Optimization of local search algorithm parameters for generating nonlinear substitutions

Nonlinear substitutions (S-boxes) are an important component of modern symmetric cryptography algorithms. They complicate symmetric transformations and introduce nonlinearity into the input-output relationship, which ensures the stability of the algorithms against some cryptanalysis methods. Generation of S-boxes can be done in different ways. However, heuristic techniques are the most promising ones. On the one hand, the generated S-boxes are in the form of random substitutions, which complicates algebraic cryptanalysis. On the other hand, heuristic search allows one to achieve high rates of nonlinearity and δ-uniformity, which complicates linear and differential cryptanalysis. This article studies the simplest local search algorithm for generating S-boxes. To assess the efficiency of the algorithm, the concept of a track of a cost function is introduced in the article. Numerous experiments are carried out, in particular, the influence of the number of internal and external loops of local search on the complexity of generating the target S-box is investigated. The optimal (from the point of view of minimum time consumption) parameters of the local search algorithm for generating S-blocks with a target nonlinearity of 104 and the number of parallel computing threads 30 are substantiated. It is shown that with the selected (optimal) parameters it is possible to reliably form S-blocks with a nonlinearity of 104.

Key Generation Using Generalized Pell’s Equation in Public Key Cryptography Based on the Prime Fake Modulus Principle to Image Encryption and Its Security Analysis

RSA is one among the most popular public key cryptographic algorithm for security systems. It is explored in the results that RSA is prone to factorization problem, since it is sharing common modulus and public key exponent. In this paper the concept of fake modulus and generalized Pell’s equation is used for enhancing the security of RSA. Using generalized Pell’s equation it is explored that public key exponent depends on several parameters, hence obtaining private key parameter itself is a big challenge. Fake modulus concept eliminates the distribution of common modulus, by replacing it with a prime integer, which will reduce the problem of factorization. It also emphasizes the algebraic cryptanalysis methods by exploring Fermat’s factorization, Wiener’s attack, and Trial and division attacks.

Algebraic Analysis of a Simplified Encryption Algorithm GOST R 34.12-2015

In January 2016, a new standard for symmetric block encryption was established in the Russian Federation. The standard contains two encryption algorithms: Magma and Kuznyechik. In this paper we propose to consider the possibility of applying the algebraic analysis method to these ciphers. To do this, we use the simplified algorithms Magma ⊕ and S-KN2. To solve sets of nonlinear Boolean equations, we choose two different approaches: a reduction and solving of the Boolean satisfiability problem (by using the CryptoMiniSat solver) and an extended linearization method (XL). In our research, we suggest using a security assessment approach that identifies the resistance of block ciphers to algebraic cryptanalysis. The algebraic analysis of an eight-round Magma (68 key bits were fixed) with the CryptoMiniSat solver demanded four known text pairs and took 3029.56 s to complete (the search took 416.31 s). The algebraic analysis of a five-round Magma cipher with weakened S-boxes required seven known text pairs and took 1135.61 s (the search took 3.36 s). The algebraic analysis of a five-round Magma cipher with disabled S-blocks (equivalent value substitution) led to getting only one solution for five known text pairs in 501.18 s (the search took 4.92 s). The complexity of the XL algebraic analysis of a four-round S-KN2 cipher with three text pairs was 236.33 s (took 1.191 Gb RAM).

Algebraic immunity of Boolean function

Алгебраический метод криптоанализа, основанный на решении систем уравнений над конечным полем, является одним из современных методов криптоанализа, широко применяющихся в процессе оценки стойкости поточного алгоритма шифрования. На практике в составе большинства поточных алгоритмов шифрования в качестве основных преобразований применяются булевы функции. Алгебраический иммунитет булевой функции — один из основных параметров, определяющих стойкость алгоритма шифрования. Для определения показателя алгебраического иммунитета булевой функции использована операция вычисления ранга специально построенной матрицы. Построен алгоритм вычисления этого показателя. Метод может быть использован в процессе алгебраического криптоанализа для оценки стойкости алгоритмов поточного шифрования. The algebraic method of cryptanalysis, based on solving systems of equations over a finite field, is one of the modern methods that is widely used in the process of assessing the strength of a stream encryption algorithm. In practice, as part of the majority of stream encryption algorithms, Boolean functions are used as the main transformations. The algebraic immunity of this Boolean function is one of the main parameters determining the strength of the encryption algorithm. In the article, a method for determining the index of algebraic immunity of a Boolean function is proposed, and an algorithm for calculating this function is constructed. To determine the index of algebraic immunity of a Boolean function, the operation of calculating the rank of a specially constructed matrix is used. A number of examples are given for calculation of the algebraic immunity of a Boolean function and construction of annihilator functions. Also, the results of the experiments are shown, by the definition of the algebraic immunity of all balanced and unbalanced Boolean functions over a finite fieldn(1 n 6). The method may be used in the process of algebraic cryptanalysis in order to assess the strength of stream encryption algorithms.