模平方根和模因式分解
Integer factoring and modular square roots
摘要：
Buresh-Oppenheim证明了寻找特殊形式的整数的非平凡因子的NP搜索问题属于Papadimitriou的PPA类，并且可以概率还原为PPP中的问题。在本文中，我们使用有界算术的思想，将这些结果推广到任意整数。我们表明，一般整数因子分解是可减少的随机多项式时间的PPA问题和PPP的问题WEAKPIGEON。在广义黎曼假设的假设下，这两种简化都可以去随机化。我们还表明（无条件），PPA包含一些相关的问题，如平方根计算模n，并找到二次nonresidues模n。
英文摘要：
Buresh-Oppenheim proved that the NP search problem to find nontrivial factors of integers of a special form belongs to Papadimitriou's class PPA, and is probabilistically reducible to a problem in PPP. In this paper, we use ideas from bounded arithmetic to extend these results to arbitrary integers. We show that general integer factoring is reducible in randomized polynomial time to a PPA problem and to the problem WEAKPIGEON in PPP. Both reductions can be derandomized under the assumption of the generalized Riemann hypothesis. We also show (unconditionally) that PPA contains some related problems, such as square root computation modulo n, and finding quadratic nonresidues modulo n.
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