Group theory in cryptography

This paper is a guide for the pure mathematician who would like to know more about cryptography based on group theory. The paper gives a brief overview of the subject, and provides pointers to good textbooks, key research papers and recent survey papers in the area.Comment: 25 pages References updated, and a few extra references added. Minor typographical changes. To appear in Proceedings of Groups St Andrews 2009 in Bath, U

A Post Quantum Key Agreement Protocol Based on a Modified Matrix Power Function over a Rectangular Matrices Semiring

We present an improved post quantum version of Sakalauskas matrix power function key agreement protocol, using rectangular matrices instead of the original square ones. Sakalauskas matrix power function is an efficient and secure way to generate a shared secret key, and using rectangular matrices provides additional flexibility and security. This method reduces the computational burden by allowing smaller random integer matrices while maintaining equal security. Another advantage of using the rank deficient rectangular matrices over key agreement protocols is that it blocks linearization attacks.Comment: 6 pages, 20 reference

Matrix Power S-box Analysis

* Work supported by the Lithuanian State Science and Studies Foundation.Construction of symmetric cipher S-box based on matrix power function and dependant on key is analyzed. The matrix consisting of plain data bit strings is combined with three round key matrices using arithmetical addition and exponent operations. The matrix power means the matrix powered by other matrix. This operation is linked with two sound one-way functions: the discrete logarithm problem and decomposition problem. The latter is used in the infinite non-commutative group based public key cryptosystems. The mathematical description of proposed S-box in its nature possesses a good “confusion and diffusion” properties and contains variables “of a complex type” as was formulated by Shannon. Core properties of matrix power operation are formulated and proven. Some preliminary cryptographic characteristics of constructed S-box are calculated

Public Key Cryptosystem Based on Polynomial Composition

The public key cryptosystem is an extension of an asymmetric key cryptosystem. The public key cryptosystems have been developed based on the concepts of matrix, polynomial and polynomial decomposition. In this study, we will introduce the public key cryptosystem over polynomial composition. This research is a literature study. The results show that the polynomial composition can be used in public-key cryptosystems by modifying special functions to apply commutative properties

Post-Quantum Key Agreement Protocol based on Non-Square Integer Matrices

We present in this paper an algorithm for exchanging session keys, coupled with a hashing encryption module. We show schemes designed for their potential invulnerability to classical and quantum attacks. In turn, if the parameters included were appropriate, brute-force attacks exceed the (five) security levels used in the NIST competition of new post-quantum standards. The original idea consists of products of rectangular matrices in Zp as public values and whose factorization is proved to be an NP-complete problem. We present running times as a function of the explored parameters and their link with operational safety. To our knowledge there are no classical and quantum attacks of polynomial complexity available at hand, remaining only the systematic exploration of the private-key space.Comment: 12 pages, 2 tables, 29 reference