Precise quantum key distribution (QKD) secure bound analysis is essential for practical QKD systems. The effect of uniformity of random number seed for privacy amplification is not considered in existing secure bound analysis. In this paper, we propose and prove the quantum leftover hash lemma with non-uniform random number seeds based on the min-entropy, and we give a precise QKD secure bound analysis with non-uniform random number seeds on this basis. We take the two-decoy BB84 protocol as an example to simulate the effect of random number seed uniformity on the secure bound of a QKD system. The experimental results indicate that when the average min-entropy of the random number generator is below 0.95, the secure bound of a QKD system will be seriously affected.
State-of-art quantum key distribution (QKD) systems are performed with several GHz pulse rates, meanwhile privacy amplification (PA) with large scale inputs has to be performed to generate the final secure keys with quantified security. In this paper, we propose a fast Fourier transform (FFT) enhanced high-speed and large-scale (HiLS) PA scheme on commercial CPU platform without increasing dedicated computational devices. The long input weak secure key is divided into many blocks and the random seed for constructing Toeplitz matrix is shuffled to multiple sub-sequences respectively, then PA procedures are parallel implemented for all sub-key blocks with correlated sub-sequences, afterwards, the outcomes are merged as the final secure key. When the input scale is 128 Mb, our proposed HiLS PA scheme reaches 71.16 Mbps, 54.08 Mbps and 39.15 Mbps with the compression ratio equals to 0.125, 0.25 and 0.375 respectively, resulting achievable secure key generation rates close to the asymptotic limit. HiLS PA scheme can be applied to 10 GHz QKD systems with even larger input scales and the evaluated throughput is around 32.49 Mbps with the compression ratio equals to 0.125 and the input scale of 1 Gb, which is ten times larger than the previous works for QKD systems. Furthermore, with the limited computational resources, the achieved throughput of HiLS PA scheme is 0.44 Mbps with the compression ratio equals to 0.125, when the input scale equals up to 128 Gb. In theory, the PA of the randomness extraction in quantum random number generation (QRNG) is same as the PA procedure in QKD, and our work can also be efficiently performed in high-speed QRNG.
We study hashing against quantum adversaries, as used in cryptographic applications, and provide bounds on the probability of failure in practical settings where seeds might be reused and might be non-ideal. Randomness extraction and privacy amplification are key components of quantum cryptographic protocols, such as quantum randomness generation or key distribution, that allow the compression of partially secure raw data into shorter output with high security. To ensure the security of any cryptographic construction that integrates them, it is desirable to establish composable security. This can be done by using methods based on universal hashing and leveraging the general leftover hash lemma, which allows extraction of near-uniform random numbers from an imperfect entropy source by sampling from an almost two-universal family of hash functions using a seed. This approach is taken in many practical implementations, commonly using Toeplitz hashing. The lemma provides an upper bound
ε on the distance of the output to uniform randomness, which holds on average over the choice of seeds. It makes no claim that any particular seed yields near-uniform random numbers. This poses a problem for practical applications where a seed is sampled once and reused for many rounds (blockwise hashing). Here, we study the subset of \textit{bad} seeds yielding far-from-uniform outputs. We show that randomness extraction against quantum side information is guaranteed to produce
δ-close-to-uniform random numbers except with failure probability
(δε)2. This result is generalised to blockwise hashing providing a guarantee of the security of
k rounds except with failure probability
(δ(2k−1)ε)2. Moreover, we extend the general leftover hash lemma to blockwise hashing and relax the assumption of almost two-universality.
Privacy amplification is an indispensable step in the post-processing of quantum key distribution, which can be used to compress the redundancy of shared key and improve the security level of the key. The commonly used privacy amplification is based on the random selection of universal hash functions, which needs the help of an additional random source, while it does not exist in general. In this paper, we propose a privacy amplification scheme based on composite coding, which is an extension of quantum CSS codes to classical linear codes. Compared with the universal hashing function, the proposed scheme does not need other random sources, and the randomness can be completely provided by the qubit string. Furthermore, the information-theoretic bound for the extraction of the key is obvious in composite coding.