Fermat Primality Test: Let n >1 be an integer. Choose a random integer a with 1 < a < n-1.
If a ^n-1≢ 1(mod n) then n is composite. If a^n-1 ≡ 1 (mod n), then n is probably prime.

 If we are careful about how we do this successive squaring, the Fermat test can be combined with the basic principle to yield the following stronger result.

Miller- Rabin Primality test:

Let n >1 be an odd integer. Write a-1 =2^km with m odd. Choose a random integer a with 1< a< n-1.  Compute b₀ ≡ a^m (mod n). If b₀ ≡ ±1 (mod n), then stop and declare that n is probably prime. Otherwise, let b₁ ≡ b₀² (mod n). If b₁ ≡ 1 (mod n), then n is composite (and gcd (b₀ -1,n) gives a nontrivial factor of n ). If b₁ ≡ -1 (mod n), then stop and declare that n is probably prime. Otherwise, let b₂ ≡ b₁² (mod n). If b₂ ≡ 1 (mod n), then n is composite. If b₂ ≡ -1 (mod n), then stop and declare that n is probably prime. Continue in this way until stopping and reaching  b_k-1. If b_k-1≢ -1 (mod n), then n is composite.

The reason why the test works is- suppose, for example that b₃≡1 (mod n). This means that b₂²≡1 (mod n). This means that b₂²≡1² (mod n). Apply the basic principle from before. Either b₂≡±1 (mod n), or b₂≢ ±1 (mod n) and n is composite. In the latter case, gcd (b₂-1, n) give a nontrivial factor of n. In the former case, the algorithm would have stopped by the previous step.

MILLER-RABIN (n, s)

1. For j← 1 to s
2.      do a ← RANDOM(1,n-1)
3.            If WITNESS (a,n)
4.                 then return COMPOSITE
5. return PRIME

WITNESS (a, n)

• Let <b_k,b_k-1…b₀ > be the binary representation of n-1.
• d←1
•     for I ← k down to 0
•          do x ← d
•                d←(d. d) mod n
•                if d=1 and x≠1 and x≠n-1
•                   then return TRUE
•               if(b_i=1) then
•                   d← (d. a) mod n
•          end for
•          if d≠1
•               then return TRUE
•          return FALSE