A Hardware Algorithm for Modular Multiplication/Division Based on the Extended Euclidean Algorithm

Abstract

A hardware algorithm for modular multiplication/division which performs modular division, Montgomery multiplication, and ordinary modular multiplication is proposed. The modular division in our algorithm is based on the extended Euclidean algorithm. We employ our newly proposed computation method that consists of processing the multiplier from the most significant digit first to calculate Montgomery multiplication. Finally, the ordinary modular multiplication is based on shift-and-add multiplication. Each of these three operations is carried out through the iteration of simple operations such as shifts and additions/subtractions. To avoid carry propagation in all additions and subtractions, the radix-2 signed-digit representation is employed. A modular multiplier/divider based on the algorithm has a linear array structure with a bit-slice feature and carries out n-bit modular multiplication/division in O(n) clock cycles, where the length of the clock cycle is constant and independent of n. This multiplier/divider can be implemented using a hardware amount only slightly larger than that of the modular divider.

Authentication and consensus overhead in vehicular ad hoc networks

“…mine a modular multiplication (n is the bit-length of x, y, or m, and b the radix) [21]. If operations (modular addition, modular multiplication, multiplication, addition, shift) on A are considered done in constant time, the time complexity is O(n) and the space complexity is O(n) [22,23].…”

A high speed Montgomery processor of 256-bit on FPGA

“…The former means is efficiently performance when the modulo is a Merssane prime [3].In the previous work , they have proposed many effective methods. In M. Kaihara et al [4] and M. Schramm et al [5], they present shift and add modular multiplication algorithm; In prime field ECC processors, carry free structure is necessary to avoid lengthy data paths caused by carry propagation. There has been redundant schemes applied to different designs, for example, Carry Save Arithmetic (CSA) or Redundant Signed Digits (RSD); There is a scalable word based structure proposed in Tenca and Koc [6].…”