New distributed algorithms for fast sign detection in residue number systems (RNS)

We identify a canonical parameter in the Chinese Remainder Theorem (CRT) and call it the ‘‘Reconstruction Coefficient’’, (denoted by ‘‘RC ’’); and introduce the notions of ‘‘Partial’’ and ‘‘Full’’ Reconstruction. If the RC can be determined efficiently, then arithmetic operations that are (relatively) harder to realize in RNS; including Sign Detection, Base change/extension and Scaling or division by a constant can also be implemented efficiently. This paper therefore focuses on and presents two distinct methods to efficiently evaluate the RC at long wordlengths. A straightforward application of these methods leads to ultra-fast sign-detection. An independent contribution of this paper is to illustrate non-trivial trade-offs between run-time computation vs. pre-computation and look-up. We show a simple method to select the moduli which leads to both the (i) number of RNS channels n; as well as (ii) the largest channel modulus mn satisfying {O(n) , O(mn)} / N ≡ the full-precision bit-length. The net result is that formany canonical operations; exhaustive look-up covering all possible input values is feasible even at long cryptographic bit-lengths N . Under fairly general and realistic assumptions about the capabilities of current hardware, the memory needed for exhaustive look-up tables is shown to be bounded by a low degree polynomial of n. Moreover, both methods to compute RC can achieve a delay of O(lg n) in a RN system with n channels. To the best of our knowledge, no other method published to date has shown a path to achieve that lower bound on the execution delay. Further, small values of channel moduli make it ideal to implement each individual RNS channel on a simple core in a many-core processor or as a distributed node, and our algorithms require a limited number of inter-channel communications, averaging O(n). Results from a multi-core GPU implementation corroborate the theory. © 2016 Elsevier Inc. All rights reserved. 1. Notation and definitions A residue number system (RNS) uses a set M of pairwise coprime positive integers called the moduli-set: M = {m1,m2, . . . ,mr , . . . ,mn} (1) ∗ Corresponding author. E-mail address: [email protected] (D.S. Phatak). http://dx.doi.org/10.1016/j.jpdc.2016.06.005 0743-7315/© 2016 Elsevier Inc. All rights reserved. where each component-modulus mr > 1 ∀r ∈ [1, n] and gcd(mi,mj) = 1 for i ≠ j. For convenience, we assume mi < mj if i < j. The number of residue channels n is the cardinality of the moduli-set: n ≡ |M| = number of moduli. The total modulusM is the product of all moduli in the moduliset: M = m1 × m2 × · · · × mn. Every integer Z ∈ [0,M − 1] can be uniquely represented by a touple/vector of its residues (remainders) w.r.t (with respect to) D.S. Phatak, S.D. Houston / J. Parallel Distrib. Comput. 97 (2016) 78–95 79 each component modulus: Z ←→ Z = [z1, z2, . . . , zn], where (2) zr = (Z mod mr), r = 1, . . . , n. (3) The RNS full-precision is defined as N base-b digits, where N = ⌈logb M⌉ and b is the radix (or base) of the original representation of the integer Z . Conversion from residues back to an integer is done using the Chinese Remainder Theorem (CRT) as follows: Z = (ZT mod M) where (4)