On quadratic residue codes and hyperelliptic curves

A long standing problem has been to develop "good" binary linear codes to be used for error-correction. This paper investigates in some detail an attack on this problem using a connection between quadratic residue codes and hyperelliptic curves. One question which coding theory is used to attack is: Does there exist a c<2 such that, for all sufficiently large $p$ and all subsets S of GF(p), we have |X_S(GF(p))| < cp?Comment: 18 pages, no figure

Families of sequences with good family complexity and cross-correlation measure

In this paper we study pseudorandomness of a family of sequences in terms of two measures, the family complexity ($f$-complexity) and the cross-correlation measure of order $\ell$. We consider sequences not only on binary alphabet but also on $k$-symbols ($k$-ary) alphabet. We first generalize some known methods on construction of the family of binary pseudorandom sequences. We prove a bound on the $f$-complexity of a large family of binary sequences of Legendre-symbols of certain irreducible polynomials. We show that this family as well as its dual family have both a large family complexity and a small cross-correlation measure up to a rather large order. Next, we present another family of binary sequences having high $f$-complexity and low cross-correlation measure. Then we extend the results to the family of sequences on $k$-symbols alphabet.Comment: 13 pages. Comments are welcome

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Algebraic Geometric Secret Sharing Schemes over Large Fields Are Asymptotically Threshold

In Chen-Cramer Crypto 2006 paper \cite{cc} algebraic geometric secret sharing schemes were proposed such that the "Fundamental Theorem in Information-Theoretically Secure Multiparty Computation" by Ben-Or, Goldwasser and Wigderson \cite{BGW88} and Chaum, Cr\'{e}peau and Damg{\aa}rd \cite{CCD88} can be established over constant-size base finite fields. These algebraic geometric secret sharing schemes defined by a curve of genus $g$ over a constant size finite field ${\bf F}_q$ is quasi-threshold in the following sense, any subset of $u \leq T-1$ players (non qualified) has no information of the secret and any subset of $u \geq T+2g$ players (qualified) can reconstruct the secret. It is natural to ask that how far from the threshold these quasi-threshold secret sharing schemes are? How many subsets of $u \in [T, T+2g-1]$ players can recover the secret or have no information of the secret? In this paper it is proved that almost all subsets of $u \in [T,T+g-1]$ players have no information of the secret and almost all subsets of $u \in [T+g,T+2g-1]$ players can reconstruct the secret when the size $q$ goes to the infinity and the genus satisfies $\lim \frac{g}{\sqrt{q}}=0$. Then algebraic geometric secret sharing schemes over large finite fields are asymptotically threshold in this case. We also analyze the case when the size $q$ of the base field is fixed and the genus goes to the infinity