On Notions of Security for Deterministic Encryption, and Efficient Constructions Without Random Oracles
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Modern Cryptography: Lecture 13 Digital Signatures
Modern Cryptography: Lecture 13 Digital Signatures Daniel Slamanig Organizational ● Where to find the slides and homework? – https://danielslamanig.info/ModernCrypto18.html ● ow to conta!t me? – [email protected] ● #utor: Karen Klein – [email protected] ● Offi!ial page at #', )o!ation et!. – https://tiss.tuwien.ac.at/!ourse/!o$rseDetails.+html?dswid=86-2&dsrid,6/0. !ourseNr,102262&semester,2018W ● #utorial, #U site – https://tiss.tuwien.ac.at/!ourse/!o$rseAnnouncement.+html?dswid=4200.dsr id,-51.!ourseNum6er,10206-.!ourseSemester,2018W ● 8+am for the se!ond part: Thursday 31.21.2210 15:00-1/:00 (Tutorial slot; 2/26 Overview Digital Signatures message m / :m( σ; secret key skA Inse!$re !hannel p$bli! key pkA p$bli! key pkA σ σ :, 7igskA:m; 6 :, =rfypkA:m( ; 3: pkA -/26 Digital Signatures: Intuitive Properties Can 6e seen as the p$6li!9key analog$e of M3Cs with p$6li! >erifiability ● Integrity protection: 3ny modifi!ation of a signed message !an 6e detected ● 7o$r!e a$thenti!ity: The sender of a signed message !an be identified ● Non9rep$diation: The signer !annot deny ha>ing signed :sent) a message 7e!$rity :intuition;: sho$ld 6e hard to !ome $p with a signat$re for a message that has not 6een signed by the holder of the pri>ate key 5/26 Digital Signatures: pplications *igital signat$res ha>e many applications and are at the heart of implementing p$6lic9key !ryptography in practice ● <ss$ing !ertifi!ates 6y CAs (?$6lic %ey <nfrastr$!t$res;: 6inding of identities to p$6lic keys ● @$ilding authenticated !hannels: a$thenti!ate parties :ser>ers; in sec$rity proto!ols :e.g.( TL7; or se!$re messaging :Whats3pp( 7ignal, ...; ● Code signing: a$thenti!ate software/firmware :$pdates; ● 7ign do!$ments :e.g.( !ontra!ts;: )egal reg$lations define when digital signat$res are eq$ivalent to handwritten signat$res ● 7ign transa!tions: $sed in the !rypto!$rren!y realm ● et!. -
Practical Homomorphic Encryption and Cryptanalysis
Practical Homomorphic Encryption and Cryptanalysis Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) an der Fakult¨atf¨urMathematik der Ruhr-Universit¨atBochum vorgelegt von Dipl. Ing. Matthias Minihold unter der Betreuung von Prof. Dr. Alexander May Bochum April 2019 First reviewer: Prof. Dr. Alexander May Second reviewer: Prof. Dr. Gregor Leander Date of oral examination (Defense): 3rd May 2019 Author's declaration The work presented in this thesis is the result of original research carried out by the candidate, partly in collaboration with others, whilst enrolled in and carried out in accordance with the requirements of the Department of Mathematics at Ruhr-University Bochum as a candidate for the degree of doctor rerum naturalium (Dr. rer. nat.). Except where indicated by reference in the text, the work is the candidates own work and has not been submitted for any other degree or award in any other university or educational establishment. Views expressed in this dissertation are those of the author. Place, Date Signature Chapter 1 Abstract My thesis on Practical Homomorphic Encryption and Cryptanalysis, is dedicated to efficient homomor- phic constructions, underlying primitives, and their practical security vetted by cryptanalytic methods. The wide-spread RSA cryptosystem serves as an early (partially) homomorphic example of a public- key encryption scheme, whose security reduction leads to problems believed to be have lower solution- complexity on average than nowadays fully homomorphic encryption schemes are based on. The reader goes on a journey towards designing a practical fully homomorphic encryption scheme, and one exemplary application of growing importance: privacy-preserving use of machine learning. -
A Survey on the Provable Security Using Indistinguishability Notion on Cryptographic Encryption Schemes
A SURVEY ON THE PROVABLE SECURITY USING INDISTINGUISHABILITY NOTION ON CRYPTOGRAPHIC ENCRYPTION SCHEMES A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF APPLIED MATHEMATICS OF MIDDLE EAST TECHNICAL UNIVERSITY BY EMRE AYAR IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN CRYPTOGRAPHY FEBRUARY 2018 Approval of the thesis: A SURVEY ON THE PROVABLE SECURITY USING INDISTINGUISHABILITY NOTION ON CRYPTOGRAPHIC ENCRYPTION SCHEMES submitted by EMRE AYAR in partial fulfillment of the requirements for the degree of Master of Science in Department of Cryptography, Middle East Technical University by, Prof. Dr. Om¨ ur¨ Ugur˘ Director, Graduate School of Applied Mathematics Prof. Dr. Ferruh Ozbudak¨ Head of Department, Cryptography Assoc. Prof. Dr. Ali Doganaksoy˘ Supervisor, Cryptography, METU Dr. Onur Koc¸ak Co-supervisor, TUB¨ ITAK˙ - UEKAE, Istanbul˙ Examining Committee Members: Assoc. Prof. Dr. Murat Cenk Cryptography, METU Assoc. Prof. Dr. Ali Doganaksoy˘ Department of Mathematics, METU Assist. Prof. Dr. Fatih Sulak Department of Mathematics, Atılım University Date: I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last Name: EMRE AYAR Signature : v vi ABSTRACT A SURVEY ON THE PROVABLE SECURITY USING INDISTINGUISHABILITY NOTION ON CRYPTOGRAPHIC ENCRYPTION SCHEMES Ayar, Emre M.S., Department of Cryptography Supervisor : Assoc. Prof. Dr. Ali Doganaksoy˘ Co-Supervisor : Dr. Onur Koc¸ak February 2018, 44 pages For an encryption scheme, instead of Shannon’s perfect security definition, Goldwasser and Micali defined a realistic provable security called semantic security. -
A Provably-Unforgeable Threshold Eddsa with an Offline Recovery Party
A Provably-Unforgeable Threshold EdDSA with an Offline Recovery Party Michele Battagliola,∗ Riccardo Longo,y Alessio Meneghetti,z and Massimiliano Salax Department of Mathematics, University Of Trento Abstract A(t; n)-threshold signature scheme enables distributed signing among n players such that any subset of size at least t can sign, whereas any subset with fewer players cannot. The goal is to produce threshold digital signatures that are compatible with an existing centralized sig- nature scheme. Starting from the threshold scheme for the ECDSA signature due to Battagliola et al., we present the first protocol that supports EdDSA multi-party signatures with an offline participant dur- ing the key-generation phase, without relying on a trusted third party. Under standard assumptions we prove our scheme secure against adap- tive malicious adversaries. Furthermore we show how our security notion can be strengthen when considering a rushing adversary. We discuss the resiliency of the recovery in the presence of a malicious party. Using a classical game-based argument, we prove that if there is an adversary capable of forging the scheme with non-negligible prob- ability, then we can build a forger for the centralized EdDSA scheme with non-negligible probability. Keywords| 94A60 Cryptography, 12E20 Finite fields, 14H52 Elliptic curves, 94A62 Authentication and secret sharing, 68W40 Analysis of algorithms 1 Introduction A(t; n)-threshold signature scheme is a multi-party computation protocol that en- arXiv:2009.01631v1 [cs.CR] 2 Sep 2020 ables a subset of at least t among n authorized players to jointly perform digital signatures. The flexibility and security advantages of threshold protocols have be- come of central importance in the research for new cryptographic primitives [9]. -
Formalizing Public Key Cryptography
Cryptography CS 555 Topic 29: Formalizing Public Key Cryptography 1 Recap • Key Management • Diffie Hellman Key Exchange • Password Authenticated Key Exchange (PAKEs) 2 Public Key Encryption: Basic Terminology • Plaintext/Plaintext Space • A message m c • Ciphertext ∈ ℳ • Public/Private Key Pair , ∈ ∈ 3 Public Key Encryption Syntax • Three Algorithms • Gen(1 , ) (Key-generation algorithm) • Input: Random Bits R Alice must run key generation • Output: , algorithm in advance an publishes the public key: pk • Enc ( ) (Encryption algorithm) pk ∈ • Decsk( ) (Decryption algorithm) • Input: Secret∈ key sk and a ciphertex c • Output: a plaintext message m Assumption: Adversary only gets to see pk (not sk) ∈ ℳ • Invariant: Decsk(Encpk(m))=m 4 Chosen-Plaintext Attacks • Model ability of adversary to control or influence what the honest parties encrypt. • Historical Example: Battle of Midway (WWII). • US Navy cryptanalysts were able to break Japanese code by tricking Japanese navy into encrypting a particular message • Private Key Cryptography 5 Recap CPA-Security (Symmetric Key Crypto) m0,1,m1,1 c1 = EncK(mb,1) m0,2,m1,2 c2 = EncK(mb,2) m0,3,m1,3 c3 = EncK(mb,3) … b’ Random bit b (negligible) s. t K = Gen(.) 1 Pr = + ( ) ∀ ∃ 2 6 ′ ≤ Chosen-Plaintext Attacks • Model ability of adversary to control or influence what the honest parties encrypt. • Private Key Crypto • Attacker tricks victim into encrypting particular messages • Public Key Cryptography • The attacker already has the public key pk • Can encrypt any message s/he wants! • CPA Security is critical! 7 CPA-Security (PubK , n ) Public Key:LR pk−cpa , A Π 1 1 = 0 1 , = 2 , 2 0 1 3 3 = 0 1 … b’ Random bit b (negligible) s. -
Securing Threshold Cryptosystems Against Chosen Ciphertext Attack∗
Securing Threshold Cryptosystems against Chosen Ciphertext Attack∗ Victor Shoupy Rosario Gennaroz September 18, 2001 Abstract For the most compelling applications of threshold cryptosystems, security against chosen ciphertext attack is a requirement. However, prior to the results presented here, there appeared to be no practical threshold cryptosystems in the literature that were provably chosen-ciphertext secure, even in the idealized random oracle model. The contribution of this paper is to present two very practical threshold cryptosystems, and to prove that they are secure against chosen ciphertext attack in the random oracle model. Not only are these protocols computationally very efficient, but they are also non-interactive, which means they can be easily run over an asynchronous communication network. 1 Introduction In a threshold cryptosystem, the secret key of a public key cryptosystem is shared among a set of decryption servers, so that a quorum of servers can act together to decrypt a given ciphertext. Just as for ordinary, non-threshold cryptosystems, a natural and very useful notion of security is that of security against chosen ciphertext attack. In this paper, we consider the problem of designing threshold cryptosystems that are secure against chosen ciphertext attack. Our goal is to design a practical scheme, and provide strong evidence that it cannot be broken. Even though the most compelling applications of threshold cryptosystems seem to require chosen-ciphertext security, prior to the results presented here, there appeared to be no practi- cal threshold cryptosystems in the literature that were provably secure | even in the random oracle model, where one models a cryptographic hash function as a random oracle. -
Probabilistic Public-Key Encryption
PROBABILISTIC PUBLIC-KEY ENCRYPTION By Azymbek Beibyt Fundamentals Semantic Security A public-key cryptosystem is semantically secure if it’s computationally infeasible for opponent to derive any information about plaintext given only ciphertext and public key. Semantic Security = Cryptosystem Indistinguishability Fundamentals Semantic Security (Cont.) Semantic security are obtained using trapdoor (one- way) functions: - prime factorization in RSA - discrete logarithm problem in ElGamal Fundamentals Semantic Security (Cont.) Why not use trapdoor functions as before? ¨ The fact that f is a trapdoor function does not rule out the possibility of computing x from f (x) when x is of a special form. ¨ The fact that f is a trapdoor function does not rule out the possibility of easily computing some partial information about x (even every other bit of x) from f(x). Probabilistic Encryption New approach introduced by Shafi Goldwasser and Silvio Micali in 1983. ¨ Replace deterministic block encryption by probabilistic encryption of single bits. ¨ Proved to be hard to extract any information about plaintext under polynomially bounded computational resources, because is based on intractability of deciding Quadratic Residuosity modulo composite numbers whose factorization is unknown. Probabilistic Public -Key Cryptosystem General Idea (P, C, K, E, D, R), where R is a set of randomizers, encryption is public and decryption is secret and following properties should be satisfied: 1. , where b Є P, r Є R and if Probabilistic Public -Key Cryptosystem General Idea (Cont.) 2. Let ϵ be specified security parameter. Define probability distribution on C, which denotes the probability that y is the ciphertext given that K is the key and x is the plaintext . -
A CCA2 Secure Public Key Encryption Scheme Based on the Mceliece Assumptions in the Standard Model
A CCA2 Secure Public Key Encryption Scheme Based on the McEliece Assumptions in the Standard Model Rafael Dowsley1, J¨ornM¨uller-Quade2, Anderson C. A. Nascimento1 1 Department of Electrical Engineering, University of Brasilia. Campus Universit´arioDarcy Ribeiro,Brasilia, CEP: 70910-900, Brazil, Email:[email protected], [email protected] 2 Universit¨atKarlsruhe, Institut f¨urAlgorithmen und Kognitive Systeme. Am Fasanengarten 5, 76128 Karlsruhe, Germany. E-mail: [email protected] We show that a recently proposed construction by Rosen and Segev can be used for obtaining the first public key encryption scheme based on the McEliece assumptions which is secure against adaptive chosen ciphertext attacks in the standard model. 1 Introduction Indistinguishability of messages under adaptive chosen ciphertext attacks is the strongest known notion of security for public key encryption schemes (PKE). Many computational assumptions have been used in the literature for obtain- ing cryptosystems meeting such a strong security requirements. Given one-way trapdoor permutations, we know how to obtain CCA2 security from any seman- tically secure public key cryptosystem [14, 20, 12]. Efficient constructions are also known based on number-theoretic assumptions [6] or on identity based encryp- tion schemes [3]. Obtaining a CCA2 secure cryptosystem (even an inefficient one) based on the McEliece assumptions in the standard model has been an open problem in this area for quite a while. Recently, Rosen and Segev proposed an elegant and simple new computa- tional assumption for obtaining CCA2 secure PKEs: correlated products [19]. They provided constructions of correlated products based on the existence of certain lossy trapdoor functions [16] which in turn can be based on the deci- sional Diffie-Hellman problem and on Paillier's decisional residuosity problem [16]. -
Arx: an Encrypted Database Using Semantically Secure Encryption
Arx: An Encrypted Database using Semantically Secure Encryption Rishabh Poddar Tobias Boelter Raluca Ada Popa UC Berkeley UC Berkeley UC Berkeley [email protected] [email protected] [email protected] ABSTRACT some of which are property-preserving by design (denoted In recent years, encrypted databases have emerged as a PPE schemes), e.g., order-preserving encryption (OPE) [8, promising direction that provides data confidentiality with- 9, 71] or deterministic encryption (DET). OPE and DET out sacrificing functionality: queries are executed on en- are designed to reveal the order and the equality relation crypted data. However, many practical proposals rely on a between data items, respectively, to enable fast order and set of weak encryption schemes that have been shown to leak equality operations. However, while these PPE schemes con- sensitive data. In this paper, we propose Arx, a practical fer protection in some specific settings, a series of recent and functionally rich database system that encrypts the data attacks [26, 37, 61] have shown that given certain auxiliary only with semantically secure encryption schemes. We show information, an attacker can extract significant sensitive in- that Arx supports real applications such as ShareLaTeX with formation from the order and equality relations revealed by a modest performance overhead. these schemes. These works demonstrate offline attacks in which the attacker steals a PPE-encrypted database and PVLDB Reference Format: analyzes it offline. Rishabh Poddar, Tobias Boelter, and Raluca Ada Popa. Arx: Leakage from queries refers to what an (online) attacker An Encrypted Database using Semantically Secure Encryption. PVLDB, 12(11): 1664-1678, 2019. -
Chapter 4 Symmetric Encryption
Chapter 4 Symmetric Encryption The symmetric setting considers two parties who share a key and will use this key to imbue communicated data with various security attributes. The main security goals are privacy and authenticity of the communicated data. The present chapter looks at privacy, Chapter ?? looks at authenticity, and Chapter ?? looks at providing both together. Chapters ?? and ?? describe tools we shall use here. 4.1 Symmetric encryption schemes The primitive we will consider is called an encryption scheme. Such a scheme speci¯es an encryption algorithm, which tells the sender how to process the plaintext using the key, thereby producing the ciphertext that is actually transmitted. An encryption scheme also speci¯es a decryption algorithm, which tells the receiver how to retrieve the original plaintext from the transmission while possibly performing some veri¯cation, too. Finally, there is a key-generation algorithm, which produces a key that the parties need to share. The formal description follows. De¯nition 4.1 A symmetric encryption scheme SE = (K; E; D) consists of three algorithms, as follows: ² The randomized key generation algorithm K returns a string K. We let Keys(SE) denote the set of all strings that have non-zero probability of be- ing output by K. The members of this set are called keys. We write K Ã$ K for the operation of executing K and letting K denote the key returned. ² The encryption algorithm E takes a key K 2 Keys(SE) and a plaintext M 2 f0; 1g¤ to return a ciphertext C 2 f0; 1g¤ [ f?g. -
Security II: Cryptography
Security II: Cryptography Markus Kuhn Computer Laboratory Lent 2012 { Part II http://www.cl.cam.ac.uk/teaching/1213/SecurityII/ 1 Related textbooks Jonathan Katz, Yehuda Lindell: Introduction to Modern Cryptography Chapman & Hall/CRC, 2008 Christof Paar, Jan Pelzl: Understanding Cryptography Springer, 2010 http://www.springerlink.com/content/978-3-642-04100-6/ http://www.crypto-textbook.com/ Douglas Stinson: Cryptography { Theory and Practice 3rd ed., CRC Press, 2005 Menezes, van Oorschot, Vanstone: Handbook of Applied Cryptography CRC Press, 1996 http://www.cacr.math.uwaterloo.ca/hac/ 2 Private-key (symmetric) encryption A private-key encryption scheme is a tuple of probabilistic polynomial-time algorithms (Gen; Enc; Dec) and sets K; M; C such that the key generation algorithm Gen receives a security parameter ` and outputs a key K Gen(1`), with K 2 K, key length jKj ≥ `; the encryption algorithm Enc maps a key K and a plaintext m message M 2 M = f0; 1g to a ciphertext message C EncK (M); the decryption algorithm Dec maps a key K and a ciphertext n C 2 C = f0; 1g (n ≥ m) to a plaintext message M := DecK (C); ` m for all `, K Gen(1 ), and M 2 f0; 1g : DecK (EncK (M)) = M. Notes: A \probabilistic algorithm" can toss coins (uniformly distributed, independent). Notation: assigns the output of a probabilistic algorithm, := that of a deterministic algorithm. A \polynomial-time algorithm" has constants a; b; c such that the runtime is always less than a · `b + c if the input is ` bits long. (think Turing machine) Technicality: we supply the security parameter ` to Gen here in unary encoding (as a sequence of ` \1" bits: 1`), merely to remain compatible with the notion of \input size" from computational ` complexity theory. -
Public Key Encryption with Equality Test in the Standard Model
Public Key Encryption with Equality Test in the Standard Model Hyung Tae Lee1, San Ling1, Jae Hong Seo2, Huaxiong Wang1, and Taek-Young Youn3 1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore fhyungtaelee,lingsan,[email protected] 2 Myongji University, Korea [email protected] 3 Electonics and Telecommunications Research Institute, Korea [email protected] Abstract. Public key encryption with equality test (PKEET) is a cryp- tosystem that allows a tester who has trapdoors issued by one or more users Ui to perform equality tests on ciphertexts encrypted using public key(s) of Ui. Since this feature has a lot of practical applications in- cluding search on encrypted data, several PKEET schemes have been proposed so far. However, to the best of our knowledge, all the existing proposals are proven secure only under the hardness of number-theoretic problems and the random oracle heuristics. In this paper, we show that this primitive can be achieved not only generically from well-established other primitives but also even with- out relying on the random oracle heuristics. More precisely, our generic construction for PKEET employs a two-level hierarchical identity-based encryption scheme, which is selectively secure against chosen plaintext attacks, a strongly unforgeable one-time signature scheme and a cryp- tographic hash function. Our generic approach toward PKEET has sev- eral advantages over all the previous works; it directly leads the first standard model construction and also directly implies the first lattice- based construction. Finally, we show how to extend our approach to the identity-based setting.