Quantum Algorithm for Attacking RSA Based on Fourier Transform and Fixed-Point

Shor in 1994 proposed a quantum polynomial-time algorithm for finding the order r of an element a in the multiplicative group Z;, which can be used to factor the integer n by computing gcd(a;±1, n),and hence break the famous RSA cryptosystem. However, the order r must be even. This restriction can be removed. So in this paper, we propose a quantum polynomial-time fixed-point attack for directly recovering the RSA plaintext M from the ciphertext C, without explicitly factoring the modulus n.Compared to Shor’s algorithm, the order r of the fixed-point C for RSA（e, n） satisfying C;≡C（mod n） does not need to be even.Moreover, the success probability of the new algorithm is at least 4φ（r）/π;r and higher than that of Shor’s algorithm, though the time complexity for both algorithms is about the same.