Computing discrete logarithms in real quadratic congruence function fields of large genus
Mathematics of Computation
Computing Riemann---Roch spaces in algebraic function fields and related topics
Journal of Symbolic Computation
Computing discrete logarithms in high-genus hyperelliptic Jacobians in provably subexponential time
Mathematics of Computation
Explict bounds and heuristics on class numbers in hyperelliptic function fields
Mathematics of Computation
ANTS-I Proceedings of the First International Symposium on Algorithmic Number Theory
An algorithm for solving the discrete log problem on hyperelliptic curves
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
An index calculus algorithm for plane curves of small degree
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
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The discrete logarithm problem in Jacobians of curves of high genus gover finite fields $\mathbb {F}_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose degrees in Xand Yare low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$, and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+ \varepsilon, o(1))$ for any positive 驴. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.