65 research outputs found
Algebraic Cryptanalysis of STARK-Friendly Designs:Application to MARVELlous and MiMC
The block cipher Jarvis and the hash function Friday, both members of the MARVELlous family of cryptographic primitives, are among the first proposed solutions to the problem of designing symmetric-key algorithms suitable for transparent, post-quantum secure zero-knowledge proof systems such as ZK-STARKs. In this paper we describe an algebraic cryptanalysis of Jarvis and Friday and show that the proposed number of rounds is not sufficient to provide adequate security. In Jarvis, the round function is obtained by combining a finite field inversion, a full-degree affine permutation polynomial and a key addition. Yet we show that even though the high degree of the affine polynomial may prevent some algebraic attacks (as claimed by the designers), the particular algebraic properties of the round function make both Jarvis and Friday vulnerable to Gröbner basis attacks. We also consider MiMC, a block cipher similar in structure to Jarvis. However, this cipher proves to be resistant against our proposed attack strategy. Still, our successful cryptanalysis of Jarvis and Friday does illustrate that block cipher designs for “algebraic platforms” such as STARKs, FHE or MPC may be particularly vulnerable to algebraic attacks
On the Rapoport-Zink space for over a ramified prime
In this work, we study the supersingular locus of the Shimura variety
associated to the unitary group over a ramified prime. We
show that the associated Rapoport-Zink space is flat, and we give an explicit
description of the irreducible components of the reduction modulo of the
basic locus. In particular, we show that these are universally homeomorphic to
either a generalized Deligne-Lusztig variety for a symplectic group or to the
closure of a vector bundle over a classical Deligne-Lusztig variety for an
orthogonal group. Our results are confirmed in the group-theoretical setting by
the reduction method \`a la Deligne and Lusztig and the study of the admissible
set
Algorithms in Intersection Theory in the Plane
This thesis presents an algorithm to find the local structure of intersections of plane curves. More precisely, we address the question of describing the scheme of the quotient ring of a bivariate zero-dimensional ideal , \textit{i.e.} finding the points (maximal ideals of ) and describing the regular functions on those points. A natural way to address this problem is via Gr\"obner bases as they reduce the problem of finding the points to a problem of factorisation, and the sheaf of rings of regular functions can be studied with those bases through the division algorithm and localisation.
Let be an ideal generated by , a subset of with and a field. We present an algorithm that features a quadratic convergence to find a Gr\"obner basis of or its primary component at the origin.
We introduce an -adic Newton iteration to lift the lexicographic Gr\"obner basis of any finite intersection of zero-dimensional primary components of if is a \textit{good} maximal ideal. It relies on a structural result about the syzygies in such a basis due to Conca \textit{\&} Valla (2008), from which arises an explicit map between ideals in a stratum (or Gr\"obner cell) and points in the associated moduli space. We also qualify what makes a maximal ideal suitable for our filtration.
When the field is \textit{large enough}, endowed with an Archimedean or ultrametric valuation, and admits a fraction reconstruction algorithm, we use this result to give a complete -adic algorithm to recover , the Gr\"obner basis of . We observe that previous results of Lazard that use Hermite normal forms to compute Gr\"obner bases for ideals with two generators can be generalised to a set of generators. We use this result to obtain a bound on the height of the coefficients of and to control the probability of choosing a \textit{good} maximal ideal to build the -adic expansion of .
Inspired by Pardue (1994), we also give a constructive proof to
characterise a Zariski open set of (with action on ) that changes coordinates in such a way as to ensure the initial term ideal of a zero-dimensional becomes Borel-fixed when is sufficiently large. This sharpens our analysis
to obtain, when or , a complexity less than cubic in terms of the dimension of and softly linear in the height of the coefficients of .
We adapt the resulting method and present the analysis to find the -primary component of . We also discuss the transition towards other primary components via linear mappings, called \emph{untangling} and \emph{tangling}, introduced by van der Hoeven and Lecerf (2017). The two maps form one isomorphism to find points with an isomorphic local structure and, at the origin, bind them. We give a slightly faster tangling algorithm and discuss new applications of these techniques. We show how to extend these ideas to bivariate settings and give a bound on the arithmetic complexity for certain algebras
- …