Abstract. This paper compares the XL algorithm with known Gröbner basis algorithms. We show that to solve a system of algebraic equations via the XL algorithm is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is also a Gröbner basis algorithm which can be represented as a redundant variant of a Gröbner basis algorithm F4. Then we compare these algorithms on semi-regular sequences, which correspond, in conjecture, to almost all polynomial systems in two cases: over the fields F2 and Fq with q n. We show that the size of the matrix constructed by XL is large compared to the ones of the F5 algorithm. Finally, we give an experimental study between XL and the Buchberger algorithm on the cryptosystem HFE and find that the Buchberger algorithm has a better behavior.

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Abstract. Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2 = x 5 + ax over given large prime fields Fp, e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l · c where l is a prime number greater than about 2 160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2 = x 5 + a where a is not square in Fp.

We clarify a relation between the XL algorithm and known Gröbner basis algorithms. The XL algorithm was proposed to be a more efficient algorithm to solve a system of algebraic equations under a special condition, without calculating a whole Gröbner basis. But in our result, it is shown that to solve a system of algebraic equations with a special condition under which the XL algorithm works is equivalent to calculate the reduced Gröbner basis of the ideal associated with the system. Moreover we show that the XL algorithm is a Gröbner basis algorithm which can be represented as a redundant variant of a known Gröbner basis algorithm F 4 .

Abstract. In 2004, a new attack against SHA-1 has been proposed by a team leaded by Wang [15]. The aim of this article 1 is to sophisticate and improve Wang's attack by using algebraic techniques. We introduce new notions, namely semi-neutral bit and adjuster and propose then an improved message modification technique based on algebraic techniques. In the case of the 58-round SHA-1, the experimental complexity of our improved attack is 2 31 SHA-1 computations, whereas Wang's method needs 2 34 SHA-1 computations. We have found many new collisions for the 58-round SHA-1. We also study the complexity of our attack for the full SHA-1.