.Akiyama and Goto have proposed a cryptosystem based on rational points on curves over function fields (stated in the equivalent form of sections of fibrations on surfaces). It is easy to construct a curve passing through a few given points, but finding the points, given only the curve, is hard. We show how to break their original cryptosystem by using algebraic points instead of rational points and discuss possibilities for changing their original system to create a secure one.
.We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields.
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Integers.We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We prove that for points on a plane curve, one of the coordinates has to have high order. We also discuss a conjecture of Poonen for subvarieties of semiabelian varieties for which our result is a weak special case. Finally, we look at some special cases where we obtain sharper bounds.
.We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.
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To appear in the Boletim da SBM.We view an algebraic curve over Q as providing a one-parameter family of number fields and obtain bounds for the average value of some standard prime ideal counting functions over these families which are better than averaging the standard estimates for these functions.
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Appeared in Bulletin of PAN.For a prime p and an absolutely irreducible modulo p polynomial f(U,V) in Z[U,V] we obtain an asymptotic formulas for the number of solutions to the congruence f(x,y) = a mod p in positive integers x < X, y < Y, with the additional condition gcd(x,y)=1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.
.Notes taken by students during a course of the same title.
.We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
.We discuss some applications of the theory of algebraic curves to the study of S-boxes in symmetric cryptography.
.We study error-correcting codes constructed from projective surfaces over finite fields using the generalized Goppa construction. We obtain bounds for the minimal distance of these codes by understanding how the zero sets of functions on a surface decompose into irreducible components. We also present a decoding algorithm for these codes based on the Luby-Mitzenmacher algorithm for LDPC codes.
.For an elliptic curve over a function field and a subgroup of rank at least six, we prove that the reduction of the subgroup modulo a place v covers the group of points of the curve modulo v for a positive proportion of v's.
.Notes taken by students during a course of the same title.
. Appeared in the Oberwolfach reportsFor infinitely many primes p, the minimal distance of the binary quadratic residue code of length p is O(p/log log p).
.We present an algorithm to compute r-th roots in a finite field with qm elements with complexity O((log m + rlog q)m2(log q)2) for certain choices of m and q.
.Notes taken by students during a course of the same title.
.In this note we give a lower bound for the minimal distance of the double circulant binary quadratic residue codes.
.We describe an algorithm that improves on the standard algorithm for computing the minimal distance of cyclic codes.
.We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational points on a family of genus 2 curves over a finite field.
.We construct (k,n)-arcs in PG(2,q) with k approximately q2/d and n approximately q/d for each divisor d of q-1.
.The main result of this paper is that, in a precise sense, a positive proportion of all hypersurfaces in Pn of degree d defined over Q are everywhere locally solvable, provided that n,d > 1 and (n,d) is not (2,2). This result is motivated by a conjecture discussed in detail in the paper about the proportion of hypersurfaces as above that are globally solvable, i.e., have a rational point.
.Let S be a subset of Fq, the field of q elements and h in Fq[x] a polynomial of degree d>1 with no roots in S. Consider the group generated by the image of {x-s | s in S} in the group of units of the ring Fq[x]/(h). In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective line over finite fields.
Contains the results of the short note "Improvements to AKS".
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.We construct irreducible plane curves over finite fields with p elements, p prime, with degree near p/2 which have d(d+p-1)/2 rational points. We also prove an irreducibility criterion for plane curves.
We give a formula as an exponential sum for a homogeneous weight on Galois rings (or equivalently, rings of Witt vectors) and use this formula to estimate the weight of codes obtained from algebraic geometric codes over rings.
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We give bounds for the minimal distance of duals of binary BCH codes. This is done by bounding the number of points on curves of the type y2-y=f(x) over finite fields of characteristic two.
We investigate some plane curves with many points over Q, finite fields and cyclotomic fields.
.In this note we give a method for computing the differential Galois group of some linear second-order ordinary differential equations using arithmetic information, namely the p-curvatures.
.We prove that a smooth surface in P3 of degree d, defined over a finite field with q elements, q prime, has at most d(d+q-1)(d+2q-2)/6 + d(11d-24)(q+1) rational points.
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pdf file .Notes taken by students during a course of the same title.
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pdf file .The math behind the puzzle Blet.