Security analysis of an encryption scheme based on nonpositional polynomial notations

Abstract

Abstract:The aim of the research was to conduct a cryptographic analysis of an encryption scheme developed on the basis of nonpositional polynomial notations to estimate the algorithm strength. Nonpositional polynomial notations (NPNs) are residue number systems (RNSs) based on irreducible polynomials over GF(2). To evaluate if the algorithms developed on the basis of NPNs are secure, mathematical models of cryptanalysis involving algebraic, linear and differential methods have been designed. The cryptanalysis is as follows. A system of nonlinear equations is obtained from a function transforming plaintext into ciphertext with a key. Next, a possibility of transition of the nonlinear system to a linear one is considered. The cryptanalysis was conducted for the cases with known: 1) ciphertext; 2) plaintext and the related ciphertext; 3) plaintext file format; and 4) ASCII-encoded plaintext.

Encryption algorithm "QAMAL NPNS" based on a nonpositional polynomial notation

“…For multiplication in an NPN, a partial attack was used. Earlier studies had been conducted in this direction [27]. The system of equations binding the key, plaintext, and ciphertext in the encryption scheme based on an NPN for one irreducible polynomial is given below: In this context, the input data are random sequences resulting from other transformations.…”

“…In the case of an algebraic attack, provided that the ciphertext and plaintext are known, the number of search operations for finding the key lies within the following interval [27]:…”

“…In recent years, the Institute of Information and Computational Technologies of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan has carried out research on the study of symmetrical block encryption algorithms for electronic messages and has developed various modifications, including those based on non-positional polynomial notations (NPNs) [15][16][17][18][19][20][21]. These works, in turn, will contribute to the creation of domestic cryptographic information protection facilities.…”

Creating an algorithm of encryption based on prime numbers in positional systems of calculating residual classes

“…In comparison of the classical system, the proposed cryptographic procedures are considered in the residual classes of the number system, where prime numbers are taken as a basis. In the encryption algorithm, the key sequence and prime numbers are taken as the full key, and as the basis of the system, all prime numbers not exceeding the length of the encrypted message from the selected bases in the order of distribution are selected [27,28].…”

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].…”