Lattice-Based Cryptography∗
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Universit`A Di Pisa Post Quantum Cryptography
Universit`adi Pisa Facolt`a di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Fisica Anno Accademico 2013/2014 Elaborato Finale Post Quantum Cryptography Candidato: Relatori: Antonio Vaira Prof. Oliver Morsch Ing. Cristiano Borrelli Alla mia mamma Abstract I started my experience with cryptography within the Airbus environment working on this master thesis. I have been asked to provide a framework, or in other words, a big picture about present-day alternatives to the most used public key crypto-system, the RSA, that are supposed to be quantum resistant. The final application of my work eventually resulted in recommendations on how to handle the quantum threat in the near future. This was quite a complex task to accomplish because it involves a huge variety of topics but by physical background was really helpful in facing it. Not because of specific and previous knowledge but for the mathematical toolsacquired during the studies and especially for that attitude that belongs to out category that make us, physicist problem solver in a large variety of fields. Indeed I also tried to go a bit further with my studies. I took one of the most promising algorithm on my opinion, but not well known yet so unfeasible for a recommendation and therefore an implementation in the close future, and I tried to figure out how to enhance it from both a security and an operational point of view (how to increase the correctness ratio of the decryption and the speed of the cryptographic operations). It followed a period of time in which I improved my skills with few computing languages and in the end I decided to implement a toy model at a high level using an interface that already implements all the mathematical and algebraical structures used to build the model. -
On Ideal Lattices and Learning with Errors Over Rings∗
On Ideal Lattices and Learning with Errors Over Rings∗ Vadim Lyubashevskyy Chris Peikertz Oded Regevx June 25, 2013 Abstract The “learning with errors” (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for lattice-based hash functions (and related primitives). We resolve this question in the affirmative by introducing an algebraic variant of LWE called ring- LWE, and proving that it too enjoys very strong hardness guarantees. Specifically, we show that the ring-LWE distribution is pseudorandom, assuming that worst-case problems on ideal lattices are hard for polynomial-time quantum algorithms. Applications include the first truly practical lattice-based public-key cryptosystem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ring-LWE. 1 Introduction Over the last decade, lattices have emerged as a very attractive foundation for cryptography. The appeal of lattice-based primitives stems from the fact that their security can often be based on worst-case hardness assumptions, and that they appear to remain secure even against quantum computers. -
Private Searching on Streaming Data∗
J. Cryptology (2007) 20: 397–430 DOI: 10.1007/s00145-007-0565-3 © 2007 International Association for Cryptologic Research Private Searching on Streaming Data∗ Rafail Ostrovsky Department of Computer Science and Department of Mathematics, University of California, Los Angeles, CA 90095, U.S.A. [email protected] William E. Skeith III Department of Mathematics, University of California, Los Angeles, CA 90095, U.S.A. [email protected] Communicated by Dan Boneh Received 1 September 2005 and revised 16 July 2006 Online publication 13 July 2007 Abstract. In this paper we consider the problem of private searching on streaming data, where we can efficiently implement searching for documents that satisfy a secret criteria (such as the presence or absence of a hidden combination of hidden keywords) under various cryptographic assumptions. Our results can be viewed in a variety of ways: as a generalization of the notion of private information retrieval (to more general queries and to a streaming environment); as positive results on privacy-preserving datamining; and as a delegation of hidden program computation to other machines. Key words. Code obfuscation, Public-key program obfuscation, Program obfus- cation, Crypto-computing, Software security, Database security, Public-key encryp- tion with special properties, Private information retrieval, Privacy-preserving keyword search, Secure algorithms for streaming data, Privacy-preserving datamining, Secure delegation of computation, Searching with privacy, Mobile code. 1. Introduction 1.1. Data Filtering for the Intelligence Community As our motivating example, we examine one of the tasks of the intelligence community, which is to collect “potentially useful” information from huge streaming sources of ∗ An abridged version appeared at CRYPTO 2005. -
Making NTRU As Secure As Worst-Case Problems Over Ideal Lattices
Making NTRU as Secure as Worst-Case Problems over Ideal Lattices Damien Stehlé1 and Ron Steinfeld2 1 CNRS, Laboratoire LIP (U. Lyon, CNRS, ENS Lyon, INRIA, UCBL), 46 Allée d’Italie, 69364 Lyon Cedex 07, France. [email protected] – http://perso.ens-lyon.fr/damien.stehle 2 Centre for Advanced Computing - Algorithms and Cryptography, Department of Computing, Macquarie University, NSW 2109, Australia [email protected] – http://web.science.mq.edu.au/~rons Abstract. NTRUEncrypt, proposed in 1996 by Hoffstein, Pipher and Sil- verman, is the fastest known lattice-based encryption scheme. Its mod- erate key-sizes, excellent asymptotic performance and conjectured resis- tance to quantum computers could make it a desirable alternative to fac- torisation and discrete-log based encryption schemes. However, since its introduction, doubts have regularly arisen on its security. In the present work, we show how to modify NTRUEncrypt to make it provably secure in the standard model, under the assumed quantum hardness of standard worst-case lattice problems, restricted to a family of lattices related to some cyclotomic fields. Our main contribution is to show that if the se- cret key polynomials are selected by rejection from discrete Gaussians, then the public key, which is their ratio, is statistically indistinguishable from uniform over its domain. The security then follows from the already proven hardness of the R-LWE problem. Keywords. Lattice-based cryptography, NTRU, provable security. 1 Introduction NTRUEncrypt, devised by Hoffstein, Pipher and Silverman, was first presented at the Crypto’96 rump session [14]. Although its description relies on arithmetic n over the polynomial ring Zq[x]=(x − 1) for n prime and q a small integer, it was quickly observed that breaking it could be expressed as a problem over Euclidean lattices [6]. -
The Cramer-Shoup Strong-RSA Signature Scheme Revisited
The Cramer-Shoup Strong-RSA Signature Scheme Revisited Marc Fischlin Johann Wolfgang Goethe-University Frankfurt am Main, Germany marc @ mi.informatik.uni-frankfurt.de http://www.mi.informatik.uni-frankfurt.de/ Abstract. We discuss a modification of the Cramer-Shoup strong-RSA signature scheme. Our proposal also presumes the strong RSA assump- tion (and a collision-intractable hash function for long messages), but |without loss in performance| the size of a signature is almost halved compared to the original scheme. We also show how to turn the signature scheme into a \lightweight" anonymous (but linkable) group identifica- tion protocol without random oracles. 1 Introduction Cramer and Shoup [CS00] have presented a signature scheme which is secure against adaptive chosen-message attacks under the strong RSA (aka. flexible RSA) assumption, and which does not rely on the random oracle model. For a 1024-bit RSA modulus and a 160-bit (hash value of a) message a signature has about 2200 bits. Cramer and Shoup also discuss a variation of their scheme which, in addition to the strong RSA assumption, requires the discrete-log as- sumption and which produces signatures of roughly half the length (about 1350 bits). Here, we show that we can achieve the same signature size under the strong RSA assumption only, even with a slightly improved performance than in the original strong-RSA-only case or the discrete-log & strong-RSA case. Our signature scheme also has the feature that for short messages, e.g., of 120 bits, a collision-intractable (or universal one-way) hash function becomes obsolete. -
Lattices That Admit Logarithmic Worst-Case to Average-Case Connection Factors
Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors Chris Peikert∗ Alon Rosen† April 4, 2007 Abstract We demonstrate an average-case problem that is as hard as finding γ(n)-approximate√ short- est vectors in certain n-dimensional lattices in the worst case, where γ(n) = O( log n). The previously best known factor for any non-trivial class of lattices was γ(n) = O˜(n). Our results apply to families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case problem we rely on is to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field. For the connection factors γ(n) we achieve, the corresponding decision problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in com- putational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices. To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3−) would yield connection factors better than O˜(n). -
Cryptography from Ideal Lattices
Introduction Ideal lattices Ring-SIS Ring-LWE Other algebraic lattices Conclusion Ideal Lattices Damien Stehl´e ENS de Lyon Berkeley, 07/07/2015 Damien Stehl´e Ideal Lattices 07/07/2015 1/32 Introduction Ideal lattices Ring-SIS Ring-LWE Other algebraic lattices Conclusion Lattice-based cryptography: elegant but impractical Lattice-based cryptography is fascinating: simple, (presumably) post-quantum, expressive But it is very slow Recall the SIS hash function: 0, 1 m Zn { } → q x xT A 7→ · Need m = Ω(n log q) to compress O(1) m n q is n , A is uniform in Zq × O(n2) space and cost ⇒ Example parameters: n 26, m n 2e4, log q 23 ≈ ≈ · 2 ≈ Damien Stehl´e Ideal Lattices 07/07/2015 2/32 Introduction Ideal lattices Ring-SIS Ring-LWE Other algebraic lattices Conclusion Lattice-based cryptography: elegant but impractical Lattice-based cryptography is fascinating: simple, (presumably) post-quantum, expressive But it is very slow Recall the SIS hash function: 0, 1 m Zn { } → q x xT A 7→ · Need m = Ω(n log q) to compress O(1) m n q is n , A is uniform in Zq × O(n2) space and cost ⇒ Example parameters: n 26, m n 2e4, log q 23 ≈ ≈ · 2 ≈ Damien Stehl´e Ideal Lattices 07/07/2015 2/32 Introduction Ideal lattices Ring-SIS Ring-LWE Other algebraic lattices Conclusion Lattice-based cryptography: elegant but impractical Lattice-based cryptography is fascinating: simple, (presumably) post-quantum, expressive But it is very slow Recall the SIS hash function: 0, 1 m Zn { } → q x xT A 7→ · Need m = Ω(n log q) to compress O(1) m n q is n , A is uniform in Zq × O(n2) -
A Weak-Randomizer Attack on RSA-OAEP with E = 3
A Weak-Randomizer Attack on RSA-OAEP with e = 3 Daniel R. L. Brown∗ July 6, 2005 Abstract Coppersmith's heuristic algorithm for finding small roots of bivariate modular equations can be applied against low-exponent RSA-OAEP if its randomizer is weak. An adversary that knows the randomizer can recover the entire plaintext message, provided it is short enough for Coppersmith's algorithm to work. In practice, messages are symmetric cipher keys and these are potentially short enough for certain sets of key sizes. Weak randomizers could arise in constrained smart cards or in kleptographic implementations. Because RSA's major use is transporting symmetric keys, this attack is a potential concern. In this respect, OAEP's design is more fragile than necessary, because a secure randomizer is critical to prevent a total loss of secrecy, not just a loss of semantic security or chosen-ciphertext security. Countermeasures and more robust designs that have little extra performance cost are proposed and discussed. 1 Introduction Evidence exists that RSA encryption when the public exponent e = 3 is weaker than the general problem of inverting the raw RSA function with e = 3 at random values. Examples are Copper- smith's [Cop96] and Hastad's broadcast attack. (Boneh [Bon99] surveys some of these results.) Security proofs aim to rule out unforeseen attacks by reducing security to a hard problem. The RSA-OAEP encryption scheme [BR94] has a proof that reduces its security to the problem of in- verting the RSA function. The proof applies even if e = 3, so therefore RSA-OAEP avoids the attacks mentioned above. -
A Decade of Lattice Cryptography
A Decade of Lattice Cryptography Chris Peikert1 February 17, 2016 1Department of Computer Science and Engineering, University of Michigan. Much of this work was done while at the School of Computer Science, Georgia Institute of Technology. This material is based upon work supported by the National Science Foundation under CAREER Award CCF-1054495, by DARPA under agreement number FA8750-11- C-0096, and by the Alfred P. Sloan Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation, DARPA or the U.S. Government, or the Sloan Foundation. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. Abstract n Lattice-based cryptography is the use of conjectured hard problems on point lattices in R as the foundation for secure cryptographic systems. Attractive features of lattice cryptography include apparent resistance to quantum attacks (in contrast with most number-theoretic cryptography), high asymptotic efficiency and parallelism, security under worst-case intractability assumptions, and solutions to long-standing open problems in cryptography. This work surveys most of the major developments in lattice cryptography over the past ten years. The main focus is on the foundational short integer solution (SIS) and learning with errors (LWE) problems (and their more efficient ring-based variants), their provable hardness assuming the worst-case intractability of standard lattice problems, and their many cryptographic applications. Contents 1 Introduction 3 1.1 Scope and Organization . .4 1.2 Other Resources . .5 2 Background 6 2.1 Notation . -
CRYSTALS-Dilithium: a Lattice-Based Digital Signature Scheme
CRYSTALS-Dilithium: A Lattice-Based Digital Signature Scheme Léo Ducas1, Eike Kiltz2, Tancrède Lepoint3, Vadim Lyubashevsky4, Peter Schwabe5, Gregor Seiler6 and Damien Stehlé7 1 CWI, Netherlands 2 Ruhr Universität Bochum, Germany 3 SRI International, USA 4 IBM Research – Zurich, Switzerland 5 Radboud University, Netherlands 6 IBM Research – Zurich and ETH Zurich, Switzerland 7 ENS de Lyon, France Abstract. In this paper, we present the lattice-based signature scheme Dilithium, which is a component of the CRYSTALS (Cryptographic Suite for Algebraic Lattices) suite that was submitted to NIST’s call for post-quantum cryptographic standards. The design of the scheme avoids all uses of discrete Gaussian sampling and is easily implementable in constant-time. For the same security levels, our scheme has a public key that is 2.5X smaller than the previously most efficient lattice-based schemes that did not use Gaussians, while having essentially the same signature size. In addition to the new design, we significantly improve the running time of the main component of many lattice-based constructions – the number theoretic transform. Our AVX2-based implementation results in a speed-up of roughly a factor of 2 over the previously best algorithms that appear in the literature. The techniques for obtaining this speed-up also have applications to other lattice-based schemes. Keywords: Lattice Cryptography · Digital Signatures · Constant-Time Implementa- tion · AVX2 1 Introduction Cryptography based on the hardness of lattice problems is seen as a very promising replacement of traditional cryptography after the eventual coming of quantum computers. In this paper, we present a new digital signature scheme Dilithium, whose security is based on the hardness of finding short vectors in lattices. -
High-Speed Signatures from Standard Lattices
High-speed signatures from standard lattices Özgür Dagdelen1, Rachid El Bansarkhani1, Florian Göpfert1, Tim Güneysu2, Tobias Oder2, Thomas Pöppelmann2, Ana Helena Sánchez3, and Peter Schwabe3 ? 1 Technische Universität Darmstadt, Germany [email protected], [email protected], [email protected] 2 Horst Görtz Institute for IT-Security, Ruhr-University Bochum, Germany [email protected] 3 Digital Security Group, Radboud University Nijmegen, The Netherlands [email protected], [email protected] Abstract. At CT-RSA 2014 Bai and Galbraith proposed a lattice-based signature scheme optimized for short signatures and with a security re- duction to hard standard lattice problems. In this work we first refine the security analysis of the original work and propose a new 128-bit se- cure parameter set chosen for software efficiency. Moreover, we increase the acceptance probability of the signing algorithm through an improved rejection condition on the secret keys. Our software implementation tar- geting Intel CPUs with AVX/AVX2 and ARM CPUs with NEON vector instructions shows that even though we do not rely on ideal lattices, we are able to achieve high performance. For this we optimize the matrix- vector operations and several other aspects of the scheme and finally compare our work with the state of the art. Keywords: Signature scheme, standard lattices, vectorization, Ivy Bridge. 1 Introduction Most practical lattice-based signatures [7, 16, 21], proposed as post-quantum [9] alternatives to RSA and ECDSA, are currently instantiated and implemented using structured ideal lattices [30] corresponding to ideals in rings of the form Z[x]=hfi, where f is a degree-n irreducible polynomial (usually f = xn +1). -
RSA–REACT: an Alternative to RSA–OAEP
Second NESSIE Workshop (september 12 { 13, 2001, Egham, UK). RSA{REACT: An Alternative to RSA{OAEP Tatsuaki Okamoto1 and David Pointcheval2 1 NTT Labs, 1-1 Hikarino-oka, Yokosuka-shi, 239-0847 Japan E-mail: [email protected] 2 D´ept d'Informatique, ENS { CNRS, 45 rue d'Ulm, 75230 Paris Cedex 05, France E-mail: [email protected] { URL: http://www.di.ens.fr/users/pointche Abstract. The last few months, several new results appeared about the OAEP con- struction, and namely the RSA{OAEP cryptosystem. Whereas OAEP was believed to provide the highest security level (IND-CCA2), with an efficient exact security level, the effective security result had been showed to be incomplete. Nevertheless, the particular instantiation with RSA (which is anyway almost the sole application) had been eventu- ally proven secure, but the security reduction appears to be quite inefficient. Therefore, with respect to the provable security result, RSA{OAEP with a 1024-bit modulus just provides a 240 security level. Several alternatives have been recently proposed, but most of them face the same prob- lem with a quadratic time security reduction. Excepted the recent generic conversion, called REACT, which admits a linear time reduction. Consequently, RSA{REACT ap- pears to be the best alternative to RSA{OAEP, granted the high security level, even with real world parameters. RSA{REACT with a 1024-bit modulus indeed guarantees a 280 security level (IND-CCA2 under the RSA assumption). Furthermore, the full construction is already proven secure when integrating symmetric encryption, which guarantees the security of the overall communication.