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Abstract. We study a generalized version of the index calculus method for the discrete logarithm problem in Fq, whenq = p n, p is a small prime and n →∞. The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database. 1.

Abstract. We consider the largest degrees that occur in the decomposi-tion of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discrete logarithm problem. In addition, we characterize the dis-tribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treat-ments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method. 1

Gau periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive rth root of unity, one obtains under certain conditions a normal basis for F q n over F q , where r is a prime and nk = r \Gamma 1 for some integer k. We generalize this construction by allowing arbitrary integers r with nk = '(r), and find in many cases smaller values of k than is possible with the previously known approach. The first two authors are with Fachbereich17 Mathematik-Informatik, Universitat-GH Paderborn, D-33095 Paderborn, Germany. The third author is with the International Computer Science Institutem Berkeley, USA 1 Introduction Let F q be a finite field with q elements. A basis of the form (ff; ff q ; : : : ; ff q n\Gamma1 ) of the vector space F q n over F q is a normal basis, and in this case ff is a normal element in F q n over F q . Gau periods have been used to construct normal bases in the following way: Let n; k 1 be integers ...

In this paper, we propose a new normal basis multiplication algorithm for GF(2 ).