Montgomery Multiplication in GF(2^k
Designs, Codes and Cryptography
Elliptic curves in cryptography
Elliptic curves in cryptography
Fast Algorithms for Digital Signal Processing
Fast Algorithms for Digital Signal Processing
Discrete Convolutions via Mersenne Transrorms
IEEE Transactions on Computers
A state-of-the-art elliptic curve cryptographic processor operating in the frequency domain
Mobile Networks and Applications
Optimal Extension Field Inversion in the Frequency Domain
WAIFI '08 Proceedings of the 2nd international workshop on Arithmetic of Finite Fields
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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(mlogmloglogm), for multiplication of m-bit integers or (m–1)st degree polynomials. However, the original DFT method bears significant overhead due to the conversions between the time and the frequency domains which makes it impractical for short operands as used in many applications. In this work, we introduce DFT modular multiplication which performs the entire modular multiplication (including the reduction step) in the frequency domain, and thus eliminates costly back and forth conversions. We show that, especially in computationally constrained platforms, multiplication of finite field elements may be achieved more efficiently in the frequency domain than in the time domain for operand sizes relevant to elliptic curve cryptography (ECC). To the best of our knowledge, this is the first work that proposes the use of frequency domain arithmetic for ECC and shows that it can be efficient.