Computing discrete logarithms in subfields of residue class rings

Recent breakthrough methods \cite{gggz,joux,bgjt} on computing discrete logarithms in small characteristic finite fields share an interesting feature in common with the earlier medium prime function field sieve method \cite{jl}. To solve discrete logarithms in a finite extension of a finite field \F, a polynomial h(x) \in \F[x] of a special form is constructed with an irreducible factor g(x) \in \F[x] of the desired degree. The special form of $h(x)$ is then exploited in generating multiplicative relations that hold in the residue class ring \F[x]/h(x)\F[x] hence also in the target residue class field \F[x]/g(x)\F[x]. An interesting question in this context and addressed in this paper is: when and how does a set of relations on the residue class ring determine the discrete logarithms in the finite fields contained in it? We give necessary and sufficient conditions for a set of relations on the residue class ring to determine discrete logarithms in the finite fields contained in it. We also present efficient algorithms to derive discrete logarithms from the relations when the conditions are met. The derived necessary conditions allow us to clearly identify structural obstructions intrinsic to the special polynomial $h(x)$ in each of the aforementioned methods, and propose modifications to the selection of $h(x)$ so as to avoid obstructions.Comment: arXiv admin note: substantial text overlap with arXiv:1312.167

Computing zeta functions of arithmetic schemes

We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a single prime p in time p^(1/2+o(1)), and another algorithm that computes zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.Comment: 23 pages, to appear in the Proceedings of the London Mathematical Societ

Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.Comment: 10 page

Heat kernel methods for Lifshitz theories

We study the one-loop covariant effective action of Lifshitz theories using the heat kernel technique. The characteristic feature of Lifshitz theories is an anisotropic scaling between space and time. This is enforced by the existence of a preferred foliation of space-time, which breaks Lorentz invariance. In contrast to the relativistic case, covariant Lifshitz theories are only invariant under diffeomorphisms preserving the foliation structure. We develop a systematic method to reduce the calculation of the effective action for a generic Lifshitz operator to an algorithm acting on known results for relativistic operators. In addition, we present techniques that drastically simplify the calculation for operators with special properties. We demonstrate the efficiency of these methods by explicit applications.Comment: 36 pages, matches journal versio