The inverse of a number modulo is a number such that . It exists (and is unique if exists) if and only and are relatively prime (that is, ). In particular, if is a prime, every non-zero element of has an inverse (thus making it an algebraic structure known as field).
Conventionally, the mathematical notation used for inverses is .
In modular arithmetic the inverse of is analogous to the number in usual real-number arithmetic. If you have a product , and one of the factors has an inverse, you can get the other factor by multiplying the product by that inverse: . Thus you can perform division in ring .
Finding the inverse
We can rewrite the defining equation of modular inverses as an equivalent linear diophantine equation: . This equation has a solution whenever , and we can find such solution by means of the extended Euclidean algorithm.
Then , and also .
The following Python code implements this algorithm.
# Iterative Algorithm (xgcd) def iterative_egcd(a, b): x,y, u,v = 0,1, 1,0 while a != 0: q,r = b//a,b%a; m,n = x-u*q,y-v*q # use x//y for floor "floor division" b,a, x,y, u,v = a,r, u,v, m,n return b, x, y # Recursive Algorithm def recursive_egcd(a, b): """Returns a triple (g, x, y), such that ax + by = g = gcd(a,b). Assumes a, b >= 0, and that at least one of them is > 0. Bounds on output values: |x|, |y| <= max(a, b).""" if a == 0: return (b, 0, 1) else: g, y, x = recursive_egcd(b % a, a) return (g, x - (b // a) * y, y) egcd = iterative_egcd # or recursive_egcd(a, m) def modinv(a, m): g, x, y = egcd(a, m) if g != 1: return None else: return x % m
And so you can utilize repeated squaring algorithm to quickly find the inverse.
This algorithm can be useful if is a fixed number in your program (so, you can hardcode a precomputed value of ), or if is a prime number, in which case . In general case, however, computing is equivalent to factoring, which is a hard problem, so prefer using the extended GCD algorithm.
Suppose we need to calculate . If and are co-primes (or if one of them is a prime), then we can calculate the modular inverse of .