Daniel Lieman

On the Distribution of the RSA Generator

Let 19, m and e be integers such that gcd(19, m) = 1. Then one can define the sequence (un) by the recurrence relation

The Distribution of the Quadratic Symbol in Function Fields and a Faster Mathematical Stream Cipher

{{u}_{n}} \equiv u_{{n - 1}}^{e}\left( {\,\bmod \,m} \right),0{{u}_{n}}m - 1,n = 1,2,...,

On The Correlation Of Binary M -sequences

(1) with theinitial value \({{u}_{0}} = \nu \).

Non-linear Complexity of the Naor-Reingold Pseudo-random Function

All the notation in this appendix will be as in the preceding paper. Let f be a Maass form which is a newform for Fo(N), with eigenvalue A and central character X, normalized so that (f, f) = 1. We have seen that the size of p(l), the first Fourier coefficient of f, is intimately related to the behavior of L(s, F) near s = 1. Here L(s, F) is the L-series of F, the adjoint square lift of f to GL(3), and the crucial question is whether or not L(s, F) vanishes when s is real and close to 1. It was shown in Theorem 0.1 that if L(s, F) is nonzero in a sufficiently wide neighborhood of 1 then Ip(1)12 0, where the implied constant depends on E and is ineffective. About a year after the preceding paper was first circulated, it developed from conversations involving the above three authors that by slightly modifying the techniques introduced in that paper the possibility of a Siegel zero could be completely eliminated in many cases. In particular, Theorem 0.1 is now true unconditionally in the generic situation when f is not a lift from GL(1), that is to say, when the L-series of f is not equal to a Hecke L-series defined over a quadratic field. If we include all cusp forms, we can still obtain the Theorem in the A-aspect, but must restrict ourselves to either an ineffective constant or a weaker effective constant in the N-aspect. We have:

Whittaker–Fourier Coefficients of Metaplectic Eisenstein Series

We present a stream cipher based on mathematical considerations which is much faster then many other mathematical ciphers. Its security is based on the uniformity of the distribution of the quadratic symbol in function fields.

An Identification Scheme Based on Sparse Polynomials

This paper introduces a new type of cryptosystem which is based on sparse polynomials over finite fields. We evaluate its theoretic characteristics and give some security analysis. Some preliminary timings are presented as well, which compare quite favourably with published optimized RSA timings. We believe that similar ideas can be used in some other settings as well.

On a New Exponential Sum

We obtain the upper bound O(214n/15 n−1/5) on the number of distinct values of all possible correlation functions between M-sequences of order n .

Cryptographic Applications of Sparse Polynomials over Finite Rings

We obtain an exponential lower bound on the non-linear complexity of the new pseudo-random function, introduced recently by M. Naor and O. Reingold. This bound is an extension of the lower bound on the linear complexity of this function that has been obtained by F. Griffin and I. E. Shparlinski.