Spectral modular arithmetic method and apparatus

US 8719324 B1

Issued: May 6, 2014

1. A method of modular exponentiation, compromising:

at a cryptographic processor, receiving a message text m, an exponent e, and a modulus n, represented as a number of words s;

determining a v0 multiple of the modulus n, representing the v0 multiple of the modulus n as a polynomial θ, and obtaining a Fourier transform of the polynomial θ, wherein v0 is an inverse of a least significant word of n modulus b=2u, and u is a word length;

obtaining a Fourier transform Δ of a polynomial representation δ associated with a square of a Montgomery coefficient modulo n;

obtaining a polynomial representation M of a Fourier transform of the message text m;

performing exponentiation of M in the Fourier transform domain by a series of j spectral modular multiplications to obtain a transform domain representation, wherein j is a number of bits in the exponent e; and

obtaining an inverse Fourier transform of the transform domain representation to obtain an integer associated with me (mod n).