论文信息 - Weak bounded arithmetic, the Diffie-Hellman problem and Constable's Class K

Weak bounded arithmetic, the Diffie-Hellman problem and Constable's Class K

The bounded arithmetic theory C/sub 2//sup 0/, which is closely related to the complexity class DLogTime-uniform TC/sup 0/, is extended by a function symbol and axioms for integer division, which is not known to be in DLogTime-uniform TC/sup 0/. About this extended theory C/sub 2//sup 0/[div], two main results are proved: (1). The Z/sub 1//sup b/-definable functions of C/sub 2//sup 0/[div] are exactly Constable's class K, a function algebra whose precise complexity-theoretic nature is yet to be determined. This also yields the new upper bound K/spl sube/uniform NC/sup 2/. (2). The /spl Delta//sub 1//sup b/-theorems C/sub 2//sup 0/[div] do not have Craig-interpolants of polynomial circuit size, unless the Diffie-Hellman key exchange protocol is insecure.

Jan Johannsen

[1] Jan Johannsen. A bounded arithmetic theory for constant depth threshold circuits , 1996 .

[2] 外史竹内. Bounded Arithmetic と計算量の根本問題 , 1996 .

[3] Peter Clote. A Note on the Relation Between Polynomial Time Functionals and Constable's Class K , 1995, CSL.

[4] Robert L. Constable,et al. Type two computational complexity , 1973, STOC.

[5] John H. Reif. Logarithmic Depth Circuits for Algebraic Functions , 1986, SIAM J. Comput..

[6] Stephen A. Cook,et al. Log Depth Circuits for Division and Related Problems , 1984, SIAM J. Comput..

[7] Jan Krajícek,et al. Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[8] Vaughan R. Pratt,et al. Every Prime has a Succinct Certificate , 1975, SIAM J. Comput..

[9] Jan Johannsen,et al. On proofs about threshold circuits and counting hierarchies , 1998, Proceedings. Thirteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.98CB36226).

[10] Jan Krajícek,et al. Some Consequences of Cryptographical Conjectures for S12 and EF , 1998, Inf. Comput..

[11] P. Clote,et al. First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes , 1995 .

[12] Ran Raz,et al. No feasible interpolation for TC/sup 0/-Frege proofs , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.