Multiprecision Multiplication on ARMv8

Multiplication of large integers is a fundamental operation for public key cryptography. In contemporary public key cryptography, the sizes of integers are typically from more than one hundred bits to even several thousands of bits. Because these sizes exceed the bit widths of all general-purpose processors, these multiplications must be performed with a multiprecision multiplication algorithm which splits the operation into multiple partial products and accumulation steps. To ensure efficiency, multiprecision multiplication algorithms must be designed with special care and optimized for the instruction sets of specific processors. Consequently, developing efficient multiprecision multiplication algorithms and optimizing them for specific platforms has been an active research topic. In this paper, we optimize multiprecision multiplication and squaring specifically for the 64-bit ARMv8 processors which are widely used, for example, in modern smart phones and tablets. We combine the subtractive Karatsuba algorithm, operand-scanning techniques (for multiplication) and sliding-block-doubling methods (for squaring) to accelerate the performance of the 256-bit multiprecision multiplication and squaring by 7.6% and 7.0% compared to the OpenSSL implementations. We focus particularly on the multiprecision multiplications that are required in elliptic curve cryptography. Our implementation supports general elliptic curves of various sizes and all source codes are available in public domain.