Circuit Principles and Weak Pigeonhole Variants

This paper considers the relational versions of the surjective and multifunction weak pigeonhole principles for PV , Σ1 and Θ b 2-formulas. We show that the relational surjective pigeonhole principle for Θ2 formulas in S 1 2 implies a circuit block-recognition principle which in turn implies the surjective weak pigeonhole principle for Σ1 formulas. We introduce a class of predicates corresponding to poly-log length iterates of polynomial-time computable predicates and show that over R 2 , the multifunction pigeonhole principle for such predicates is equivalent to an “iterative” circuit block-recognition principle. A consequence of this is that if R 3 proves this circuit iteration principle then RSA is vulnerable to quasi-polynomial time attacks. Id: php.tex,v 1.137 2004/08/30 18:29:27 ndanner Exp