We propose the first general multiplication algorithm in GF (2k) with a subquadratic area complexity of O(k8/5) = O(k1.6). We represent the elements of GF (2k) according to 2n pairwise prime trinomials, T1, . . . , T2n, of degree d, such that nd ≥ k. Our algorithm is based on Montgomery’s multiplication applied to the ring formed by the direct product of the n first trinomials.

In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works.

We consider trinomials of the form zn+azm+b, where 0<m<n are relatively prime integers and a b nonzero complex numbers. Typically (but not exclusively), will be an integer with |a|≥2, while b=±1. Our main results cover irreducibility, Mahler measure house such trinomials.

Swan (1962) gives conditions under which the trinomial x+x+1 over F2 is reducible. Vishne (1997) extends this result to trinomials over extensions of F2. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field Fq, where q is a power of an odd prime.

Recently, the multipliers with subquadratic space complexity for trinomials and some specific pentanomials have been proposed. For such kind of multipliers, alternatively, we use double basis multiplication which combines the polynomial basis and the modified polynomial basis to develop a new efficient digit-serial systolic multiplier. The proposed multiplier depends on trinomials and almost eq...

Pseudo-random numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime. These generators are based on “almost primitive” trinomials, that is trinomials ha...

In this manuscript we study the counting problem for harmonic trinomials of form a?n+b??m+c, where n,m?N, n>m, and a, b c are non-zero complex numbers. As a consequence, obtain Fundamental Theorem Algebra Wilmshurst conjecture trinomials. The proof relies on Bohl method introduced in [1].

The standard algorithm for testing reducibility of a trinomial of prime degree r over GF(2) requires 2r+O(1) bits of memory and Θ(r) bit-operations. We describe an algorithm which requires only 3r/2 + O(1) bits of memory and significantly fewer bit-operations than the standard algorithm. Using the algorithm, we have found 18 new irreducible trinomials of degree r in the range 100151 ≤ r ≤ 70005...