Private Key Extension of Polly Cracker Cryptosystems

In 1993 Koblitz and Fellows proposed a public key cryptosystem, Polly Cracker, based on the problem of solving multivariate systems of polynomial equations, which was soon generalized to a Dröbner basis formulation. Since then a handful of improvements of this construction has been proposed. In this paper it is suggested that security, and possibly efficiency, of any Polly Cracker-type cryptosystem could be increased by altering the premises regarding private- and public information

Exponential Time Complexity of the Permanent and the Tutte Polynomial

We show conditional lower bounds for well-studied #P-hard problems: (a) The number of satisfying assignments of a 2-CNF formula with n variables cannot be counted in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. (b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). (c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting

Exponential time complexity of the permanent and the Tutte polynomial

We show conditional lower bounds for well-studied #P-hard problems: ◦ The number of satisfying assignments of a 2-CNF formula with n variables cannot be computed in time exp(o(n)), and the same is true for computing the number of all independent sets in an n-vertex graph. ◦ The permanent of an n × n matrix with entries 0 and 1 cannot be computed in time exp(o(n)). ◦ The Tutte polynomial of an n-vertex multigraph cannot be computed in time exp(o(n)) at most evaluation points (x, y) in the case of multigraphs, and it cannot be computed in time exp(o(n / poly log n)) in the case of simple graphs. Our lower bounds are relative to (variants of) the Exponential Time Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF formulas cannot be decided in time exp(o(n)). We relax this hypothesis by introducing its counting version #ETH, namely that the satisfying assignments cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds, we transfer the sparsification lemma for d-CNF formulas to the counting setting