9,771 research outputs found
Last fall degree, HFE, and Weil descent attacks on ECDLP
Weil descent methods have recently been applied to attack the Hidden Field Equation (HFE) public key systems and solve the elliptic curve discrete logarithm problem (ECDLP) in small characteristic. However the claims of quasi-polynomial time attacks on the HFE systems and the subexponential time algorithm for the ECDLP depend on various heuristic assumptions.
In this paper we introduce the notion of the last fall degree of a polynomial system, which is independent of choice of a monomial order. We then develop complexity bounds on solving polynomial systems based on this last fall degree.
We prove that HFE systems have a small last fall degree, by showing that one can do division with remainder after Weil descent. This allows us to solve HFE systems unconditionally in polynomial time if the degree of the defining polynomial and the cardinality of the base field are fixed.
For the ECDLP over a finite field of characteristic 2, we provide computational evidence that raises doubt on the validity of the first fall degree assumption, which was widely adopted in earlier works and which promises sub-exponential algorithms for ECDLP. In addition, we construct a Weil descent system from a set of summation polynomials in which the first fall degree assumption is unlikely to hold. These examples suggest that greater care needs to be exercised when applying this heuristic assumption to arrive at complexity estimates.
These results taken together underscore the importance of rigorously bounding last fall degrees of Weil descent systems, which remains an interesting but challenging open problem
Twists of X(7) and primitive solutions to x^2+y^3=z^7
We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent
argument involving the simple group of order 168 reduces the problem to the
determination of the set of rational points on a finite set of twists of the
Klein quartic curve X. To restrict the set of relevant twists, we exploit the
isomorphism between X and the modular curve X(7), and use modularity of
elliptic curves and level lowering. This leaves 10 genus-3 curves, whose
rational points are found by a combination of methods.Comment: 47 page
- …