Chris Charnes

Attacking the SL2 Hashing Scheme

Tillich and Zemor proposed a definition of a new hash function which uses SL2(2, 2n), the group of unimodular two-dimensional matrices with entries in GF(2n, to compute the hash values of binary strings. This hashing scheme has several attractive features; hash values can be computed quickly and small modifications to the input text can be detected. However, this scheme is weak. Using a group theoretic argument we are able to produce two distinct binary strings of small length which hash to the same value in SL2(2,2n) for the range specified by Tillich and Zemor.

Conditionally secure secret sharing schemes with disenrollment capability

The paper describes an implementation of Shamir secret sharing schemes based on exponentiation in Galois fields. It is shown how to generate shares so the scheme has the disenrollment capability. Next a family of conditionally secure Shamir schemes is defined and the disenrollment capability is investigated for the family. The paper also examines a problem of covert channels which are present in any secret sharing scheme.

Homogeneous bent functions of degree n in 2n variables do not exist for n > 3

We prove that homogeneous bent functions f:GF(2)^2n --> GF(2) of degree n do not exist for n>3. Consequently homogeneous bent functions must have degree 3.

The translation planes of order 49 and their automorphism groups

Using isomorphism invariants, we enumerate the translation planes of order 49 and determine their automorphism groups.

Linear Nonequivalence versus Nonlinearity

The choice of a collection of cryptographically strong Boolean functions is crucial in designing a strong hashing algorithm. The paper shows that it is possible to obtain five linearly nonequivalent functions with five Boolean variables which are cryptographically strong and easy to implement. They can be readily used to design hashing algorithms (of the MD5 structure).

On Homogeneous Bent Functions

A new surprising connection between invariant theory and the theory of bent functions is established. This enables us to construct Boolean function having a prescribed symmetry given by a group action. Besides the quadratic bent functions the only other known homogeneous bent functions are the six variable degree three functions constructed in [14]. We show that these bent functions arise as invariants under an action of the symmetric group on four letters. Extending to more variables we apply the machinery of invariant theory to construct previously unknown homogeneous bent functions of degree three in 8 and 10 variables. This approach gives a great computational advantage over the unstructured search problem. We finally consider the question of linear equivalence of the constructed bent functions.

Secret sharing in hierarchical groups

We introduce the idea of hierarchical delegation within a secret sharing scheme and consider solutions with both conditional and unconditional security.

Cryptosystems for hierarchical groups

This paper addresses the problem of information protection in hierarchical groups. Higher level groups of participants can control the information flow (the decryption ability) to lower level groups. If a higher level group decides to allow a lower level group to read the message, it passes a go ahead ticket so the lower level group can decrypt the cryptogram and read the message. The formal model of top-down hierarchical cryptosystems is given.

Families of threshold schemes

The notion of families of ideal threshold schemes (ITS) is introduced, their properties and applications are investigated. We consider secret sharing schemes whose parameters can be adjusted. Our model is a threshold scheme family (TSF) whose threshold scheme parameters can be modified dynamically. Some applications of this model includes schemes which have disenrollment capability, and a scheme to resist cheating.<<ETX>>

The eight dimensional ovoids over GF(5)

In this article we outline a computer assisted classification of the ovoids in an orthogonal space of the type Ω + (8,5).