quantum Orthogonal state qubit-string Coding with orthogonal qubits ciphertext Qubit-string Encrypt with quantum operation Figure 2. The plaintexts are classical bits. Firstly we encodes them with orthogonal quantum states, then we encrypt the quantum states with quantum operation which is determined by the classical key, at last the ciphertexts are also quantum states. 3.2 Quantum unicity distance There are many differences between the unicity distance for the classical cryptography and the quantum cryp- tography. Firstly we cannot measure one quantum state repeatedly, so while one classical ciphertext can be used to verify any bit-string to see if it is the key, one quantum ciphertext can only be used to verify one bit-string, the quantum unicity distance shall be much bigger than the classical one; secondly any bit-string may have nonzero probability to decrypt ciphertexts into meaningful plaintexts under quantum mechanical. If we define the quantum spurious key referring to Shannon’s unicity distance, we can get this result: the spurious key of a quantum ciphertext Y may be defined as the bit-strings with which decrypting Y can get a meaningful plaintext. But under quantum mechanical, if we decrypt Y with any bit-strings except the key, the result may also be a superposition state which has some meaningful components and others meaningless. So we need to give a new definition for quantum spurious. We define the spurious key as follow: Definition 3.1. The quantum spurious key of a ciphertext Y is the bit-string except the unique key with which decrypting Y may result a meaningful plaintext with probability more than 1 − δ. Let’s define set Z as the set of meaningful plaintexts, spurious key of Y can be represented as follow: K(Y )={k ∈ K|P (Ek(Y ) ∈ Z) > 1 − δ}. For a quantum encryption protocol, if there exists a N satisfying that while the length of ciphertext is bigger than N, the number of quantum spurious keys is 0, we will call N the quantum unicity distance of the quantum cryptography. Notice that a classical ciphertext can be used any times to verify any bit-string to see that if it is the unique key, but a quantum ciphertext can only be used once, so quantum unicity distance should contain two components: one is the length of ciphertext with which we can verify if a bit-string is a wrong key and the other is the size of the key space. This definition is suited for calculating the quantum unicity distance of a specific quantum cryptography, but it is difficult to calculate the quantum unicity distance of abstract model of quantum cryptography. Proc. of SPIE Vol. 8440 84400T-3 3.3 Information-theoretic security of quantum cryptography15 We suggest here a definition of information-theoretic security of quantum encryption as follows: Definition 3.2. A quantum encryption is information-theoretically secure if for every quantum circuit family {Cn}, every positive polynomial p(·),allsufficientlylargen, and every x, y in plaintext space: n n 1 Pr[Cn(G(1 ),EG(1n)(x)) = 1] − Pr[Cn(G(1 ),EG(1n)(y)) = 1] < , (2) p(n) where the encryption algorithm E is a quantum algorithm, and the ciphertext E(x),E(y) are quantum states. Then we give the sufficient condition of the information-theoretic security which has been proved in paper.15 Theorem 3.3. For every plaintexts x and y, let the density operators of cipher states G(1n)(x) and G(1n)(y) are ρx and ρy, respectively. A quantum encryption protocol is said to be information-theoretically secure if for every positive polynomial p(·) and every sufficiently large n, 1 D(ρx,ρy) < . (3) p(n) 4. QUANTUM PROBABILISTIC ENCRYPTION PROTOCOL 4.1 Probabilistic encryption Probabilistic encryption16 is also called randomized encryption. A process of probabilistic encryption E can be shown as follow: E : Vn × K × R → Vm, where Vn and Vm are plaintext space and ciphertext space with m>n, K is the key space and R is a set of random numbers. The probabilistic cryptography maps one plaintext into different ciphertexts with the same key, so it can resist the Chosen Plaintext Attack better, and increase the effective size of the plaintext space.
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