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The arithmetic of Jacobian groups of superelliptic cubics. (English) Zbl 1094.14049
Let \(K\) denote a perfect field of characteristic different from 3. A superelliptic cubic over \(K\) has affine equation \(Y^3=f(X)\), where \(f\in K[X]\) is a separable polynomial with degree at least 4 and not a multiple of 3. The authors give algorithms for addition in the Jacobian of such a curve. Because of the existence of subexponential attacks on the discrete logarithm problem in the Jacobian for curves of moderately high genus, the authors are mainly interested in curves of genus 3 and 4 (so \(\deg f=4\) or 5). Let \(C\) be a superelliptic cubic of genus \(g\) over \({\mathbb F}_q\). A divisor on \(C\) is called typical if it corresponds to an ideal of the form \((u,Y-v)\), where \(u,v\in K[X], \deg v<\deg u\leq g\) and \(u| v^3-f\). (The key here is that the second generator is linear in \(Y\).) The authors show that the probability that a randomly chosen reduced divisor is typical is \(1-(1/q)O(1)\). Thus, if \(q\) is large, then all but a negligible proportion of reduced divisors are typical. They also show that if \(g=3\) or 4, then a typical divisor \((u,Y-v)\) is reduced whenever \(\deg u<g\) or \(\deg u=g\) and \(\deg v=g-1\). By restricting to typical divisors, the authors can give addition algorithms that are simpler than the more general algorithms devised by M. L. Bauer [Math. Comput. 73, No. 245, 387–413 (2004; Zbl 1053.11087)] and R. Scheidler [J. Théor. Nombres Bordx. 13, No. 2, 609–631 (2001; Zbl 0995.11064)]. The authors give two algorithms for reducing the composition of two divisors: one inspired by D. Cantor’s algorithm for hyperelliptic curves [Math. Comput. 48, 95–101 (1987; Zbl 0613.14022)], and a second based on Gröbner bases, which uses the FGLM algorithm for switching between Gröbner bases of a zero-dimensional ideal with respect to different orderings. This use of Gröbner bases allows one to carry out the computations symbolically, thus obtaining explicit formulas for the reduced ideal in terms of the coefficients of the input polynomials. Another algorithm using Gröbner bases has been given by S. Arita [Discrete Appl. Math. 130, No. 1, 13–31 (2003; Zbl 1037.14010)], and another addition algorithm in the case of genus 3 has been given by S. Flon and R. Oyono [Fast arithmetic on Jacobians of Picard curves. in: Public key cryptography – PKC 2004, Lect Notes Comput. Sci. 2947, 55–68 (2004)]. An example is given here for a curve of genus 3 over the field \({\mathbb F}_{2003}\), and more explicit results on the implementation of their work in genus 3 appear in their paper [in: Algorithmic number theory. 6th int. symp., ANTS-VI, Burlington, VT, USA, June 13–18, 2004. Lect. Notes Comput. Sci. 3076, 87–101 (2004; Zbl 1148.14304)]

MSC:
14Q05 Computational aspects of algebraic curves
11G20 Curves over finite and local fields
14H40 Jacobians, Prym varieties
14H45 Special algebraic curves and curves of low genus
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
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