José Felipe Voloch

Diagonal equations over function fields

LetK be a function field in one variable over ℂ anda1,...,am,b non-zero elements ofK, such thatb is linearly independent froma1,...,am over ℂ. We show that forn sufficiently large, the equation ∑i=1maixin has no non-constant solutions inK.

Efficient Computation of Roots in Finite Fields

We present an algorithm to compute rth roots in

The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields

\mathbb{F}_{q^m}

THE CHARACTERIZATION OF ELLIPTIC CURVES OVER FINITE FIELDS

with complexity Õ[(log m + r log q) m log q] if (m,q) = 1 and either (q(q−1),r) = 1 or r|(q−1) and ((q−1)/r,r) = 1. This compares well to previously known algorithms, which need O(rm3 log3q) steps.

MULTIPLICATIVE ORDER OF GAUSS PERIODS

At Crypto 2010, Brier et al. proposed the first construction of a hash function into ordinary elliptic curves that was indifferentiable from a random oracle, based on Icart’s deterministic encoding from Crypto 2009. Such a hash function can be plugged into any cryptosystem that requires hashing into elliptic curves, while not compromising proofs of security in the random oracle model. However, the proof relied on relatively involved tools from algebraic geometry, and only applied to Icart’s deterministic encoding from Crypto 2009. In this paper, we present a new, simpler technique based on bounds of character sums to prove the indifferentiability of similar hash function constructions based on essentially any deterministic encoding to elliptic curves or curves of higher genus, such as the algorithms by Shallue, van de Woestijne and Ulas, or the Icart-like encodings recently presented by Kammerer, Lercier and Renault. In particular, we get the first constructions of well-behaved hash functions to Jacobians of hyperelliptic curves. Our technique also provides more precise estimates on the statistical behavior of those deterministic encodings and the hash function constructions based on them. Additionally, we can derive pseudorandomness results for partial bit patterns of such encodings.

Euclidean weights of codes from elliptic curves over rings

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.

DIOPHANTINE APPROXIMATION ON ABELIAN VARIETIES IN CHARACTERISTIC p

Consider hypersurfaces of fixed degree d in a fixed projective space Pn over ℚ. We present a conjecture about the fraction of these that have rational points, and present evidence for the conjecture, including a proof that a positive fraction of the hypersurfaces have points over every completion of ℚ, provided that n, d ≥ 2 and (n, d) ≠ (2, 2). Generalizations to number fields are discussed. One of our proofs uses a result of Colliot-Thelene, proved in an appendix, that there is no Brauer-Manin obstruction to the Hasse principle for smooth complete intersections of dimension ≥ 3 in projective space over number fields. Colliot-Thelene’s proof, in turn, uses a consequence of the Weak Lefschetz Theorem proved in an appendix by Katz.

On hashing into elliptic curves

J. W. P. HIRSCHFELD and J. F. VOLOCH(Received 11 March 1987)Communicated by R Lid. lAbstractIn a finite Desarguesian plan oef odd order i,t was shown by Segre thirty years ago tha a set tof maximum size wit aht most two point os n a line is a conic. Here, in a plane of odd or evenorder, sufficient condition are givesn for a set with at most three point os n a line to be a cubiccurve. The case of an elliptic curv ies of particular interest.1980 Mathematics subject classification (Amer. Math. Soc.) (198 Revision):5 51 E 15, 14 H 25.0. NotationGF(q) th finite field oef q elementsPG(2, q) th projectivee plane over GF(q)PG(V(2, q) the se linet ofs in PG(2, q)P(X) th point of PC?(2e, q) with coordinate vecto X rUQ P(l,0,0),UI P(0,l,0)U