No abstract
Nonlinear substitutions (S-boxes) are used in most modern symmetric cryptoalgorithms. They are designed to mix input data and play a significant role in ensuring resistance against known cryptanalytic attacks (differential, linear, algebraic and other cryptanalysis methods). However, random generation of nonlinear substitutions with the desired indicators is an extremely difficult mathematical problem. This article explores the heuristic techniques for S-boxes informed search, in particular, discusses various cost functions used in most of the known algorithms (for example, local search, hill climbing, simulated annealing, genetic search, etc.). The aim of the study is to determine the specific parameters of heuristic functions, which, on the one hand, do not reduce the degree of awareness of the search nodes, and on the other hand, do not require significant computational costs. The article examines the influence of individual parameters on the value of the cost function and complexity of its calculation. It also provides specific recommendations for the formation of parameters for heuristic search for S-boxes, which significantly affect the efficiency of generating nonlinear substitutions for symmetric cryptography.
To ensure the security of information technology, cryptographic information protection tools are used, in particular block and stream encryption algorithms with a symmetric key. Reliability and cryptographic strength of cryptoalgorithms is provided by the properties of the applied primitives. For example, non-linear substitutions (S-boxes) are used as the main component of modern symmetric ciphers. Therefore, generation of substitutions is an important scientific task directly related to the security of information technology and improvement of modern symmetric ciphers. The paper investigates the properties of iterative algorithms for generating non-linear substitutions and special cost functions, which play a decisive role in the heuristic search for S-boxes with the required properties. We consider the cost function of the WCF (Cost Function of the content of the Walsh-Hadamard spectrum) and optimize its parameters. The obtained optimization results in combination with the Hill Climbing iterative search algorithm can reduce significantly the number of iterations. In particular, we show that for a substitution search with a non-linearity of 104, on average, we reduce the computational complexity of generation by more than 20%. In addition, it is possible to increase the success rate of the heuristic search. In particular, for the selected settings, in 100% of cases, a beaktive S-box with a non-linearity of 104 was found.
Development and investigation of electronic signatures on algebraic lattices is one of the promising directions in post-quantum cryptography. Cryptosystems CRYSTALS-Dilithium and Falcon represent lattice cryptography in the category of electronic signatures in the NIST PQC open competition among the finalists. Most operations in these cryptosystems are reduced to addition and multiplication of polynomials in a finite field with a generating cyclotomic polynomial xN + 1. Using such a field allows the use of a number-theoretic transformation (NTT) to create fast and reliable software implementations. In practice, vectorized set (SIMD) instructions are used to achieve good performance. AVX2 instructions are most often used among existing implementations. At the same time, the possibility of using AVX512 instructions remains little explored. The purpose of this work is to investigate the feasibility of applying AVX512 instructions to optimization of the NTT, used in modern EPs on algebraic lattices. In particular, the paper presents a method for implementing a number-theoretic transformation using AVX512 for CRYSTALS-Dilithium and Falcon. An increase in performance is shown in comparison with the reference optimized author's implementations.
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