Abstract
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.
Highlights
With the rapid developments in quantum algorithms and computations, research in lattice-based cryptography has attracted considerable attention because lattice-based cryptosystems are likely to be effective against quantum computing attacks in the future. e first lattice-based cryptosystem is proposed by Ajtai and Dwork [1] in 1997 which is known as the first cryptosystem that achieves worst case to average case reduction
We propose another way to increase the success probability of rejection sampling without loss of security. is idea is obtained by firstly proposing a more practical rejection sampling theorem with masking vector sampled from bounded uniform distribution, where a tighter bound is achieved by using Renyi divergence rather than statistical distance
By choosing proper parameters, the efficiency of sign algorithm in CRYSTALS-Dilithium can be further improved depending on the security levels
Summary
With the rapid developments in quantum algorithms and computations, research in lattice-based cryptography has attracted considerable attention because lattice-based cryptosystems are likely to be effective against quantum computing attacks in the future. e first lattice-based cryptosystem is proposed by Ajtai and Dwork [1] in 1997 which is known as the first cryptosystem that achieves worst case to average case reduction. Many variants based on GGH structure concentrate on improving the security against the attack proposed in [17] As for another basic type, Fiat–Shamir structure is first used to design practical latticebased signature scheme in [18]. Many practical schemes choose to sample from a bounded uniform distribution for the masking vector including the two NIST candidates, CRYSTALS-Dilithium [22] and qTESLA [23].
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