Parameter Determination for Complex Number-Theoretic Transforms Using Cyclotomic Polynomials

Faster arithmetic for number-theoretic transforms

We show how to improve the efficiency of the computation of fast Fourier transforms over Fp where p is a word-sized prime. Our main technique is optimisation of the basic arithmetic, in effect decreasing the total number of reductions modulo p, by making use of a redundant representation for integers modulo p. We give performance results showing a significant improvement over Shoup’s NTL library.

Randomization Using Quasigroups, Hadamard and Number Theoretic Transforms

Good symmetric encryption schemes as well as randomization and hashing techniques are based on effective techniques of confusion and diffusion [1]. Quasigroups provide an excellent way to generate an astronomical number of keys and therefore they are excellent at confusion [2] but they are not equally good at diffusing the statistics of the plaintext. Specifically, the quasigroup transformation...

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Number-theoretic transforms of prescribed length

A new constructive method for finding all convenient moduli m for a number-theoretic transform with given length N and given primitive Nth root of unity modulo m is presented. This method is based on the prime factorization of cyclotomic polynomials with integer-valued argument or on the primitive divisors of integers. Many known results can be obtained as simple corollaries.

Cyclotomic Polynomials