Development of an Encryption Algorithm Based on Nonpositional Polynomial Notations

Abstract

Abstract-Cryptographic systems, developed on the basis of nonpositional polynomial notations, are called nonconventional or modular. In this paper modelling of the encryption algorithm based on nonpositional polynomial notations is described. The development of the model of block cipher system comprises the construction of the modified nonpositional block cipher algorithm, using an analog of the Feistel scheme and a mode of application for this modified algorithm.

Symmetric Encryption Algorithms in a Polynomial Residue Number System

“…So, to find the number of irreducible polynomials f x = a n x n + a n−1 x n−1 + a n−2 x n−2 + ⋯+a 0 x 0 over GF p , you need to determine all divisors of degree n, calculate the Möbius function for each of them, substitute them into Formula (10), and sum them. For example, for irreducible polynomials over GF p of degree n = 3 with divisors 1 (the value of the Möbius function is 1) and 3 (the value of the Möbius function is −1), their number can be found by the following formula:…”

“…The requirements for symmetric methods have become more stringent in terms of ensuring their cryptographic strength due to the rapid development of computing tools and their increased speed. Polynomial algorithms are an alternative to modern numerical cryptoalgorithms [10][11][12]. In the ring Z x , as in any other ring of polynomials, basic cryptographic operations are performed: addition, multiplication, and division with remainder [13][14][15].…”

The Device for Multiplying Polynomials Modulo an Irreducible Polynomial

“…In [14][15][16][17] described approaches to the development of a block symmetric encryption algorithm based on polynomial RNS, where secrecy is determined by the so-called "full key", which consists of a secret key (pseudo-random) sequence and the selected moduli system. Resistance against exhaustive search in this case depends not only on the length of the secret sequence but also on the composition of selected system of polynomial bases, and on the number of possible permutations of bases in that system [18].…”