Finite Field Polynomial Multiplication in the Frequency Domain with Application to Elliptic Curve Cryptography

We introduce an efficient method for computing Montgomery products of polynomials in the frequency domain. The discrete Fourier transform (DFT) based method originally proposed for integer multiplication provides an extremely efficient method with the best asymptotic complexity, i.e. O(m log m log log m), for multiplication of m-bit integers or (m−1)st degree polynomials. Unfortunately, the DFT method bears significant overhead due to the conversions between the time and the frequency domains. This makes the original DFT method impractical for multiplication of short operands as used in many applications. In this work, we introduce an algorithm which performs the entire modular multiplication (including the reduction step) in the frequency domain, and thus eliminates costly back and forth conversions between the frequency and the time domains. Furthermore, with careful selection of parameters, we show that in computational platforms where multiplication operation is expensive frequency domain multiplication of finite field elements may be realized more efficiently than multiplication in the time domain for operand sizes relevant to elliptic curve cryptography.