Chapter 9 - Number Theoretic Algorithms

Abstract

This chapter discusses several fundamental number theoretic algorithms, such as the greatest common divisor, least common multiple, and Jacobi symbol computation. These algorithms arise as essential components in several key cryptographic algorithms, such as the RSA public key algorithm and various sieve-based factoring algorithms. Furthermore, the algorithm mp_gcd produces the greatest common divisor of two mp_ints a and b. It was originally based on Algorithm B, of Knuth, but has been modified to explain in a simpler way. In theory, it achieves the same asymptotic working time as Algorithm B, and in practice this appears to be true. The first two steps handle the cases where either one or both inputs are zero. If either input is zero, the greatest common divisor is the largest input or zero if they are both zero.