12 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
On the first fall degree of summation polynomials
We improve on the first fall degree bound of polynomial systems that arise
from a Weil descent along Semaev's summation polynomials relevant to the
solution of the Elliptic Curve Discrete Logarithm Problem via Gr\"obner basis
algorithms.Comment: 12 pages, fina
On Index Calculus Algorithms for Subfield Curves
In this paper we further the study of index calculus methods for solving the elliptic curve discrete logarithm problem (ECDLP). We focus on the index calculus for subfield curves, also called Koblitz curves, defined over Fq with ECDLP in Fqn. Instead of accelerating the solution of polynomial systems during index calculus as was predominantly done in previous work, we define factor bases that are invariant under the q-power Frobenius automorphism of the field Fqn, reducing the number of polynomial systems that need to be solved. A reduction by a factor of 1/n is the best one could hope for. We show how to choose factor bases to achieve this, while simultaneously accelerating the linear algebra step of the index calculus method for Koblitz curves by a factor n2. Furthermore, we show how to use the Frobenius endomorphism to improve symmetry breaking for Koblitz curves. We provide constructions of factor bases with the desired properties, and we study their impact on the polynomial system solving costs experimentally.SCOPUS: cp.kinfo:eu-repo/semantics/publishe
Stronger bounds on the cost of computing Groebner bases for HFE systems
We give upper bounds for the solving degree and the last fall degree of the
polynomial system associated to the HFE (Hidden Field Equations) cryptosystem.
Our bounds improve the known bounds for this type of systems. We also present
new results on the connection between the solving degree and the last fall
degree and prove that, in some cases, the solving degree is independent of
coordinate changes.Comment: 15 page
Recent progress on the elliptic curve discrete logarithm problem
International audienceWe survey recent work on the elliptic curve discrete logarithm problem. In particular we review index calculus algorithms using summation polynomials, and claims about their complexity
Two philosophies for solving non-linear equations in algebraic cryptanalysis
Algebraic Cryptanalysis [45] is concerned with solving of particular systems of multivariate non-linear equations which occur in cryptanalysis. Many different methods for solving such problems have been proposed in cryptanalytic literature: XL and XSL method, Gröbner bases, SAT solvers, as well as many other. In this paper we survey these methods and point out that the main working principle in all of them is essentially the same. One quantity grows faster than another quantity which leads to a “phase transition” and the problem becomes efficiently solvable. We illustrate this with examples from both symmetric and asymmetric cryptanalysis. In this paper we point out that there exists a second (more) general way of formulating algebraic attacks through dedicated coding techniques which involve redundancy with addition of new variables. This opens numerous new possibilities for the attackers and leads to interesting optimization problems where the existence of interesting equations may be somewhat deliberately engineered by the attacker
An upper bound for the solving degree in terms of the degree of regularity
The solving degree is an important parameter for estimating the complexity of
solving a system of polynomial equations. In this paper, we provide an upper
bound for the solving degree in terms of the degree of regularity. We also show
that this bound is optimal. As a direct consequence, we prove an upper bound
for the last fall degree and a Macaulay bound.Comment: 7 page
On Splitting a Point with Summation Polynomials in Binary Elliptic Curves
Recent research for efficient algorithms for solving the discrete logarithm (DL) problem on elliptic curves depends on the difficult
question of the feasibility of index calculus which would consist of splitting EC points into sums of points lying in a certain subspace. A natural algebraic approach towards this goal is through solving systems of non linear multivariate equations derived from the so called summation polynomials which method have been proposed by Semaev in 2004 [12].
In this paper we consider simplified variants of this problem with splitting in two or three parts in binary curves. We propose three algorithms with running time of the order of 2^n/3 for both problems. It is not clear how to interpret these results but they do in some sense violate the generic group model for these curves
Solving degree, last fall degree, and related invariants
In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Groebner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo-Mumford regularity