Computational Analysis of Quantitative Characteristics of some Residual Properties of Solvable Baumslag-Solitar Groups

Let $G_{k}$ be defined as $G_{k} = \langle a, b;\ a^{-1}ba = b^{k} \rangle$, where $k \ne 0$. It is known that, if $p$ is some prime number, then $G_{k}$ is residually a finite $p$-group if and only if $p \mid k - 1$. It is also known that, if $p$ and $q$ are primes not dividing $k - 1$, $p < q$, and $\pi = \{p,\,q\}$, then $G_{k}$ is residually a finite $\pi$-group if and only if $(k, q) = 1$, $p \mid q - 1$, and the order of $k$ in the multiplicative group of the field $\mathbb{Z}_{q}$ is a $p$\-number. This paper examines the question of the number of two-element sets of prime numbers that satisfy the conditions of the last criterion. More precisely, let $f_{k}(x)$ be the number of sets $\{p,\,q\}$ such that $p < q$, $p \nmid k - 1$, $q \nmid k - 1$, $(k, q) = 1$, $p \mid q - 1$, the order of $k$ modulo $q$ is a $p$\-number, and $p$, $q$ are chosen among the first $x$ primes. We state that, if $2 \leq |k| \leq 10000$ and $1 \leq x \leq 50000$, then, for almost all considered $k$, the function $f_{k}(x)$ can be approximated quite accurately by the function $\alpha_{k}x^{0.85}$, where the coefficient $\alpha_{k}$ is different for each $k$ and $\{\alpha_{k} \mid 2 \leq |k| \leq 10000\} \subseteq (0.28;\,0.31]$. We also investigate the dependence of the value $f_{k}(50000)$ on $k$ and propose an effective algorithm for checking a two-element set of prime numbers for compliance with the conditions of the last criterion. The results obtained may have applications in the theory of computational complexity and algebraic cryptography.

Mixed signal testing system technique with algebraic signature analyzer without carry propagation

Signature Analyzer is an analyzer which is widely used for mixed-signal system testing. But its hardware has high complexity in implementation as the application technique is a system with rules of an arithmetic finite field with arbitrary radix. It’s a challenging task. To avoid this complexity here the project is made based on Algebraic Signature Analyzer that can be used for mixed signal testing and the analyzer doesn’t contain carry propagation circuitry. It improves performance and fault tolerance. This technique is simple and applicable to systems of any size or radix. The hardware complexity is very low compared to the conventional one and can be used in arithmetic/ algebraic cryptography as well as coding

Mixed signal testing system technique with algebraic signature analyzer without carry propagation

Signature Analyzer is an analyzer which is widely used for mixed-signal system testing. But its hardware has high complexity in implementation as the application technique is a system with rules of an arithmetic finite field with arbitrary radix. It’s a challenging task. To avoid this complexity here the project is made based on Algebraic Signature Analyzer that can be used for mixed signal testing and the analyzer doesn’t contain carry propagation circuitry. It improves performance and fault tolerance. This technique is simple and applicable to systems of any size or radix. The hardware complexity is very low compared to the conventional one and can be used in arithmetic/ algebraic cryptography as well as coding

Two general schemes of algebraic cryptography

In this paper, we introduce two general schemes of algebraic cryptography. We show that many of the systems and protocols considered in literature that use two-sided multiplications are specific cases of the first general scheme. In a similar way, we introduce the second general scheme that joins systems and protocols based on automorphisms or endomorphisms of algebraic systems. Also, we discuss possible applications of the membership search problem in algebraic cryptanalysis. We show how an efficient decidability of the underlined membership search problem for an algebraic system chosen as the platform can be applied to show a vulnerability of both schemes. Our attacks are based on the linear or on the nonlinear decomposition method, which complete each other. We give a couple of examples of systems and protocols known in the literature that use one of the two introduced schemes with their cryptanalysis. Mostly, these protocols simulate classical cryptographic schemes, such as Diffie–Hellman, Massey–Omura and ElGamal in algebraic setting. Furthermore, we show that, in many cases, one can break the schemes without solving the algorithmic problems on which the assumptions are based.