Digital Signatures Based on the Hardness of Ideal Lattice Problems in all Rings

Many practical lattice-based schemes are built upon the Ring-SIS or Ring-LWE problems, which are problems that are based on the presumed difficulty of finding low-weight solutions to linear equations over polynomial rings $Z_q[x]/\langle f(x) \rangle$. Our belief in the asymptotic computational hardness of these problems rests in part on the fact that there are reduction showing that solving them is as hard as finding short vectors in all lattices that correspond to ideals of the polynomial ring $Z[x]/\langle f(x) \rangle$. These reductions, however, do not give us an indication as to the effect that the polynomial $f(x)$, which defines the ring, has on the average-case or worst-case problems. \\ As of today, there haven\u27t been any weaknesses found in Ring-SIS or Ring-LWE problems when one uses an $f(x)$ which leads to a meaningful worst-case to average-case reduction, but there have been some recent algorithms for related problems that heavily use the algebraic structures of the underlying rings. It is thus conceivable that some rings could give rise to more difficult instances of Ring-SIS and Ring-LWE than other rings. A more ideal scenario would therefore be if there would be an average-case problem, allowing for efficient cryptographic constructions, that is based on the hardness of finding short vectors in ideals of $Z[x]/\langle f(x)\rangle$ for \emph{every} $f(x)$.\\ In this work, we show that the above may actually be possible. We construct a digital signature scheme based (in the random oracle model) on a simple adaptation of the Ring-SIS problem which is as hard to break as worst-case problems in every $f(x)$ whose degree is bounded by the parameters of the scheme. Up to constant factors, our scheme is as efficient as the highly practical schemes that work over the ring $Z[x]/\langle x^n+1\rangle$

Hard isogeny problems over RSA moduli and groups with infeasible inversion

We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders. Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.Comment: Significant revision of the article previously titled "A Candidate Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the constructions by giving toy examples, added "The Parallelogram Attack" (Sec 5.3.2). 54 pages, 8 figure

Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory

The present survey reports on the state of the art of the different cryptographic functionalities built upon the ring learning with errors problem and its interplay with several classical problems in algebraic number theory. The survey is based to a certain extent on an invited course given by the author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other authors/ comment of the author: quotation has been added to Theorem 5.

A framework for cryptographic problems from linear algebra

We introduce a general framework encompassing the main hard problems emerging in lattice-based cryptography, which naturally includes the recently proposed Mersenne prime cryptosystem, but also problems coming from code-based cryptography. The framework allows to easily instantiate new hard problems and to automatically construct plausibly post-quantum secure primitives from them. As a first basic application, we introduce two new hard problems and the corresponding encryption schemes. Concretely, we study generalisations of hard problems such as SIS, LWE and NTRU to free modules over quotients of Z[X] by ideals of the form (f,g), where f is a monic polynomial and g∈Z[X] is a ciphertext modulus coprime to f. For trivial modules (i.e. of rank one), the case f=Xn+1 and g=q∈Z>1 corresponds to ring-LWE, ring-SIS and NTRU, while the choices f=Xn−1 and g=X−2 essentially cover the recently proposed Mersenne prime cryptosystems. At the other extreme, when considering modules of large rank and letting deg(f)=1, one recovers the framework of LWE and SIS

09221 Abstracts Collection -- Algorithms and NumberTheory

From 24.05. to 29.05.2009, the Dagstuhl Seminar 09221 ``Algorithms and Number Theory \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available