solving ax ≡ b (mod n)

gcd(n,a) = 1?

Find a-1 mod n. Solution is x = ba-1 (mod n).

Find d = gcd(n,a). Does d divide b?

Divide everything (a, b and n) by d to obtain a new equation. Solve it using above method to obtain x0.

Solutions to original equation will be x0, x0 + n/d, x0 + 2(n/d), and so on. There will be d solutions altogether.

Equation has no solutions

Y N

Y N

solving x2 ≡ a (mod n)

Is n prime?

a = 1? Know factors of n?

n ≡ 3 (mod 4) ? x = {1,n-1}

Either x ≡ ±a(n+1)/4 (mod n), or there are no solutions.

Use trial and error. (An efficient algorithm exists, but we won’t learn it in this course).

Split into multiple congruences of relatively prime moduli, solve using CRT

No known efficient algorithm exists. Believed to be a hard problem.

Y

Y Y

Y

N

N

N

N