Modular Root

Related Concepts

Euler’s Totient Function; Integer Factoring; Modular Arithmetic; Number Theory; Quadratic Residue; RSA Problem

Definition

In the congruence ${x}^{e} \equiv y{\rm mod}\,\,n$, x is said to be the ${e}^{\mathrm{th}}$ modular root of y with respect to modulus n.

Background

The cases that are of interest to cryptography have gcd(x, n) = gcd(y, n) = 1. Algorithms for finding modular roots are relevant to the security of the RSA cryptosystem.

Theory

Computing modular roots is no more difficult than finding the order of the multiplicative group modulo n. In number theoretic terminology, this value is known as Euler’s totient function, $\phi (n)$, which is defined to be the number of integers in {1, 2, …, n − 1} that are relatively prime to n. If $\gcd (e,\phi (n)) = 1$, then there is either one or zero solutions, depending upon whether y is in the multiplicative subgroup generated by x. Assuming it is, the solution is obtained by raising both sides of the congruence...