Fast Truncated Multiplication and its Applications in Cryptography

Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 long integers. We review a few truncated multiplication algorithms and adapt them to integers. They are a constant times faster than n-digit full multiplications of time complexity O(n), with 1< α ≤ 2, important in cryptography. For example, the least significant half products with Karatsuba multiplication need 80% of the full multiplication time. The faster the multiplication, the less relative time saving we can achieve. Several improved long integer arithmetic algorithms are presented, including integer reciprocals and divisions, 2n-digit modular multiplication on HW for n-digit half products, Barrettand Montgomerymultiplications. They get further accelerated with application of fast truncated multiplication, like Montgomery multiplication performed in 2.6 Karatsuba multiplications time.