8 research outputs found
The second moment of twisted modular L-functions
We prove an asymptotic formula with a power saving error term for the (pure
or mixed) second moment of central values of L-functions of any two (possibly
equal) fixed cusp forms f, g twisted by all primitive characters modulo q,
valid for all sufficiently factorable q including 99.9% of all admissible
moduli. The two key ingredients are a careful spectral analysis of a
potentially highly unbalanced shifted convolution problem in Hecke eigenvalues
and power-saving bounds for sums of products of Kloosterman sums where the
length of the sum is below the square-root threshold of the modulus.
Applications are given to simultaneous non-vanishing and lower bounds on higher
moments of twisted L-functions.Comment: 64 page
Some Facets of Complexity Theory and Cryptography: A Five-Lectures Tutorial
In this tutorial, selected topics of cryptology and of computational
complexity theory are presented. We give a brief overview of the history and
the foundations of classical cryptography, and then move on to modern
public-key cryptography. Particular attention is paid to cryptographic
protocols and the problem of constructing the key components of such protocols
such as one-way functions. A function is one-way if it is easy to compute, but
hard to invert. We discuss the notion of one-way functions both in a
cryptographic and in a complexity-theoretic setting. We also consider
interactive proof systems and present some interesting zero-knowledge
protocols. In a zero-knowledge protocol one party can convince the other party
of knowing some secret information without disclosing any bit of this
information. Motivated by these protocols, we survey some complexity-theoretic
results on interactive proof systems and related complexity classes.Comment: 57 pages, 17 figures, Lecture Notes for the 11th Jyvaskyla Summer
Schoo