We apply the framework of algebraic feedback shift registers to polynomial rings over finite fields. This gives a construction of new pseudorandom sequences (over non-prime finite fields), which satisfy Golomb’s three randomness criteria.

The discrete logarithm problem over finite fields serves as the source of security for several cryptographic primitives. The fastest known algorithms for solving the discrete logarithm problem require solutions of large sparse linear systems over large prime fields, and employ iterative solvers for this purpose. The published results on this topic are mainly focused on systems over binary field...

Using number theory on function fields and algebraic number fields we prove results about Chebyshev polynomials over finite prime fields to investigate reversibility of two-dimensional additive cellular automata on finite square grids. For example, we show that there are infinitely many primitive irreversible additive cellular automata on square grids when the base field has order two or three.

We show that the classifying topos for the theory of fields does not satisfy De Morgan’s law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.

in a previous paper, the second author established that, given finite fields $f < e$ and certain subgroups $c leq e^times$, there is a galois connection between the intermediate field lattice ${l mid f leq l leq e}$ and $c$'s subgroup lattice. based on the galois connection, the paper then calculated the irreducible, complex character degrees of the semi-direct product $c rtimes {rm gal} (e/f)$...

For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of K2OE . For quadratic fields whose discriminant has arbitarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form Q[ √ p1 · · · pk], where the primes pi are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrel...

In this note we address the question whether for a given prime number p, the zeta-function of a number field always determines the p-part of its class number. The answer is known to be no for p = 2. Using torsion points on elliptic curves we give for each odd prime p an explicit family of pairs of non-isomorphic number fields of degree 2p + 2 which have the same zeta-function and which satisfy ...

Prime numbers are considered the basic building blocks of the counting numbers, and thus a natural question is: Are there infinitely many primes? Around 300BC, Euclid demonstrated, with a proof by contradiction, that infinitely many prime numbers exist. Since his work, the development of various fields of mathematics has produced subsequent proofs of the infinitude of primes. Each new and uniqu...

11.1 Prime fields of odd characteristic 201 Representations and reductions • Multiplication • Inversion and division • Exponentiation • Squares and square roots 11.2 Finite fields of characteristic 2 213 Representation • Multiplication • Squaring • Inversion and division • Exponentiation • Square roots and quadratic equations 11.3 Optimal extension fields 229 Introduction • Multiplication • Exp...

Let χ be a non-principal Dirichlet character modulo a prime p. Let q1 < q2 denote the two smallest prime non-residues of χ. We give explicit upper bounds on q2 that improve upon all known results. We also provide a good upper estimate on the product q1q2 which has an upcoming application to the study of norm-Euclidean Galois fields.