On computing the degree of a Chebyshev Polynomial from its value

Abstract

Abstract Algorithms for interpolating a polynomial f from its evaluation points whose running time depends on the sparsity t of the polynomial when it is represented as a linear combination of t Chebyshev Polynomials of the First Kind with non-zero scalar coefficients are given by Lakshman and Saunders (1995) , Kaltofen and Lee (2003) and Arnold and Kaltofen (2015) . The term degrees are computed from values of Chebyshev Polynomials of those degrees. We give an algorithm that computes those degrees in the manner of the Pohlig and Hellman algorithm ( 1978 ) for computing discrete logarithms modulo a prime number p when the factorization of p − 1 (or p + 1 ) has small prime factors, that is, when p − 1 (or p + 1 ) is smooth. Our algorithm can determine the Chebyshev degrees modulo such primes in bit complexity log \u2061 ( p ) O ( 1 ) times the squareroot of the largest prime factor of p − 1 (or p + 1 ).