Applications of Fast Truncated Multiplication in Cryptography

Truncated multiplications compute truncated products, contiguous subsequences of the digits of integer products. For an n-digit multiplication algorithm of time complexity O(nα), with 1 < α ≤ 2, there is a truncated multiplication algorithm, which is constant times faster when computing a short enough truncated product. Applying these fast truncated multiplications, several cryptographic long integer arithmetic algorithms are improved, including integer reciprocals, divisions, Barrett and Montgomery multiplications, 2n-digit modular multiplication on hardware for n-digit half products. For example, Montgomery multiplication is performed in 2.6 Karatsuba multiplication time.