Notes on Lattice-Based Cryptography

Asymmetrisk kryptering er avhengig av antakelsen om at noen beregningsproblemer er vanskelige å løse. I 1994 viste Peter Shor at de to mest brukte beregningsproblemene, nemlig det diskrete logaritmeproblemet og primtallsfaktorisering, ikke lenger er vanskelige å løse når man bruker en kvantedatamaskin. Siden den gang har forskere jobbet med å finne nye beregningsproblemer som er motstandsdyktige mot kvanteangrep for å erstatte disse to. Gitterbasert kryptografi er forskningsfeltet som bruker kryptografiske primitiver som involverer vanskelige problemer definert på gitter, for eksempel det korteste vektorproblemet og det nærmeste vektorproblemet. NTRU-kryptosystemet, publisert i 1998, var et av de første som ble introdusert på dette feltet. Problemet Learning With Error (LWE) ble introdusert i 2005 av Regev, og det regnes nå som et av de mest lovende beregningsproblemene som snart tas i bruk i stor skala. Å studere vanskelighetsgraden og å finne nye og raskere algoritmer som løser den, ble et ledende forskningstema innen kryptografi. Denne oppgaven inkluderer følgende bidrag til feltet: - En ikke-triviell reduksjon av Mersenne Low Hamming Combination Search Problem, det underliggende problemet med et NTRU-lignende kryptosystem, til Integer Linear Programming (ILP). Særlig finner vi en familie av svake nøkler. - En konkret sikkerhetsanalyse av Integer-RLWE, en vanskelig beregningsproblemvariant av LWE, introdusert av Gu Chunsheng. Vi formaliserer et meet-in-the-middle og et gitterbasert angrep for denne saken, og vi utnytter en svakhet ved parametervalget gitt av Gu, for å bygge et forbedret gitterbasert angrep. - En forbedring av Blum-Kalai-Wasserman-algoritmen for å løse LWE. Mer spesifikt, introduserer vi et nytt reduksjonstrinn og en ny gjetteprosedyre til algoritmen. Disse tillot oss å utvikle to implementeringer av algoritmen, som er i stand til å løse relativt store LWE-forekomster. Mens den første effektivt bare bruker RAM-minne og er fullt parallelliserbar, utnytter den andre en kombinasjon av RAM og disklagring for å overvinne minnebegrensningene gitt av RAM. - Vi fyller et tomrom i paringsbasert kryptografi. Dette ved å gi konkrete formler for å beregne hash-funksjon til G2, den andre gruppen i paringsdomenet, for Barreto-Lynn-Scott-familien av paringsvennlige elliptiske kurver.Public-key Cryptography relies on the assumption that some computational problems are hard to solve. In 1994, Peter Shor showed that the two most used computational problems, namely the Discrete Logarithm Problem and the Integer Factoring Problem, are not hard to solve anymore when using a quantum computer. Since then, researchers have worked on finding new computational problems that are resistant to quantum attacks to replace these two. Lattice-based Cryptography is the research field that employs cryptographic primitives involving hard problems defined on lattices, such as the Shortest Vector Problem and the Closest Vector Problem. The NTRU cryptosystem, published in 1998, was one of the first to be introduced in this field. The Learning With Error (LWE) problem was introduced in 2005 by Regev, and it is now considered one of the most promising computational problems to be employed on a large scale in the near future. Studying its hardness and finding new and faster algorithms that solve it became a leading research topic in Cryptology. This thesis includes the following contributions to the field: - A non-trivial reduction of the Mersenne Low Hamming Combination Search Problem, the underlying problem of an NTRU-like cryptosystem, to Integer Linear Programming (ILP). In particular, we find a family of weak keys. - A concrete security analysis of the Integer-RLWE, a hard computational problem variant of LWE introduced by Gu Chunsheng. We formalize a meet-in-the-middle attack and a lattice-based attack for this case, and we exploit a weakness of the parameters choice given by Gu to build an improved lattice-based attack. - An improvement of the Blum-Kalai-Wasserman algorithm to solve LWE. In particular, we introduce a new reduction step and a new guessing procedure to the algorithm. These allowed us to develop two implementations of the algorithm that are able to solve relatively large LWE instances. While the first one efficiently uses only RAM memory and is fully parallelizable, the second one exploits a combination of RAM and disk storage to overcome the memory limitations given by the RAM. - We fill a gap in Pairing-based Cryptography by providing concrete formulas to compute hash-maps to G2, the second group in the pairing domain, for the Barreto-Lynn-Scott family of pairing-friendly elliptic curves.Doktorgradsavhandlin

Lattice-Based proof of a shuffle

In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally hiding under the RLWE assumption, thus achieving security in a post-quantum scenario. Furthermore we provide a new definition for a secure mixing node (mix-node) and prove that our construction satisfies this definition.Peer ReviewedPostprint (author's final draft

Reduction algorithms for the cryptanalysis of lattice based asymmetrical cryptosystems

Thesis (Master)--Izmir Institute of Technology, Computer Engineering, Izmir, 2008Includes bibliographical references (leaves: 79-91)Text in English; Abstract: Turkish and Englishxi, 119 leavesThe theory of lattices has attracted a great deal of attention in cryptology in recent years. Several cryptosystems are constructed based on the hardness of the lattice problems such as the shortest vector problem and the closest vector problem. The aim of this thesis is to study the most commonly used lattice basis reduction algorithms, namely Lenstra Lenstra Lovasz (LLL) and Block Kolmogorov Zolotarev (BKZ) algorithms, which are utilized to approximately solve the mentioned lattice based problems.Furthermore, the most popular variants of these algorithms in practice are evaluated experimentally by varying the common reduction parameter delta in order to propose some practical assessments about the effect of this parameter on the process of basis reduction.These kind of practical assessments are believed to have non-negligible impact on the theory of lattice reduction, and so the cryptanalysis of lattice cryptosystems, due to thefact that the contemporary nature of the reduction process is mainly controlled by theheuristics

On the Quantum Complexity of the Continuous Hidden Subgroup Problem

The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor's celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms. The latest successful generalization (Eisentrager et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space Rm , even for large dimensions m . It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices. The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits. This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Incidentally, our work clarifies certain claims from the extended abstract of Eisentrager et al

Subfield Algorithms for Ideal- and Module-SVP Based on the Decomposition Group

Whilst lattice-based cryptosystems are believed to be resistant to quantum attack, they are often forced to pay for that security with inefficiencies in implementation. This problem is overcome by ring- and module-based schemes such as Ring-LWE or Module-LWE, whose keysize can be reduced by exploiting its algebraic structure, allowing for neater and faster computations. Many rings may be chosen to define such cryptoschemes, but cyclotomic rings, due to their cyclic nature allowing for easy multiplication, are the community standard. However, there is still much uncertainty as to whether this structure may be exploited to an adversary\u27s benefit. In this paper, we show that the decomposition group of a cyclotomic ring of arbitrary conductor may be utilised in order to significantly decrease the dimension of the ideal (or module) lattice required to solve a given instance of SVP. Moreover, we show that there exist a large number of rational primes for which, if the prime ideal factors of an ideal lie over primes of this form, give rise to an ``easy\u27\u27 instance of SVP. However, it is important to note that this work does not break Ring-LWE or Module-LWE, since the security reduction is from worst case ideal or module SVP to average case Ring-LWE or Module-LWE respectively, and is one way

Usability of structured lattices for a post-quantum cryptography: practical computations, and a study of some real Kummer extensions

Lattice-based cryptography is an excellent candidate for post-quantum cryptography, i.e. cryptosystems which are resistant to attacks run on quantum computers. For efficiency reason, most of the constructions explored nowadays are based on structured lattices, such as module lattices or ideal lattices. The security of most constructions can be related to the hardness of retrieving a short element in such lattices, and one does not know yet to what extent these additional structures weaken the cryptosystems. A related problem – which is an extension of a classical problem in computational number theory – called the Short Principal Ideal Problem (or SPIP), consists of finding a short generator of a principal ideal. Its assumed hardness has been used to build some cryptographic schemes. However it has been shown to be solvable in quantum polynomial time over cyclotomic fields, through an attack which uses the Log-unit lattice of the field considered. Later, practical results showed that multiquadratic fields were also weak to this strategy. The main general question that we study in this thesis is To what extent can structured lattices be used to build a post-quantum cryptography