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VII. HYPERELLIPTIC CURVES in .NET Encoder QR-Code in .NET VII. HYPERELLIPTIC CURVES




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NET ulus Algorithm for Hyperelliptic Curves The original algorithm by Adleman, DeMarrais and Huang is highly heuristic and more theoretical than of practical use. Their idea was to work with functions on the curve. This idea was pushed further by Flassenberg and Paulus [121] who were able to design a sieving technique, and they did the rst implementation of this kind of algorithm.

Later on, M ller, Stein and u Thiel [257] and Enge [116] managed to get rid of the heuristics, by working with elements of the Jacobian instead of functions, so that Theorem VII.8 could be applied. The method we shall now describe is the further improvement by Enge and Gaudry [117].

We give a version which is close to what is really done when it is implemented; there exists also a more complicated version that has a heuristic-free complexity. We rst present a sketch of the algorithm, and then details about each phase are given. Algorithm VII.

4: Hyperelliptic Index-Calculus Algorithm A divisor D1 in JC (Fq ) with known order N = ord(D1 ), and a divisor D2 D1 OUTPUT: An integer such that D2 = D1 INPUT: 1. 2. 3.

4. 5. 6.

Fix a smnoothness bound and construct the factor basis. While not enough relations have been found do: Pick a random element R = D1 + D2 . If R is smooth, record the corresponding relation.

Solve the linear algebra system over Z/N Z. Return ..

VII.5.1.

Construc tion of the Factor Basis. First a smoothness bound B is chosen. In the complexity analysis, we shall explain how to get the best value for B.

Then the factor basis F contains all the prime reduced divisors of weight at most B: F = {P JC : P is prime, wt(P ) B}. This set can be constructed in a naive way: for each monic polynomial of degree at most B, check if it is irreducible and if it is the a-polynomial of a reduced divisor. In that case, nd all the compatible b-polynomials and add the corresponding divisors to F.

For convenience, we give names to the elements of F: F = {gi : i [1, #F]}.. VII.5. INDEX-CALCULUS ALGORITHM FOR HYPERELLIPTIC CURVES VII.5.2.

A Pseudo -Random Walk. Selecting a random element R = D1 + D2 is costly: the values of and are randomly chosen in the interval [1, N ] and then two scalar multiplications have to be done. Using the binary powering method, the cost is O(log N ) group operations.

We use a pseudo-random walk instead, so that the construction of each new random element costs just one group operation. The pseudo-random walk is exactly the same as the one which is used in [323] in discrete log algorithms based on the birthday paradox. For j from 1 to 20 we randomly choose aj and bj in [1, N ] and compute the multiplier Tj aj D1 + bj D2 .

In our case where the group is traditionally written additively, summand would be a better name but we stick to the classical name. The pseudo-random walk is then de ned as follows: R0 is given by 0 D1 + 0 D2 where 0 and 0 are randomly chosen in [1, N ]. Ri+1 is de ned by adding to Ri one of the multipliers Tj .

The index j is given by the value of a hash function evaluated at Ri . Then the representation of Ri+1 in terms of D1 and D2 is deduced from the corresponding represenation of Ri : we set i+1 = i + aj and i+i = i + bj for the same j. Practical experiments [323] suggest that by taking 20 multipliers the pseudo-random walk behaves almost like a purely random walk.

In our case, it is not really necessary to take j deterministic in terms of Ri : picking a random multiplier at each step would do the job. The determinism in the pseudo-random walk was necessary in [323] to use distinguished points, but it is not used in our algorithm. VII.

5.3. Collection of Relations.

Each time we produce a new random element R = D1 + D2 , we test it for smoothness. If R is not smooth, then we continue the pseudo-random walk to get another element. The smoothness test is done by factoring the a-polynomial of R.

If all its irreducible factors have degree at most B, then Lemma VII.4 allows us to write R as a sum of elements of F. The kth smooth element Sk = k D1 + k D2 that is found is stored in the kth column of a matrix M = (mik ) which has #F rows: Sk =.

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