Lattice-Based Cryptography - a Comparative Description and Analysis of Proposed Schemes
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Etsi Gr Qsc 001 V1.1.1 (2016-07)
ETSI GR QSC 001 V1.1.1 (2016-07) GROUP REPORT Quantum-Safe Cryptography (QSC); Quantum-safe algorithmic framework 2 ETSI GR QSC 001 V1.1.1 (2016-07) Reference DGR/QSC-001 Keywords algorithm, authentication, confidentiality, security ETSI 650 Route des Lucioles F-06921 Sophia Antipolis Cedex - FRANCE Tel.: +33 4 92 94 42 00 Fax: +33 4 93 65 47 16 Siret N° 348 623 562 00017 - NAF 742 C Association à but non lucratif enregistrée à la Sous-Préfecture de Grasse (06) N° 7803/88 Important notice The present document can be downloaded from: http://www.etsi.org/standards-search The present document may be made available in electronic versions and/or in print. The content of any electronic and/or print versions of the present document shall not be modified without the prior written authorization of ETSI. In case of any existing or perceived difference in contents between such versions and/or in print, the only prevailing document is the print of the Portable Document Format (PDF) version kept on a specific network drive within ETSI Secretariat. Users of the present document should be aware that the document may be subject to revision or change of status. Information on the current status of this and other ETSI documents is available at https://portal.etsi.org/TB/ETSIDeliverableStatus.aspx If you find errors in the present document, please send your comment to one of the following services: https://portal.etsi.org/People/CommiteeSupportStaff.aspx Copyright Notification No part may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm except as authorized by written permission of ETSI. -
NSS: an NTRU Lattice –Based Signature Scheme
ISSN(Online) : 2319 - 8753 ISSN (Print) : 2347 - 6710 International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 4, Issue 4, April 2015 NSS: An NTRU Lattice –Based Signature Scheme S.Esther Sukila Department of Mathematics, Bharath University, Chennai, Tamil Nadu, India ABSTRACT: Whenever a PKC is designed, it is also analyzed whether it could be used as a signature scheme. In this paper, how the NTRU concept can be used to form a digital signature scheme [1] is described. I. INTRODUCTION Digital Signature Schemes The notion of a digital signature may prove to be one of the most fundamental and useful inventions of modern cryptography. A digital signature or digital signature scheme is a mathematical scheme for demonstrating the authenticity of a digital message or document. The creator of a message can attach a code, the signature, which guarantees the source and integrity of the message. A valid digital signature gives a recipient reason to believe that the message was created by a known sender and that it was not altered in transit. Digital signatures are commonly used for software distribution, financial transactions etc. Digital signatures are equivalent to traditional handwritten signatures. Properly implemented digital signatures are more difficult to forge than the handwritten type. 1.1A digital signature scheme typically consists of three algorithms A key generationalgorithm, that selects a private key uniformly at random from a set of possible private keys. The algorithm outputs the private key and a corresponding public key. A signing algorithm, that given a message and a private key produces a signature. -
Scalable Verification for Outsourced Dynamic Databases Hwee Hwa PANG Singapore Management University, [email protected]
Singapore Management University Institutional Knowledge at Singapore Management University Research Collection School Of Information Systems School of Information Systems 8-2009 Scalable Verification for Outsourced Dynamic Databases Hwee Hwa PANG Singapore Management University, [email protected] Jilian ZHANG Singapore Management University, [email protected] Kyriakos MOURATIDIS Singapore Management University, [email protected] DOI: https://doi.org/10.14778/1687627.1687718 Follow this and additional works at: https://ink.library.smu.edu.sg/sis_research Part of the Databases and Information Systems Commons, and the Numerical Analysis and Scientific omputC ing Commons Citation PANG, Hwee Hwa; ZHANG, Jilian; and MOURATIDIS, Kyriakos. Scalable Verification for Outsourced Dynamic Databases. (2009). Proceedings of the VLDB Endowment: VLDB'09, August 24-28, Lyon, France. 2, (1), 802-813. Research Collection School Of Information Systems. Available at: https://ink.library.smu.edu.sg/sis_research/876 This Conference Proceeding Article is brought to you for free and open access by the School of Information Systems at Institutional Knowledge at Singapore Management University. It has been accepted for inclusion in Research Collection School Of Information Systems by an authorized administrator of Institutional Knowledge at Singapore Management University. For more information, please email [email protected]. Scalable Verification for Outsourced Dynamic Databases HweeHwa Pang Jilian Zhang Kyriakos Mouratidis School of Information Systems Singapore Management University fhhpang, jilian.z.2007, [email protected] ABSTRACT receive updated price quotes early could act ahead of their com- Query answers from servers operated by third parties need to be petitors, so the advantage of data freshness would justify investing verified, as the third parties may not be trusted or their servers may in the computing infrastructure. -
A Study on the Security of Ntrusign Digital Signature Scheme
A Thesis for the Degree of Master of Science A Study on the Security of NTRUSign digital signature scheme SungJun Min School of Engineering Information and Communications University 2004 A Study on the Security of NTRUSign digital signature scheme A Study on the Security of NTRUSign digital signature scheme Advisor : Professor Kwangjo Kim by SungJun Min School of Engineering Information and Communications University A thesis submitted to the faculty of Information and Commu- nications University in partial fulfillment of the requirements for the degree of Master of Science in the School of Engineering Daejeon, Korea Jan. 03. 2004 Approved by (signed) Professor Kwangjo Kim Major Advisor A Study on the Security of NTRUSign digital signature scheme SungJun Min We certify that this work has passed the scholastic standards required by Information and Communications University as a thesis for the degree of Master of Science Jan. 03. 2004 Approved: Chairman of the Committee Kwangjo Kim, Professor School of Engineering Committee Member Jae Choon Cha, Assistant Professor School of Engineering Committee Member Dae Sung Kwon, Ph.D NSRI M.S. SungJun Min 20022052 A Study on the Security of NTRUSign digital signature scheme School of Engineering, 2004, 43p. Major Advisor : Prof. Kwangjo Kim. Text in English Abstract The lattices have been studied by cryptographers for last decades, both in the field of cryptanalysis and as a source of hard problems on which to build encryption schemes. Interestingly, though, research about building secure and efficient -
(Not) to Design and Implement Post-Quantum Cryptography
SoK: How (not) to Design and Implement Post-Quantum Cryptography James Howe1 , Thomas Prest1 , and Daniel Apon2 1 PQShield, Oxford, UK. {james.howe,thomas.prest}@pqshield.com 2 National Institute of Standards and Technology, USA. [email protected] Abstract Post-quantum cryptography has known a Cambrian explo- sion in the last decade. What started as a very theoretical and mathe- matical area has now evolved into a sprawling research ˝eld, complete with side-channel resistant embedded implementations, large scale de- ployment tests and standardization e˙orts. This study systematizes the current state of knowledge on post-quantum cryptography. Compared to existing studies, we adopt a transversal point of view and center our study around three areas: (i) paradigms, (ii) implementation, (iii) deployment. Our point of view allows to cast almost all classical and post-quantum schemes into just a few paradigms. We highlight trends, common methodologies, and pitfalls to look for and recurrent challenges. 1 Introduction Since Shor's discovery of polynomial-time quantum algorithms for the factoring and discrete logarithm problems, researchers have looked at ways to manage the potential advent of large-scale quantum computers, a prospect which has become much more tangible of late. The proposed solutions are cryptographic schemes based on problems assumed to be resistant to quantum computers, such as those related to lattices or hash functions. Post-quantum cryptography (PQC) is an umbrella term that encompasses the design, implementation, and integration of these schemes. This document is a Systematization of Knowledge (SoK) on this diverse and progressive topic. We have made two editorial choices. -
Round5: a KEX Based on Learning with Rounding Over the Rings
Round5: a KEX based on learning with rounding over the rings Zhenfei Zhang [email protected] September 19, 2018 Round5 September 19, 2018 1 / 49 Areas I have been working on Theoretical results Signature schemes: Falcon (NTRUSign+GPV), pqNTRUSign Security proofs: Computational R-LWR problem Fully homomorphic encryptions Raptor: lattice based linkable ring signature (Blockchains!) Practical instantiations NTRU, Round5 Cryptanalysis and parameter derivation for lattices Efficient implementations: AVX-2 Constant time implementations Standardization: NIST, IETF, ETSI, ISO, CACR PQC process Deployment: enabling PQC for TLS, Tor, libgcrypt Under the radar: lattice based DAA, NIZK Round5 September 19, 2018 2 / 49 This talk Theoretical results Signature schemes: Falcon (NTRUSign+GPV), pqNTRUSign Security proofs: Computational R-LWR problem Fully homomorphic encryptions Raptor: lattice based linkable ring signature (Blockchains!) Practical instantiations NTRU, Round5 Cryptanalysis and parameter derivation for lattices Efficient implementations: AVX-2 Constant time implementations Standardization: NIST, IETF, ETSI, ISO, CACR PQC process Deployment: enabling PQC for TLS, Tor, libgcrypt Under the radar: lattice based DAA, NIZK Round5 September 19, 2018 3 / 49 Diffie-Hellman Alice:(a; A = g a) BoB(b; B = g b) A B g ab ab ∗ A; B and g are group elements over Zq. Round5 September 19, 2018 4 / 49 RLWE-KEX Alice:(a; A = Ga + ea) BoB(b; B = Gb + eb) A B aGb + aeb ≈ bGa + bea . Every element is a ring element over R .= Zq[x]=f (x). Round5 September 19, 2018 5 / 49 RLWE-KEX Alice:(a; X = Ga + ea) BoB(b; B = Gb + eb) A B Reconciliation r Extract(aGb + aeb; r) = Extract(bGa + bea; r) . -
Fault Analysis of the Ntrusign Digital Signature Scheme
Cryptogr. Commun. (2012) 4:131–144 DOI 10.1007/s12095-011-0061-3 Fault analysis of the NTRUSign digital signature scheme Abdel Alim Kamal · Amr M. Youssef Received: 11 December 2010 / Accepted: 14 December 2011 / Published online: 6 January 2012 © Springer Science+Business Media, LLC 2012 Abstract We present a fault analysis of the NTRUSign digital signature scheme. The utilized fault model is the one in which the attacker is assumed to be able to fault a small number of coefficients in a specific polynomial during the signing process but cannot control the exact location of the injected transient faults. For NTRUsign with parameters (N, q = pl, B, standard, N ), when the attacker is able to skip the norm-bound signature checking step, our attack needs one fault, ≈ − 1 (( )t) succeeds with probability 1 p and requires O qN steps when the number of faulted polynomial coefficients is upper bounded by t. The attack is also applicable to NTRUSign utilizing the transpose NTRU lattice but it requires double the number of fault injections. Different countermeasures against the proposed attack are investigated. Keywords Side channel attacks · Lattice-based public key cryptosystems · Fault analysis and countermeasures · Digital signature schemes · NTRU Mathematics Subject Classification (2010) 94A60 1 Introduction NTRUSign [1–4] is a parameterized family of lattice-based public key digital signa- ture schemes that is currently under consideration for standardization by the IEEE P1363 working group. It was proposed after the original NTRU signature scheme (NSS) [5] was broken [6, 7]. Similar to the GGH cryptosystem [8], the security of A. A. -
Public-Key Cryptography
Public-key Cryptogra; Theory and Practice ABHIJIT DAS Department of Computer Science and Engineering Indian Institute of Technology Kharagpur C. E. VENIMADHAVAN Department of Computer Science and Automation Indian Institute ofScience, Bangalore PEARSON Chennai • Delhi • Chandigarh Upper Saddle River • Boston • London Sydney • Singapore • Hong Kong • Toronto • Tokyo Contents Preface xiii Notations xv 1 Overview 1 1.1 Introduction 2 1.2 Common Cryptographic Primitives 2 1.2.1 The Classical Problem: Secure Transmission of Messages 2 Symmetric-key or secret-key cryptography 4 Asymmetric-key or public-key cryptography 4 1.2.2 Key Exchange 5 1.2.3 Digital Signatures 5 1.2.4 Entity Authentication 6 1.2.5 Secret Sharing 8 1.2.6 Hashing 8 1.2.7 Certification 9 1.3 Public-key Cryptography 9 1.3.1 The Mathematical Problems 9 1.3.2 Realization of Key Pairs 10 1.3.3 Public-key Cryptanalysis 11 1.4 Some Cryptographic Terms 11 1.4.1 Models of Attacks 12 1.4.2 Models of Passive Attacks 12 1.4.3 Public Versus Private Algorithms 13 2 Mathematical Concepts 15 2.1 Introduction 16 2.2 Sets, Relations and Functions 16 2.2.1 Set Operations 17 2.2.2 Relations 17 2.2.3 Functions 18 2.2.4 The Axioms of Mathematics 19 Exercise Set 2.2 20 2.3 Groups 21 2.3.1 Definition and Basic Properties . 21 2.3.2 Subgroups, Cosets and Quotient Groups 23 2.3.3 Homomorphisms 25 2.3.4 Generators and Orders 26 2.3.5 Sylow's Theorem 27 Exercise Set 2.3 29 2.4 Rings 31 2.4.1 Definition and Basic Properties 31 2.4.2 Subrings, Ideals and Quotient Rings 34 2.4.3 Homomorphisms 37 2.4.4 -
The Whole Is Less Than the Sum of Its Parts: Constructing More Efficient Lattice-Based Akes?
The Whole is Less than the Sum of its Parts: Constructing More Efficient Lattice-Based AKEs? Rafael del Pino1;2;3, Vadim Lyubashevsky4, and David Pointcheval3;2;1 1 INRIA, Paris 2 Ecole´ Normale Sup´erieure,Paris 3 CNRS 4 IBM Research Zurich Abstract. Authenticated Key Exchange (AKE) is the backbone of internet security protocols such as TLS and IKE. A recent announcement by standardization bodies calling for a shift to quantum-resilient crypto has resulted in several AKE proposals from the research community. Be- cause AKE can be generically constructed by combining a digital signature scheme with public key encryption (or a KEM), most of these proposals focused on optimizing the known KEMs and left the authentication part to the generic combination with digital signatures. In this paper, we show that by simultaneously considering the secrecy and authenticity require- ments of an AKE, we can construct a scheme that is more secure and with smaller communication complexity than a scheme created by a generic combination of a KEM with a signature scheme. Our improvement uses particular properties of lattice-based encryption and signature schemes and consists of two parts { the first part increases security, whereas the second reduces communication complexity. We first observe that parameters for lattice-based encryption schemes are always set so as to avoid decryption errors, since many observations by the adversary of such failures usually leads to him recovering the secret key. But since one of the requirements of an AKE is that it be forward-secure, the public key must change every time. The intuition is therefore that one can set the parameters of the scheme so as to not care about decryption errors and everything should still remain secure. -
How to Share a Lattice Trapdoor: Threshold Protocols for Signatures and (H)IBE
How to Share a Lattice Trapdoor: Threshold Protocols for Signatures and (H)IBE Rikke Bendlin∗ Sara Krehbiely Chris Peikertz Abstract We develop secure threshold protocols for two important operations in lattice cryptography, namely, generating a hard lattice Λ together with a “strong” trapdoor, and sampling from a discrete Gaussian distribution over a desired coset of Λ using the trapdoor. These are the central operations of many crypto- graphic schemes: for example, they are exactly the key-generation and signing operations (respectively) for the GPV signature scheme, and they are the public parameter generation and private key extraction operations (respectively) for the GPV IBE. We also provide a protocol for trapdoor delegation, which is used in lattice-based hierarchical IBE schemes. Our work therefore directly transfers all these systems to the threshold setting. Our protocols provide information-theoretic (i.e., statistical) security against adaptive corruptions in the UC framework, and they are private and robust against an optimal number of semi-honest or malicious parties. Our Gaussian sampling protocol is both noninteractive and efficient, assuming either a trusted setup phase (e.g., performed as part of key generation) or a sufficient amount of interactive but offline precomputation, which can be performed before the inputs to the sampling phase are known. ∗Department of Computer Science, Aarhus University. Email: [email protected]. Supported by the Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, within which part of this work was performed; and also from the CFEM research center (supported by the Danish Strategic Research Council). -
Low-Latency Elliptic Curve Scalar Multiplication
Noname manuscript No. (will be inserted by the editor) Low-Latency Elliptic Curve Scalar Multiplication Joppe W. Bos Received: date / Accepted: date Abstract This paper presents a low-latency algorithm designed for parallel computer architectures to compute the scalar multiplication of elliptic curve points based on approaches from cryptographic side-channel analysis. A graph- ics processing unit implementation using a standardized elliptic curve over a 224-bit prime field, complying with the new 112-bit security level, computes the scalar multiplication in 1.9 milliseconds on the NVIDIA GTX 500 archi- tecture family. The presented methods and implementation considerations can be applied to any parallel 32-bit architecture. Keywords Elliptic Curve Cryptography · Elliptic Curve Scalar Multiplica- tion · Parallel Computing · Low-Latency Algorithm 1 Introduction Elliptic curve cryptography (ECC) [30,37] is an approach to public-key cryp- tography which enjoys increasing popularity since its invention in the mid 1980s. The attractiveness of small key-sizes [32] has placed this public-key cryptosystem as the preferred alternative to the widely used RSA public-key cryptosystem [48]. This is emphasized by the current migration away from 80-bit to 128-bit security where, for instance, the United States' National Se- curity Agency restricts the use of public key cryptography in \Suite B" [40] to ECC. The National Institute of Standards and Technology (NIST) is more flexible and settles for 112-bit security at the lowest level from the year 2011 on [52]. At the same time there is a shift in architecture design towards many-core processors [47]. Following both these trends, we evaluate the performance of a Laboratory for Cryptologic Algorithms Ecole´ Polytechnique F´ed´eralede Lausanne (EPFL) Station 14, CH-1015 Lausanne, Switzerland E-mail: joppe.bos@epfl.ch 2 Joppe W. -
Post-Quantum Key Exchange for the Internet and the Open Quantum Safe Project∗
Post-Quantum Key Exchange for the Internet and the Open Quantum Safe Project∗ Douglas Stebila1;y and Michele Mosca2;3;4;z 1 Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada [email protected] 2 Institute for Quantum Computing and Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, Canada 3 Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada 4 Canadian Institute for Advanced Research, Toronto, Ontario, Canada [email protected] July 28, 2017 Abstract Designing public key cryptosystems that resist attacks by quantum computers is an important area of current cryptographic research and standardization. To retain confidentiality of today's communications against future quantum computers, applications and protocols must begin exploring the use of quantum-resistant key exchange and encryption. In this paper, we explore post-quantum cryptography in general and key exchange specifically. We review two protocols for quantum-resistant key exchange based on lattice problems: BCNS15, based on the ring learning with errors problem, and Frodo, based on the learning with errors problem. We discuss their security and performance characteristics, both on their own and in the context of the Transport Layer Security (TLS) protocol. We introduce the Open Quantum Safe project, an open-source software project for prototyping quantum-resistant cryptography, which includes liboqs, a C library of quantum-resistant algorithms, and our integrations of liboqs into popular open-source applications and protocols, including the widely used OpenSSL library. ∗Based on the Stafford Tavares Invited Lecture at Selected Areas in Cryptography (SAC) 2016 by D. Stebila. ySupported in part by Australian Research Council (ARC) Discovery Project grant DP130104304, Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery grant RGPIN-2016-05146, and an NSERC Discovery Accelerator Supplement grant.