finite field | Academic Journals and Conferences

Program Implementation of Cryptographic Primitives

Developed on the C# platform (.NET Framework 5.0), which provides high flexibility in work, a program for performing operations (addition, multiplication, raising to the power of a large natural number, finding the inverse relatively to multiplication) on elements of extended finite fields and general linear groups over such fields. The general linear group is one of the wellknown non-Abelian groups, the application of which is actively studied in the field of postquantum cryptography.

Encryption Method Based on Codes

This paper proposes an improvement of the McEliece asymmetric cryptosystem based on code-based cryptography by replacing the permutation matrix with a modulo operation and using a finite field $GF(q)$.

Program model of Reed-Solomon codes

Software is designed for modeling of Reed-Solomon codes on a base of object-oriented technology. Input data for system are blocks of bytes for transmitting through communication channel, where errors can occur in the blocks. Designed program realizes codes of (255,239) and (255,223) type for finite field GF(28) with standard generating polynomials x8+x4+x3+x2+1 and x8+x7+x2+x+1. Moreover, a possibility is provided to add other types of codes and generating polynomials.

Elements of high multiplicative order in extended finite fields on a base of modified GAO approach

The Gao approach to construction of high order elements in arbitrary finite fields is to choose a convenient polynomial, which defines an extension of an initial prime field. This choice depends on one polynomial-parameter. That is why the mentioned approach can be considered as using of a finite field description with one degree of freedom. We explore in the paper the possibility of improvement of lower bound on element orders in finite fields of general form with using of two degrees of freedom.

Computer verification of gao assumption, related with obtaining of high order elements in finite fields

We have performed computer calculations in Maple environment for verification of Gao assumption for finite fields of characteristic 2, 3, 5 and presented correspondent results. If the assumption is true, then it is possible to construct explicitly in these fields in polynomial time elements of high multiplicative order that are used in cryptography (Diffie-Hellman protocol, El-Gamal public key cryptosystem, El-Gamal digital signature).

Extended Finite Fields in Cryptographic Information Protection

The use of extended finite fields for cryptographic information protection is focused on. In particular, explicit construction in finite fields elements of high multiplicative order is described. The obtained correspondent lower bounds on the order are provided.