Root Extraction in Finite Abelian Groups

We formulate the Root Extraction problem in finite Abelian $p$-groups and then extend it to generic finite Abelian groups. We provide algorithms to solve them. We also give the bounds on the number of group operations required for these algorithms. We observe that once a basis is computed and the discrete logarithm relative to the basis is solved, root extraction takes relatively fewer "bookkeeping" steps. Thus, we conclude that root extraction in finite Abelian groups is no harder than solving discrete logarithms and computing basis

Structure computation and discrete logarithms in finite abelian p-groups

We present a generic algorithm for computing discrete logarithms in a finite abelian p-group H, improving the Pohlig-Hellman algorithm and its generalization to noncyclic groups by Teske. We then give a direct method to compute a basis for H without using a relation matrix. The problem of computing a basis for some or all of the Sylow p-subgroups of an arbitrary finite abelian group G is addressed, yielding a Monte Carlo algorithm to compute the structure of G using O(|G|^0.5) group operations. These results also improve generic algorithms for extracting pth roots in G.Comment: 23 pages, minor edit

Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Arithmetic and geometric structures in cryptography

We explore a few algebraic and geometric structures, through certain questions posed by modern cryptography. We focus on the cases of discrete logarithms in finite fields of small characteristic, the structure of isogeny graphs of ordinary abelian varieties, and the geometry of ideals in cyclotomic rings. The presumed difficulty of computing discrete logarithms in certain groups is essential for the security of a number of communication protocols deployed today. One of the most classic choices for the underlying group is the multiplicative group of a finite field. Yet this choice is showing its age, and particularly when the characteristic of the field is small: recent algorithms allow to compute logarithms efficiently in these groups. However, these methods are only heuristic: they seem to always work, yet we do not know how to prove it. In the first part, we propose to study these methods in the hope to get a better understanding, notably by revealing the geometric structures at play. A more modern choice is the group of rational points of an elliptic curve defined over a finite field. There, the difficulty of the discrete logarithm problem seems at its peak. More generally, the group of rational points of an abelian variety (notably the Jacobian of a curve of small genus) could be appropriate. One of the main tools for studying discrete logarithms on such objects is the notion of isogeny: a morphism from a variety to another one, which allows, among other things, to transfer the computation of a logarithm. Whereas the theory for elliptic curves is already mature, little is known about the structures formed by these isogenies (the isogeny graphs) for varieties of higher dimension. In the second part, we study the structure of isogeny graphs of absolutely simple, ordinary abelian varieties, with a few consequences regarding discrete logarithms on Jacobians of hyperelliptic curves of genus 2, the main object of concern of so-called hyperelliptic cryptography. The security of quite a few protocols, notably those relying on discrete logarithms, would collapse in front of an adversary equipped with a large-scale quantum computer. This perspective motivates cryptographers to study problems that would resist this technological feat. One of the major directions is cryptography based on Euclidean lattices, relying on the difficulty to find short vectors in a given lattice. For efficiency, one benefits from considering lattices endowed with more structure, such as the ideals of a cyclotomic field. In the third part, we study the geometry of these ideals, and show that a quantum computer allows to efficiently find much shorter vectors in these ideals than is currently possible in generic lattices

Discrete Logarithms in Generalized Jacobians

D\'ech\`ene has proposed generalized Jacobians as a source of groups for public-key cryptosystems based on the hardness of the Discrete Logarithm Problem (DLP). Her specific proposal gives rise to a group isomorphic to the semidirect product of an elliptic curve and a multiplicative group of a finite field. We explain why her proposal has no advantages over simply taking the direct product of groups. We then argue that generalized Jacobians offer poorer security and efficiency than standard Jacobians

A kilobit hidden SNFS discrete logarithm computation

We perform a special number field sieve discrete logarithm computation in a 1024-bit prime field. To our knowledge, this is the first kilobit-sized discrete logarithm computation ever reported for prime fields. This computation took a little over two months of calendar time on an academic cluster using the open-source CADO-NFS software. Our chosen prime $p$ looks random, and $p--1$ has a 160-bit prime factor, in line with recommended parameters for the Digital Signature Algorithm. However, our p has been trapdoored in such a way that the special number field sieve can be used to compute discrete logarithms in $\mathbb{F}\_p^*$ , yet detecting that p has this trapdoor seems out of reach. Twenty-five years ago, there was considerable controversy around the possibility of back-doored parameters for DSA. Our computations show that trapdoored primes are entirely feasible with current computing technology. We also describe special number field sieve discrete log computations carried out for multiple weak primes found in use in the wild. As can be expected from a trapdoor mechanism which we say is hard to detect, our research did not reveal any trapdoored prime in wide use. The only way for a user to defend against a hypothetical trapdoor of this kind is to require verifiably random primes