Montgomery Reduction Algorithm for Modular Multiplication Using Low-Weight Polynomial Form Integers

Abstract

Abstract. We extend low-weight polynomial form integers (LWPFIs) presented in [5]. An LWPFI p is an integer expressed as a degree-l, monic polynomial such that p = t l + f l−1 t l + · · · + f1t + f0, where t can be any positive integer. In [5], fi's are limited to 0 and ±1, but here we let |fi| ≤ ξ for some small positive integer ξ. In modular multiplication based on LWPFI, elements in Zp are expressed in polynomial in t and multiplication is performed in Z[t]/f (t). The coefficients must be reduced for subsequent modular multiplications. In [5], a coefficient reduction algorithm based on a division algorithm derived from the Barrett reduction algorithm is presented. In this report, we present a coefficient reduction algorithm based on the Montgomery reduction algorithm and its detailed analysis results. Bounds on the input and output of our coefficient reduction algorithm is carefully analyzed. We give conditions for eliminating the final subtractions at the end of the Montgomery reduction algorithm. In addition, we present efficient modular addition and subtraction methods using LWPFI moduli.

Generalised Mersenne numbers revisited

“…This interpretation of F p for p a cyclotomic prime is implicit within the arithmetic we develop here, albeit only insofar as it provides a theoretical context for it; this perspective offers no obvious insight into how to perform arithmetic efficiently and the algorithms we develop make no use of it at all. Similarly, the method of Chung and Hasan [15] upon which our residue representation is based can be seen as arising in exactly the same way for the much larger set of primes they consider, with the field modelled as a quotient of the ring of integers of a suitable number field by a degree one prime ideal, just as for the cyclotomic primes.…”

“…Chung-Hasan arithmetic. We now describe the ideas behind Chung-Hasan arithmetic [13,14,15]. The arithmetic was developed for a class of integers they term low-weight polynomial form integers (LWPFIs), whose definition we now recall.…”

“…Chung and Hasan refer to this issue as the coefficient reduction problem (CRP), and developed three solutions in their series of papers on LWPFI arithmetic [13,14,15]. Each of these solutions is based on an underlying lattice, although this was only made explicit in [15]. Since the lattice interpretation is the most elegant and simplifies the exposition, in the sequel we opt to develop the necessary theory for GRP arithmetic in this setting.…”

“…The idea described in [14] was based on an analogue of Barrett reduction [2]. The method we shall use, from [15], is based on Montgomery reduction [49] and for t not a power of 2 is the most efficient of the three Chung-Hasan methods.…”

“…In [15], an algorithm was given for computing u and w in (3.4) and (3.5) respectively, for an arbitrary LWPFI f (t). The number of word-by-word multiply instructions in the algorithm -which is the dominant cost -is ≈ nq 2 , where n is the degree of f (t), and R = b q .…”