Shrinking generators and statistical leakage

AbstractShrinking is a newly proposed technique for combining a pair of pseudo random binary sequences, (a,s), to form a new sequence, z, with better randomness, where randomness here stands for difficulty of prediction. The ones in the second sequence s are used to point out the bits in the sequence a to be included in z. The generator that performs this process is known as the shrinking generator (SG). In this paper, it is shown for the existing combining method that deviation from randomness in the statistics of a leads to the leakage of this statistics into z. We also show that it is sufficient for constructing a statistically balanced SG to at least have one statistically balanced generator. A new shrinking rule that yields statistically balanced output, even if a and s are not balanced, is then proposed. Self-shrinking in which a single pseudo random bit generator (PRBG) shrinks itself is also investigated and a modification of the existing shrinking rule is proposed. Simulation results show the robustness of the proposed methods. For self-shrinking, in particular, results show that the proposed shrinking rule yields sequences with balanced statistics even for extremely biased generators. This suggests possible application of the new rule to strengthen running key generators

Linear solutions for cryptographic nonlinear sequence generators

This letter shows that linear Cellular Automata based on rules 90/150 generate all the solutions of linear difference equations with binary constant coefficients. Some of these solutions are pseudo-random noise sequences with application in cryptography: the sequences generated by the class of shrinking generators. Consequently, this contribution show that shrinking generators do not provide enough guarantees to be used for encryption purposes. Furthermore, the linearization is achieved through a simple algorithm about which a full description is provided

Moduli of roots of line bundles on curves

We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.Comment: Final version, added references. To appear in Trans. Amer. Math. So

Toric algebra of hypergraphs

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in algebraic statistics and to combinatorial discrepancy. Section 6 (open problems) has been moderately revise