DOCSLIB.ORG

Efficient Cryptosystem for Universally Verifiable Mixnets

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Efficient Cryptosystem for Universally Verifiable Mixnets EFFICIENT CRYPTOSYSTEM FOR UNIVERSALLY VERIFIABLE MIXNETS Sandra Guasch Castelló Company Supervisor: Jordi Puiggalí Allepuz Academic Supervisor: Miquel Soriano Ibáñez Bachelor Degree of Telecomunication Engineering Universitat Politècnica de Catalunya, ETSETB Scytl Secure Electronic Voting December 2009 This PFC has been done by means of a university-company collaboration program in Scytl Secure Electronic Voting. 1 2 Agraïments M’agradaria primer de tot agrair an es meus pares i an es meu germà es recolzament que m’han donat aquests últims dies de projecte, tot i que no acabéssin d’entendre sa feina i presió que comporta. Sobretot a sa meua mare, que m’ha fet correccions d’anglès a sa introducció i a ses conclusions “perquè lo altre, que és més tècnic, ja ho sabràs tu millor”. Després d’ells, es següent que és mereix es meu agraïment és en Ferran, que ha tengut (i té) infinita paciència amb es meus histerismes, dubtes i queixes. Amb ell he compartit s’últim any de sa carrera i es procés de realització des projecte, i poder comptar sempre amb una persona que estava en sa mateixa situació que jo m’ha ajudat molt. Grasis! També m’agradaria agrair as meus companys de classe de sa enginyeria superior es anys de carrera, durs però entretenguts, i a ses meues amigues de sa tècnica i d’Eivissa poder-nos ajuntar de quan en quan per fer uns cafès, contar-nos ses nostres experiències en es camins diferents que hem près, i riure’ns un poc de tot. No puc deixar d’anomenar aquí a la Paz Morillo, que ha estat sempre allí per a qualsevol dubte que he tengut. Finalment, m’agradaria agraïr al Miquel Soriano i al Jordi Puggalí sa oportunitat que m’han donat de entrar a treballar a Scytl per fer allí es meu PFC, i an es meus companys de feina, sobretot a en Miquel Comas i a en Jaume Poch, per fer de sa feina un lloc més agradable i entretengut. 3 4 Index of Contents 1. Introduction .......................................................................................................................... 9 1.1. Democracy and electoral processes: ............................................................................. 9 1.2. Security in remote e-Voting ........................................................................................ 10 1.3. Securing elections using basic cryptography .............................................................. 11 1.3.1. Vote encryption ................................................................................................... 11 1.3.2. Digital signature .................................................................................................. 12 2. e-Voting protocols focused in the protection of the voter privacy .................................... 15 2.1. Protocols that implement anonymity in the pre-election phase ................................ 15 2.2. Protocols that implement anonymity in the voting phase ......................................... 18 2.3. Protocols that implement anonymity in the counting phase ..................................... 20 2.3.1. Homomorphic tally protocols ............................................................................. 20 2.3.2. Mixnets ................................................................................................................ 25 3. Building Blocks .................................................................................................................... 35 3.1. Definition of a Vote anonymization system ................................................................ 35 3.2. Definition of the Integrity Proofs and the Verification process .................................. 37 3.3. Cryptosystem Definition .............................................................................................. 38 3.3.1. RSA ...................................................................................................................... 39 3.3.2. ElGamal ............................................................................................................... 42 3.3.3. Paillier .................................................................................................................. 43 4. Design of a Cryptosystem for an Efficient and Universally Verifiable Mixnet .................... 47 4.1. Initial Assumptions ...................................................................................................... 47 4.2. Analysis Model ............................................................................................................ 48 4.3. ElGamal Cryptosystem ................................................................................................ 49 4.3.1. Conventional ElGamal encryption ....................................................................... 50 4.3.2. ElGamal homomorphisms ................................................................................... 51 4.3.3. Definition of ElGamal for semantic security in voting applications .................... 52 4.4. Proposals for a new cryptosystem based on ElGamal encryption .............................. 54 4.4.1. Proposal 0: ElGamal conventional cryptosystem ................................................ 55 4.4.2. Proposal 1 ............................................................................................................ 56 4.4.3. Proposal 2 ............................................................................................................ 57 4.4.4. Proposal 3 (our first proposal) ............................................................................ 59 4.4.5. The re-encryption process .................................................................................. 64 4.4.6. Proposal 4 (our second proposal) ....................................................................... 65 5 4.4.7. Problems found in the previous proposals ......................................................... 71 4.4.8. Proposal 5 (our third proposal) ........................................................................... 75 4.4.9. Proposal 6 ............................................................................................................ 90 4.4.10. Conclusions about the proposed cryptosystems .............................................. 105 4.5. Elliptic Curve Cryptography: EC ElGamal Cryptosystem ........................................... 107 4.5.1. Elliptic Curve Cryptography ............................................................................... 107 4.5.2. EC Homomorphic Cryptosystems ...................................................................... 112 4.5.3. Use of the EC ElGamal cryptosystem in homomorphic schemes. Proposal for a mixing process ................................................................................................................... 114 5. Zero Knowledge Proofs in ElGamal cryptosystems ........................................................... 121 5.1. Schnorr Identification Protocol ................................................................................. 122 5.1.1. Protocol: ............................................................................................................ 123 5.1.2. Schnorr Signature .............................................................................................. 123 5.2. Proof of Discrete-Log equality - Chaum-Pedersen Protocol ..................................... 123 5.3. Proof of Plaintext Knowledge .................................................................................... 125 5.4. Proof of Plaintext Equivalence .................................................................................. 126 5.5. Proof of Subset Membership .................................................................................... 128 5.6. Proof of Correct Decryption using Schnorr Identification Protocol .......................... 130 6. Case Study: use of the proposed encryption algorithm in a Universally Verifiable Mixnet 133 6.1. Defining the cryptosystem ........................................................................................ 134 6.2. Voting Phase .............................................................................................................. 135 6.2.1. Vote casting ....................................................................................................... 135 6.2.2. Vote reception ................................................................................................... 135 6.3. Mixing Phase ............................................................................................................. 136 6.4. Decryption and Counting Phase ................................................................................ 141 7. Conclusions ....................................................................................................................... 145 8. References ......................................................................................................................... 147 6 Index of Figures Fig. 1 - 1 Electronic voting using basic cryptography .................................................................. 13 Fig. 2 - 1 Example of paper ballot in Pollsterless ......................................................................... 16 Fig. 2 - 2 Communications in a two agencies model ................................................................... 19 Fig. 2 - 3 Example of an additive homomorphic tally .................................................................. 21 Fig. 2 - 4 Example of a multiplicative homomorphic
Load more

Recommended publications
  • Authenticating Computation on Groups: New Homomorphic Primitives and Applications
    Universita` degli Studi di Catania Dipartimento di Matematica ed Informatica Dottorato di Ricerca in Matematica Pura ed Applicata XXVI ciclo Orazio Puglisi Authenticating Computation on Groups: New Homomorphic Primitives and Applications Advisor: Ch.mo Prof. Dario Catalano ANNO ACCADEMICO 2013-2014 Contents Contents i Acknowledgments iv 1 Introduction 1 1.1 Afewwordsaboutcryptographyhistory . 1 1.2 From Encryption to Homomorphic Encryption . 3 1.3 Whatabouthomomorphicsignatures? . 4 1.4 Fromaconcreteproblemtoanewprimitive . 6 1.5 Organizationofthisthesis . 8 2 Preliminaries and notations 10 2.1 BasicNotations .......................... 10 2.1.1 Probabilisticnotation. 10 2.1.2 NumberTheory...................... 11 2.1.3 Pairings .......................... 11 2.1.4 Computationalassumptions . 12 2.2 PrimitivesandSecurity. 13 2.2.1 Usersandprimitives . 13 2.2.2 Indistinguishability under CPA and CCA . 15 2.2.3 Theasymptoticapproach . 16 2.2.4 Primitives ......................... 17 2.2.4.1 HashFunction . 17 2.2.4.2 ChameleonHashFunction . 18 2.2.4.3 PublicKeyEncryption. 19 i CONTENTS ii 2.2.4.3.1 Security for Public Key Encryption Schemes. ............... 20 2.2.4.3.2 Paillier Encryption Scheme . 22 2.2.4.4 Signatures. 23 2.2.4.4.1 Security for Digital Signatures Schemes 24 2.2.4.4.2 WatersSignature . 26 2.2.4.5 AuthenticatedEncryption . 27 2.2.4.6 SigmaProtocol . 28 2.2.4.6.1 SchnorrSigmaProtocol . 30 2.2.5 Homomorphicprimitives . 30 3 A linearly homomorphic signature scheme to sign elements in bilinear groups 32 3.1 Linear Network Coding and Linearly Homomorphic Signatures 33 3.2 HomomorphicSignaturesscheme . 34 3.2.1 Correctness and Security for Homomorphic Signatures 35 3.3 LHSG:Definition ........................
    [Show full text]
  • The Twin Diffie-Hellman Problem and Applications
    The Twin Diffie-Hellman Problem and Applications David Cash1 Eike Kiltz2 Victor Shoup3 February 10, 2009 Abstract We propose a new computational problem called the twin Diffie-Hellman problem. This problem is closely related to the usual (computational) Diffie-Hellman problem and can be used in many of the same cryptographic constructions that are based on the Diffie-Hellman problem. Moreover, the twin Diffie-Hellman problem is at least as hard as the ordinary Diffie-Hellman problem. However, we are able to show that the twin Diffie-Hellman problem remains hard, even in the presence of a decision oracle that recognizes solutions to the problem — this is a feature not enjoyed by the Diffie-Hellman problem in general. Specifically, we show how to build a certain “trapdoor test” that allows us to effectively answer decision oracle queries for the twin Diffie-Hellman problem without knowing any of the corresponding discrete logarithms. Our new techniques have many applications. As one such application, we present a new variant of ElGamal encryption with very short ciphertexts, and with a very simple and tight security proof, in the random oracle model, under the assumption that the ordinary Diffie-Hellman problem is hard. We present several other applications as well, including: a new variant of Diffie and Hellman’s non-interactive key exchange protocol; a new variant of Cramer-Shoup encryption, with a very simple proof in the standard model; a new variant of Boneh-Franklin identity-based encryption, with very short ciphertexts; a more robust version of a password-authenticated key exchange protocol of Abdalla and Pointcheval.
    [Show full text]
  • Blind Schnorr Signatures in the Algebraic Group Model
    An extended abstract of this work appears in EUROCRYPT’20. This is the full version. Blind Schnorr Signatures and Signed ElGamal Encryption in the Algebraic Group Model Georg Fuchsbauer1, Antoine Plouviez2, and Yannick Seurin3 1 TU Wien, Austria 2 Inria, ENS, CNRS, PSL, Paris, France 3 ANSSI, Paris, France first.last@{tuwien.ac.at,ens.fr,m4x.org} January 16, 2021 Abstract. The Schnorr blind signing protocol allows blind issuing of Schnorr signatures, one of the most widely used signatures. Despite its practical relevance, its security analysis is unsatisfactory. The only known security proof is rather informal and in the combination of the generic group model (GGM) and the random oracle model (ROM) assuming that the “ROS problem” is hard. The situation is similar for (Schnorr-)signed ElGamal encryption, a simple CCA2-secure variant of ElGamal. We analyze the security of these schemes in the algebraic group model (AGM), an idealized model closer to the standard model than the GGM. We first prove tight security of Schnorr signatures from the discrete logarithm assumption (DL) in the AGM+ROM. We then give a rigorous proof for blind Schnorr signatures in the AGM+ROM assuming hardness of the one-more discrete logarithm problem and ROS. As ROS can be solved in sub-exponential time using Wagner’s algorithm, we propose a simple modification of the signing protocol, which leaves the signatures unchanged. It is therefore compatible with systems that already use Schnorr signatures, such as blockchain protocols. We show that the security of our modified scheme relies on the hardness of a problem related to ROS that appears much harder.
    [Show full text]
  • Implementation and Performance Evaluation of XTR Over Wireless Network
    Implementation and Performance Evaluation of XTR over Wireless Network By Basem Shihada [email protected] Dept. of Computer Science 200 University Avenue West Waterloo, Ontario, Canada (519) 888-4567 ext. 6238 CS 887 Final Project 19th of April 2002 Implementation and Performance Evaluation of XTR over Wireless Network 1. Abstract Wireless systems require reliable data transmission, large bandwidth and maximum data security. Most current implementations of wireless security algorithms perform lots of operations on the wireless device. This result in a large number of computation overhead, thus reducing the device performance. Furthermore, many current implementations do not provide a fast level of security measures such as client authentication, authorization, data validation and data encryption. XTR is an abbreviation of Efficient and Compact Subgroup Trace Representation (ECSTR). Developed by Arjen Lenstra & Eric Verheul and considered a new public key cryptographic security system that merges high level of security GF(p6) with less number of computation GF(p2). The claim here is that XTR has less communication requirements, and significant computation advantages, which indicate that XTR is suitable for the small computing devices such as, wireless devices, wireless internet, and general wireless applications. The hoping result is a more flexible and powerful secure wireless network that can be easily used for application deployment. This project presents an implementation and performance evaluation to XTR public key cryptographic system over wireless network. The goal of this project is to develop an efficient and portable secure wireless network, which perform a variety of wireless applications in a secure manner. The project literately surveys XTR mathematical and theoretical background as well as system implementation and deployment over wireless network.
    [Show full text]
  • Secure E-Voting System by Utilizing Homomorphic Properties of the Encryption Algorithm
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by TELKOMNIKA (Telecommunication Computing Electronics and Control) TELKOMNIKA, Vol.16, No.2, April 2018, pp. 862~867 ISSN: 1693-6930, accredited A by DIKTI, Decree No: 58/DIKTI/Kep/2013 DOI: 10.12928/TELKOMNIKA.v16i2.8420 862 Secure E-voting System by Utilizing Homomorphic Properties of the Encryption Algorithm Rifki Suwandi*, Surya Michrandi Nasution, Fairuz Azmi Electrical Engineering Faculty, Telkom University, Bandung, Indonesia *Corresponding author, e-mail: [email protected], [email protected], [email protected] Abstract The use of cryptography in the e-voting system to secure data is a must to ensure the authenticity of the data. In contrast to common encryption algorithms, homomorphic encryption algorithms had unique properties that can perform mathematical operations against ciphertext. This paper proposed the use of the Paillier and Okamoto-Uchiyama algorithms as two homomorphic encryption algorithms that have the additional properties so that it can calculate the results of voting data that has been encrypted without having to be decrypted first. The main purpose is to avoid manipulation and data falsification during vote tallying process by comparing the advantages and disadvantages of each algorithm. Keywords: cryptography, encryption, homomorphic, Paillier, Okamoto-Uchiyama Copyright © 2018 Universitas Ahmad Dahlan. All rights reserved. 1. Introduction There are numerous issues with the electronic vote casting system, together with system errors, network safety, information security, and so on. One of the primary issues is cheating committed by either insider or outsider, especially for some important role holders with authority that can access the system itself.
    [Show full text]
  • Lecture 14: Elgamal Encryption, Hash Functions from DL, Prgs from DDH 1 Topics Covered 2 Recall 3 Key Agreement from the Diffie
    CS 7880 Graduate Cryptography October 29, 2015 Lecture 14: ElGamal Encryption, Hash Functions from DL, PRGs from DDH Lecturer: Daniel Wichs Scribe: Biswaroop Maiti 1 Topics Covered • Public Key Encryption • A Public Key Encryption from the DDH Assumption • El Gamal Encryption • CRHF from Discrete Log • PRG from DDH 2 Recall Recall the three number theoretic assumptions we saw last time. We will build Crypto- graphic schemes or protocols based on the hardness of these problems. Definition 1 (G; g; q) Groupgen(1n) } Assumption 1 DL Given g; gX , it is hard to find X. Assumption 2 Computational Diffie Hellman Given g; gX ; gY , it is hard to find gXY . Assumption 3 Decisional Diffie Hellman Given g; gX ; gY , it is hard to distinguish between gXY and gZ , where Z is chosen at random. (g; gX ; gY ; gXY ) ≈ (g; gX ; gY ; gZ ) 3 Key Agreement from the Diffie Helman scheme AB x Zq X hA = g −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−£ Y hB = g ¡−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Y Zq X XY Y XY hB = g hA = g The keys agreed upon by A and B is gXY . Lecture 14, Page 1 It is interesting to note that in this scheme, A and B were able to agree upon a key without communicating about it. Each party generates a puzzle uniformly at random: A X Y generates hA = g , and B generates hB = g . Then, they send their puzzles to each other, and establish the key to be gXY . Proving this scheme is secure is equivalent to showing that the DDH assumption holds. 4 Public Key Encryption The general syntax of Public Key Encryption is the following.
    [Show full text]
  • Two-Party ECDSA from Hash Proof Systems and Efficient Instantiations
    Two-Party ECDSA from Hash Proof Systems and Efficient Instantiations Guilhem Castagnos, Dario Catalano, Fabien Laguillaumie, Federico Savasta, Ida Tucker To cite this version: Guilhem Castagnos, Dario Catalano, Fabien Laguillaumie, Federico Savasta, Ida Tucker. Two-Party ECDSA from Hash Proof Systems and Efficient Instantiations. CRYPTO 2019 - 39th Annual Inter- national Cryptology Conference, Aug 2019, Santa Barbara, United States. pp.191-221, 10.1007/978- 3-030-26954-8_7. hal-02281931 HAL Id: hal-02281931 https://hal.archives-ouvertes.fr/hal-02281931 Submitted on 9 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Two-Party ECDSA from Hash Proof Systems and Efficient Instantiations Guilhem Castagnos1, Dario Catalano2, Fabien Laguillaumie3, Federico Savasta2;4, and Ida Tucker3 1 Université de Bordeaux, INRIA, CNRS, IMB UMR 5251, F-33405 Talence, France. 2 Università di Catania, Italy. 3 Univ Lyon, EnsL, UCBL, CNRS, Inria, LIP, F-69342, LYON Cedex 07, France. 4 Scuola Superiore di Catania, Italy Abstract. ECDSA is a widely adopted digital signature standard. Unfortunately, efficient distributed variants of this primitive are notoriously hard to achieveand known solutions often require expensive zero knowledge proofs to deal with malicious adversaries.
    [Show full text]
  • Implementing Several Attacks on Plain Elgamal Encryption Bryce D
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2008 Implementing several attacks on plain ElGamal encryption Bryce D. Allen Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Mathematics Commons Recommended Citation Allen, Bryce D., "Implementing several attacks on plain ElGamal encryption" (2008). Graduate Theses and Dissertations. 11535. https://lib.dr.iastate.edu/etd/11535 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Implementing several attacks on plain ElGamal encryption by Bryce Allen A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Mathematics Program of Study Committee: Clifford Bergman, Major Professor Paul Sacks Sung-Yell Song Iowa State University Ames, Iowa 2008 Copyright c Bryce Allen, 2008. All rights reserved. ii Contents CHAPTER 1. OVERVIEW . 1 1.1 Introduction . .1 1.2 Hybrid Cryptosystems . .2 1.2.1 Symmetric key cryptosystems . .2 1.2.2 Public key cryptosystems . .3 1.2.3 Combining the two systems . .3 1.2.4 Short messages and public key systems . .4 1.3 Implementation . .4 1.4 Splitting Probabilities . .5 CHAPTER 2. THE ELGAMAL CRYPTOSYSTEM . 7 2.1 The Discrete Log Problem . .7 2.2 Encryption and Decryption . .8 2.3 Security . .9 2.4 Efficiency .
    [Show full text]
  • Power Analysis on NTRU Prime
    Power Analysis on NTRU Prime Wei-Lun Huang, Jiun-Peng Chen and Bo-Yin Yang Academia Sinica, Taiwan, [email protected], [email protected], [email protected] Abstract. This paper applies a variety of power analysis techniques to several imple- mentations of NTRU Prime, a Round 2 submission to the NIST PQC Standardization Project. The techniques include vertical correlation power analysis, horizontal in- depth correlation power analysis, online template attacks, and chosen-input simple power analysis. The implementations include the reference one, the one optimized using smladx, and three protected ones. Adversaries in this study can fully recover private keys with one single trace of short observation span, with few template traces from a fully controlled device similar to the target and no a priori power model, or sometimes even with the naked eye. The techniques target the constant-time generic polynomial multiplications in the product scanning method. Though in this work they focus on the decapsulation, they also work on the key generation and encapsulation of NTRU Prime. Moreover, they apply to the ideal-lattice-based cryptosystems where each private-key coefficient comes from a small set of possibilities. Keywords: Ideal Lattice Cryptography · Single-Trace Attack · Correlation Power Analysis · Template Attack · Simple Power Analysis · NTRU Prime 1 Introduction Due to Shor’s algorithm [Sho97], quantum computing is a potential threat to all public-key cryptosystems based on the hardness of integer factorization and discrete logarithms, in- cluding RSA [RSA78], Diffie-Hellman key agreement [DH76], ElGamal encryption [Gam85], and ECDSA [JMV01]. Recently, quantum computers have been estimated as arriving in 10 to 15 years [Sni16, WLYZ18], which identifies an urgent need for the examination and standardization of post-quantum cryptosystems.
    [Show full text]
  • Comparison of Two Public Key Cryptosystems
    Journal of Optoelectronical Nanostructures Summer 2018 / Vol. 3, No. 3 Islamic Azad University Comparison of two Public Key Cryptosystems 1 *,2 1 Mahnaz Mohammadi , Alireza Zolghadr , Mohammad Ali Purmina 1 Department of Electrical and Electronic Eng., Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Faculty of Computer and Electrical Eng., Department of Communication and Electronics, Shiraz University, Shiraz, Iran (Received 10 Jun. 2018; Revised 11 Jul. 2018; Accepted 26 Aug. 2018; Published 15 Sep. 2018) Abstract: Since the time public-key cryptography was introduced by Diffie and Hellman in 1976, numerous public-key algorithms have been proposed. Some of these algorithms are insecure and the others that seem secure, many are impractical, either they have too large keys or the cipher text they produce is much longer than the plaintext. This paper focuses on efficient implementation and analysis of two most popular of these algorithms, RSA and ElGamal for key generation and the encryption scheme (encryption/decryption operation). RSA relies on the difficulty of prime factorization of a very large number, and the hardness of ElGamal algorithm is essentially equivalent to the hardness of finding discrete logarithm modulo a large prime. These two systems are compared to each other from different parameters points of view such as performance, security, speed and applications. To have a good comparison and also to have a good level of security correspond to users need the systems implemented are designed flexibly in terms of the key size. Keywords: Cryptography, Public Key Cryptosystems, RSA, ElGamal. 1. INTRODUCTION The security is a necessary for the network to do all the works related to ATM cards, computer passwords and electronic commerce.
    [Show full text]
  • Timing Attacks and the NTRU Public-Key Cryptosystem
    Eindhoven University of Technology BACHELOR Timing attacks and the NTRU public-key cryptosystem Gunter, S.P. Award date: 2019 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain Timing Attacks and the NTRU Public-Key Cryptosystem by Stijn Gunter supervised by prof.dr. Tanja Lange ir. Leon Groot Bruinderink A thesis submitted in partial fulfilment of the requirements for the degree of Bachelor of Science July 2019 Abstract As we inch ever closer to a future in which quantum computers exist and are capable of break- ing many of the cryptographic algorithms that are in use today, more and more research is being done in the field of post-quantum cryptography.
    [Show full text]
  • Lecture 19: Public-Key Cryptography (Diffie-Hellman Key Exchange & Elgamal Encryption)
    Lecture 19: Public-key Cryptography (Diffie-Hellman Key Exchange & ElGamal Encryption) Public-key Cryptography Recall In private-key cryptography the secret-key sk is always established ahead of time The secrecy of the private-key cryptography relies on the fact that the adversary does not have access to the secret key sk For example, consider a private-key encryption scheme $ 1 The Alice and Bob generate sk Gen() ahead of time 2 Later, when Alice wants to encrypt and send a message to Bob, she computes the cipher-text c = Encsk(m) 3 The eavesdropping adversary see c but gains no additional information about the message m 4 Bob can decrypt the message me = Decsk(c) 5 Note that the knowledge of sk distinguishes Bob from the eavesdropping adversary Public-key Cryptography Perspective If jskj >jmj, then we can construct private-key encryption schemes (like, one-time pad) that is secure even against adversaries with unbounded computational power If jskj = O(jmj"), where " 2 (0; 1) is a constant, then we can construction private-key encryption schemes using pseudorandom generators (PRGs) What if, jskj = 0? That is, what if Alice and Bob never met? How is “Bob” any different from an “adversary”? Public-key Cryptography In this Lecture We shall introduce the Decisional Diffie-Hellmann (DDH) Assumption and the Diffie-Hellman key-exchange protocol, We shall introduce the El Gamal (public-key) Encryption Scheme, and Finally, abstract out the principal design principles learned. Public-key Cryptography Decisional Diffie-Hellman (DDH) Computational Hardness AssumptionI Let (G; ◦) be a group of size N that is generated by g.
    [Show full text]