The locally repairable code (LRC) studied in this paper is an [n, k] linear code of which the value at each coordinate can be recovered by a linear combination of at most r other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters n 1 > n 2 , where n 1 = n r+1 and n 2 = n 1 (r + 1) − n. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field F 2 m , where m ≥ n 1 r. Based on these results, the largest possible minimum distance for all LRCs with r ≤ √ n − 1 has been definitely determined, which is of great significance in practical use.
The current widely used public‐key cryptosystems are vulnerable to quantum attacks. To prepare for cybersecurity in the quantum era, some projects have been launched to call for post‐quantum alternatives. Due to solid security and desirable performance, lattice‐based cryptosystems are viewed as promising candidates in the upcoming standardisation of post‐quantum cryptography. This study surveys the lattice‐based cryptosystems in the post‐quantum standardisation processes including the NIST Post‐Quantum Cryptography Standardisation and the Chinese Cryptographic Algorithm Design Competition, from both design and security aspects. We present generic design paradigms of lattice‐based schemes and describe several representative proposals and recent progress. We also recap some main cryptanalytic results and methods for estimating the concrete security of lattice‐based schemes.
Rejection sampling technology is a core tool in the design of lattice-based signatures with ‘Fiat–Shamir with Aborts’ structure, and it is related to signing efficiency and signature, size as well as security. In the rejection sampling theorem proposed by Lyubashevsky, the masking vector of rejection sampling is chosen from discrete Gaussian distribution. However, in practical designs, the masking vector is more likely to be chosen from bounded uniform distribution due to better efficiency and simpler implementation. Besides, as one of the third-round candidate signatures in the NIST postquantum cryptography standardization process, the 3rd round version of CRYSTALS-Dilithium has proposed a new method to decrease the rejection probability in order to achieve better efficiency and smaller signature size by decreasing the number of nonzero coefficients of the challenge polynomial according to the security levels. However, it is seen that small entropies in this new method may lead to higher risk of forgery attack compared with former schemes proposed in its 2nd version. Thus, in this paper, we first analyze the complexity of forgery attack for small entropies and then introduce a new method to decrease the rejection probability without loss of security including the security against forgery attack. This method is achieved by introducing a new rejection sampling theorem with tighter bound by utilizing Rényi divergence where masking vector follows uniform distribution. By observing large gaps between the security claim and actual security bound in CRYSTALS-Dilithium, we propose two series of adapted parameters for CRYSTALS-Dilithium. The first set can improve the efficiency of the signing process in CRYSTALS-Dilithium by factors of 61.7 % and 41.7 % , according to the security levels, and ensure the security against known attacks, including forgery attack. And, the second set can reduce the signature size by a factor of 14.09 % with small improvements in efficiency at the same security level.
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