We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger bounds, e.g., 248832 via a famous general result of Khovanski. Our bound is sharp, allows real exponents, and extends to certain systems of n-variate fewnomials,...

This work presents novel multipliers for Montgomery multiplication defined on binary fields GF(2). Different to state of the art Montgomery multipliers, this work uses a Linear Feedback Shift Register (LFSR) as the main building block. We studied different architectures for bit-serial and digit-serial Montgomery multipliers using the LFSR and the Montgomery factors x and xm−1. The proposed mult...

We give a simple condition for a linear recurrence (mod 2w) of degree r to have themaximal possible period 2w 1(2r 1). It follows that the period is maximal in the casesof interest for pseudo-random number generation, i.e. for 3-term linear recurrences de nedby trinomials which are primitive (mod 2) and of degree r > 2. We consider the enumera-tion of certain exceptional pol...

We present a new formulation of the Mastrovito multiplication matrix for the field GF (2) generated by an arbitrary irreducible polynomial. We study in detail several specific types of irreducible polynomials, e.g., trinomials, all-one-polynomials, and equally-spaced-polynomials, and obtain the time and space complexity of these designs. Particular examples, illustrating the properties of the p...

This paper describes an efficient FPGA implementation for modular multiplication in the finite field GF(2) that is suitable for implementing Elliptic Curve Cryptosystems. We have developed a systolic array implementation of a Montgomery modular multiplication. Our solution is efficient for large finite fields (m=160-193) that offer a high security level, and it can be scaled easily to larger va...

Marsaglia recently introduced a class of “xorshift” random number generators (RNGs) with periods 2n − 1 for n = 32, 64, etc. Here Marsaglia’s xorshift generators are generalised to obtain fast and highquality RNGs with extremely long periods. Whereas RNGs based on primitive trinomials may be unsatisfactory, because a trinomial has very small weight, these new generators can be chosen so that th...