Systems of Random Equations over Finite Algebraic Structures

Abstract

519.21The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.Keywords: system of random equations over finite algebraic structure (finite field, finite ring, finite Abelian group), Boolean system of equations, certainly compatible system of random equations, system of random equations with independent left-and right-hand sides, system of random equations with distorted right-hand sides.This review of the results from the studies on systems of random equations over finite algebraic structures is timed to the 70th anniversary of Igor N. Kovalenko, a prominent Ukrainian mathematician, Doctor of engineering, physics, and mathematics, Academician of the National Academy of Sciences of Ukraine. He may rightfully be considered a founder of this complex and fascinating division in discrete mathematics. His fundamental and mathematically beautiful study [33] (1967) on the invariance of the limit behavior of the number of solutions of one class of systems of random equations over a GF(2) field became a basis for disciples of I. N. Kovalenko's school in extending invariance theory to random systems over arbitrary finite fields and finite rings. Over nearly 40 years since the paper [33] was published, the theory of systems of random equations over finite algebraic structures has been enriched with new problems, results, and methods. To efficiently develop this theory, we today need to sort out and systematize everything accumulated in this area. The author of the present review, a follower of I. N. Kovalenko, has tried to do this with gratitude and deep respect for the teacher.The studies by V. L. Goncharov on random substitutions and series in random (0,1)-sequences, published in the 40s of the last century, have initiated a systematic analysis of combinatory objects using methods of probability theory. Thus, they promoted the development of a new division in discrete mathematics, probabilistic combinatorial analysis, where the theory of systems of random equations over finite algebraic structures is a relatively small subdivision, despite its important applications in the theory of finite state automata, the theory of coding, the theory of random graphs, cryptography, etc. there is an explanation of this. The use of random systems as probability-theoretic models of combinatorial processes occurring in information-protection systems has vetoed publications on this subject in the available scientific literature for many years. The situation began to change for the better only in the early 90s of the last century. Indeed, a large number of papers on random systems over finite algebraic structures have appeared in the last 10 to 14 years in large mathematical journals. It is pleasant to point out that whereas this subjects was considered especially "Russian" when the review [40,41] devoted to random linear systems over finite algebraic structures was written (since the corresponding publications were mainly due to Soviet ma...

Random networks tossing biased coins

“…Indeed, because of its simple formulation, the potential for further analytical calculations is large. For example, one can evaluate the kernel of A, which is useful in connection with problems of the Satisfiability class, which have seldom been analyzed on non-Poisson random graphs [31][32][33][34]. Note that, since the finite n formula for the mean is known exactly, the finite size scaling can be computed analytically, simply by isolating the leading terms in the approach to the asymptotic limit.…”

The normal limit distribution of the normalized number of false solutions of a one system of nonlinear random equations over the field GF(2)

“…Most attention was paid to finding conditions for the convergence of the distribution of the random variable ν n to a Poisson distribution as n → ∞ in the previously published papers (see review article [2]). We are interested in the conditions under which in appropriate way normalized random variable ν n has a normal limit (n → ∞) distribution and n − ρ(n) → ∞ (n → ∞), f (n) ≥ 2.…”

“…Therefore, conditions of the Lemma 2 in [5] checked up and with the help of this Lemma, (2) and (25) max 0≤t≤(1+ω)λ * |P {ν n ≥ t} − P {Y ≥ t}| → 0 as n → ∞.…”

Solving the problem of invariance of probabilistic characteristics for a priori solvable systems of random nonlinear equations over a finite commutative ring with unity

“…They may be thought of as foundational for developing a new line of investigation of systems of random equations that has led to the creation of the invariance theory for systems of random equations over finite algebraic structures. Theoretical implications of this theory are considered in more detail in [2].This work is devoted to the solution of invariance problems for probabilistic characteristics of a class of nonlinear random systems. The subjects of investigation are the so-called a priori solvable systems of random nonlinear equations over an arbitrary commutative ring R (with unity) of cardinality | | R = m and of the following form:…”