By M. Brown

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M b1 m b2 m If two such fractions cannot be found, return FAIL. 4. Let t = lcm(b1 , b2 ). If t > 5. If f (0) = f (t), return FAIL. 6. Return t. m , 2 return FAIL. 4. 12. 11 finds the correct value of r with probability at least 32 . π4 If it does not return FAIL, it returns a multiple of r. Proof. 8 twice independently, obtaining results yk1 and yk2 . 7, y ki m − ki r ≤ 1 m k1 r and k2 , r respectively. with probability at least 8 π2 for i = 1, and indepen- dently for i = 2. The probability that the inequality is satisfied for both i = 1 and i = 2 is therefore at least 64 .

Otherwise, take r to be the minimum non-FAIL output. 4. If r is odd, return FAIL. 5. Let t = gcd(ar/2 − 1, n). If t = 1, return FAIL. 6. Return t. 14. 13 correctly returns a non-trivial factor of n with probability at least 31 . Proof. For any integer a ∈ {0, 1, . . , n − 1} which is coprime with n and whose order in 34 CHAPTER 3. INTRODUCTION TO QUANTUM ALGORITHMS Z∗n is r, we know that ar ≡ 1 (mod n) ar − 1 ≡ 0 (mod n) (ar/2 − 1)(ar/2 + 1) ≡ 0 (mod n) (if r is even). Since r is the order of a, we know that ar/2 − 1 ≡ 0 (mod n).

It is easily verified that the resulting integers are indeed the four square roots of c. Note that if one or both of p or q is congruent to 1 mod 4, the square roots can still be efficiently computed, but the algorithm is more complicated [MvOV96]. For this reason, during the key generation procedure it makes sense to choose the primes p and q to be congruent to 3 mod 4. It is also interesting to note as in [MvOV96] that Rabin encryption is more efficient than RSA encryption, since it requires a single modular squaring operation.