A New Index Calculus Algorithm with Complexity $$L(1/4+o(1))$$ in Small Characteristic

In this paper, we describe a new algorithm for discrete logarithms in small characteristic. This algorithm is based on index calculus and includes two new contributions. The first is a new method for generating multiplicative relations among elements of a small smoothness basis. The second is a new descent strategy that allows us to express the logarithm of an arbitrary finite field element in terms of the logarithm of elements from the smoothness basis. For a small characteristic finite field of size $$Q=p^n$$ , this algorithm achieves heuristic complexity $$L_Q1/4+o1.$$ For technical reasons, unless $$n$$ is already a composite with factors of the right size, this is done by embedding $${\mathbb F}_{Q}$$ in a small extension $${\mathbb F}_{Q^e}$$ with $$e\le 2\lceil \log _p n \rceil $$ .

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