【量子计算-算法】shor大数分解

shor大数分解算法

【量子计算-算法】shor大数分解

Classical part[edit]

Pick a random number a < N.
Compute gcd(a, N). This may be done using the Euclidean algorithm.
If gcd(a, N) ≠ 1, then this number is a nontrivial factor of N, so we are done.
Otherwise, use the period-finding subroutine (below) to find r, the period of the following function:
f(x)=axmodN,{\displaystyle f(x)=a^{x}{\bmod {N}},}
i.e. the order r{\displaystyle r} of a{\displaystyle a} in (ZN)×{\displaystyle (\mathbb {Z} _{N})^{\times }}, which is the smallest positive integer r for which f(x+r)=f(x){\displaystyle f(x+r)=f(x)}, or f(x+r)=ax+rmodN≡axmodN.{\displaystyle f(x+r)=a^{x+r}{\bmod {N}}\equiv a^{x}{\bmod {N}}.}
If r is odd, go back to step 1.
If a ^r /2 ≡{\displaystyle \equiv } −1 (mod N), go back to step 1.
gcd(a^r/2 + 1, N) and gcd(a^r/2 - 1, N) are both nontrivial factors of N. We are done.

For example: N=15,a=7,r=4{\displaystyle N=15,a=7,r=4} 【量子计算-算法】shor大数分解 , gcd(72±1,15)=gcd(49±1,15){\displaystyle \mathrm {gcd} (7^{2}\pm 1,15)=\mathrm {gcd} (49\pm 1,15)}, where gcd(48,15)=3{\displaystyle \mathrm {gcd} (48,15)=3}, and gcd(50,15)=5{\displaystyle \mathrm {gcd} (50,15)=5}.

【量子计算-算法】shor大数分解

参考文献：

[1] Shor P W. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer[M]. Society for Industrial and Applied Mathematics, 1997.

【量子计算-算法】shor大数分解

Classical part[edit]

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