Fast Truncated Multiplication for Cryptographic Applications

Abstract

Abstract. The Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 integers. A few truncated polynomial multiplication algorithms are presented and adapted to integers. They are based on the most often used n-digit full multiplication algorithms of time complexity O(n α ), with 1< α ≤ 2, but a constant times faster. For example, the least significant half products with Karatsuba multiplication need only 80% of the full multiplication time. The faster the multiplication, the less relative time saving can be achieved. [x] denotes the integer part of x, and 0 ≤ {x} < 1 is the fractional part, such that x = [x] + {x} lg n = log 2 n = log n / log 2 LS stands for Least Significant, the low order bit/s or digit/s of a number MS stands for Most Significant, the high order bit/s or digit/s of a number (Grammar) School multiplication, division: the digit-by-digit multiplication and division algorithms, as taught in elementary schools A B, A B denote the MS or LS half of the digit-sequence of A×B (or A·B), respectively A⊗B denotes the middle third of the digit-sequence of A×B M α (n) the time complexity of the Toom-Cook type full multiplication, O(n α ), with 1< α ≤ 2 γ α = the speedup factor of the half multiplication, relative to M α (n) δ α = the speedup factor of the middle-third product, relative to M α (n)

Efficient Modular Arithmetic for SIMD Devices

“…Due to the missing carry bits from the lower half product, a method similar to Algorithm 6 is hard to achieve for the more significant bits of a multiplication. First attempts where to calculate some additional guard bits to correct the error of missing carry bits [13].…”