Besides Karatsuba algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF (2) subquadratic multipliers. However, when GF (2) elements are represented using a shifted polynomial basis, this approach is currently appliable only to GF (2)s generated by all irreducible trinomials and a special type of irreducible pentanomials, not all general irreducible p...

The first author had shown earlier that for a standard graded ring R and ideal I in characteristic p > 0 , with ? ( / ) < ? there exists compactly supported continuous function f whose Riemann integral is the HK multiplicity e H K . We explore further some other invariants, namely shape of graph m (where maximal maximum support (denoted as ? In case domain dimension d ? 2 we prove regular if on...

Elements of a finite field, GF ð2Þ, are represented as elements in a ring in which multiplication is more time efficient. This leads to faster multipliers with a modest increase in the number of XOR and AND gates needed to construct the multiplier. Such multipliers are used in error control coding and cryptography. We consider rings modulo trinomials and 4-term polynomials. In each case, we sho...

It is shown how correlations in the generalized feedback shift-register (GFSR) random-number generator are greatly diminished when the number of feedback taps is increased from two to four (or more) and the tap offsets are lengthened. Simple formulas for producing maximal-cycle four-tap rules from available primitive trinomials are given, and explicit threeand four-point correlations are found ...

Let Fq be a finite field of odd order q. In this paper, the irreducible factorization of x2 pr − 1 over Fq is given in a very explicit form, where a, b, c are positive integers and p, r are odd prime divisors of q−1. It is shown that all the irreducible factors of x2 pr − 1 over Fq are either binomials or trinomials. In general, denote by vp(m) the degree of prime p in the standard decompositio...

finite fields, irreducible polynomials A table of low-weight irreducible polynomials over the finite field F2 is presented. For each integer n in the range 2 = n = 10,000, a binary irreducible polynomial f(x) of degree n and minimum posible weight is listed. Among those of minimum weight, the polynomial listed is such that the degree of f(x) – xn is lowest (similarly, subsequent lower degrees a...

Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic which yields similar applications. All known pesudo-planar f...