DOCSLIB.ORG

RSA and PUBLIC-KEY CRYPTOGRAPHY

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RSA and PUBLIC-KEY CRYPTOGRAPHY RSA and PUBLIC-KEY CRYPTOGRAPHY © 2003 by CRC Press LLC DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. AT&T Labs Abstract Algebra Applications with Maple, Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon Algebraic Number Theory, Richard A. Mollin An Atlas of The Smaller Maps in Orientable and Nonorientable Surfaces, David M. Jackson and Terry I. Visentin An Introduction to Crytography, Richard A. Mollin Combinatorial Algorithms: Generation Enumeration and Search, Donald L. Kreher and Douglas R. Stinson The CRC Handbook of Combinatorial Designs, Charles J. Colbourn and Jeffrey H. Dinitz Cryptography: Theory and Practice, Second Edition, Douglas R. Stinson Design Theory, Charles C. Lindner and Christopher A. Rodgers Frames and Resolvable Designs: Uses, Constructions, and Existence, Steven Furino, Ying Miao, and Jianxing Yin Fundamental Number Theory with Applications, Richard A. Mollin Graph Theory and Its Applications, Jonathan Gross and Jay Yellen Handbook of Applied Cryptography, Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone Handbook of Constrained Optimization, Herbert B. Shulman and Venkat Venkateswaran Handbook of Discrete and Combinatorial Mathematics, Kenneth H. Rosen Handbook of Discrete and Computational Geometry, Jacob E. Goodman and Joseph O’Rourke Introduction to Information Theory and Data Compression, Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson © 2003 by CRC Press LLC Continued Titles Network Reliability: Experiments with a Symbolic Algebra Environment, Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt RSA and Public-Key Cryptography Richard A. Mollin Quadratics, Richard A. Mollin Verificaton of Computer Codes in Computational Science and Engineering, Patrick Knupp and Kambiz Salari © 2003 by CRC Press LLC DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN RSA and PUBLIC-KEY CRYPTOGRAPHY Richard A. Mollin CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. © 2003 by CRC Press LLC C3383 disclaimer Page 1 Monday, October 7, 2002 1:33 PM Library of Congress Cataloging-in-Publication Data Mollin, Richard A., 1947- RSA and public-key cryptography / Richard A. Mollin. p. cm. — (Discrete mathematics and its applicatoins) Includes bibliographical references and index. ISBN 1-58488-338-3 1. Coding theory. 2. Public key cryptography. I. Title. II. Series. QA268 .M655 2002 652′.8—dc21 2002031096 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2003 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58488-338-3 Library of Congress Card Number 2002031096 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper © 2003 by CRC Press LLC Preface This book is intended for a second course in cryptography at the undergrad- uate level, where the student is assumed to have had a course in introductory number theory. Also, the book is intended as a source book for those in the cryptography business, who will find collected together herein numerous facts that are currently scattered throughout the literature on public-key cryptogra- phy and related issues described in the Table of Contents. The impetus for the writing of this text arose from the author’s involvement in the establishment of an iCORE (Informatics Circle of Research Excellence) Chair in cryptography at the University of Calgary in September of 2001, and the launching of an associ- ated Centre for Information Security and Cryptography (CISAC). In addition to the education of numerous graduate students and postdoctoral fellows in cryp- tology, we have an ongoing commitment to the development of a new stream of cryptography courses for the Mathematics Department. This text will serve as the text for one of them at the senior undergraduate level. No suitable text for that course was on the market, nor is there one at the time of this writing, hence the appearance of this one. N Features of This Text • The book is ideal for the student since it offers a wealth of exercises with 350 problems. The more challenging exercises are marked with a ✰. Also, complete and detailed solutions to all of the odd-numbered exercises are provided at the back of the text. Complete and detailed solutions of the even-numbered exercises are included in a Solutions Manual, which is available from the publisher for the instructor who adopts the text for a course. Moreover, the exercises are presented at the end of each section, rather than at the end of each chapter. • The text is accessible to anyone from the senior undergraduate to the research scientist, and all levels of readers will find challenging and inspira- tional data. To ensure that the book is as self-contained as possible, we have three appendices of relevant background information. Appendix A has a brief, but highly informative, overview of letter frequency analysis (of the English language) to assist in our cryptanalytic travels. Appendix B has a solid back- ground review and analysis of elementary complexity theory to provide us with the necessary tools for algorithmic analysis and related phenomena. Lastly, Appendix C contains the fundamentals of the number-theoretic results used in the text together with any other relevant information that we will need such as vector space basics, matrix theory fundamentals, and some facts on continued fractions. • There are over 100 footnotes containing nearly 40 biographies of the indi- viduals who helped develop cryptologic concepts, together with historical data of interest, as well as other information which the discerning reader may want to explore at leisure. These are woven throughout the text, to give a human face v © 2003 by CRC Press LLC vi RSA and Public-Key Cryptography to the cryptology being presented. A knowledge of the lives of these individ- uals can only deepen our appreciation of the development of PKC and related concepts. The footnote presentation of this material allows the reader to have immediate information at will, or to treat them as digressions, and access them later without significantly interfering with the main discussion. • There are optional topics, denoted by , which add additional material for the more advanced reader or the reader requiring more challenging material which goes beyond the basics presented in the core data. • There are more than 60 examples,diagrams,figures , and tables throughout the text to illustrate the concepts presented. • For ease of search, the reader will find consecutive numbering, namely object N.m is the mth object in Chapter N (or Appendix N), exclusive of footnotes and exercises, which are numbered separately and consecutively unto themselves. Thus, for instance, Diagram 3.5 is the 5th numbered object in Chapter Three; exclusive of footnotes and exercise Exercise 4.37 is the 37 th exercise in Chapter Four; and Footnote 9.2 is the second footnote in Chapter Nine. • The bibliography contains nearly 250 references for further reading. • The index has more than 2,000 entries, and has been devised in such a way to ensure that there is maximum ease in getting information from the text. • The webpage cited below will contain a file for comments, and any ty- pos/errors that are found. Furthermore, comments via the e-mail address below are also welcome. Lastly, if the instructor for a given course, using this text, has any difficulty in getting the solutions manual for the even-numbered exercises, contacting the author will expedite matters. N Acknowledgments First of all, thanks go to Hugh Williams for discus- sions prior to the writing of this book that inspired my creating the original template for a table of contents. As iCORE Chair in cryptography at U. of C., he has brought a new dimension and exciting prospects for the future. The author is grateful for the proofreading done by the following people, each of whom lent his own (largely non-intersecting) expertise and valuable time: John Brillhart (U.S.A.), John Burke (U.S.A.), Jacek Fabrykowski (U.S.A.), Bart God- dard (U.S.A.), Franz Lemmermeyer (U.S.A.), John Robertson (U.S.A.), Kjell Wooding (Canada), graduate student in cryptography at U. of C., Thomas Zaplachinski (Canada), a former student, now cryptographer, and Robert Zuc- cherato (U.S.A.). Richard Mollin, Calgary, September 1, 2002 website: http://www.math.ucalgary.ca/˜ramollin/ e-mail: [email protected] © 2003 by CRC Press LLC The Author vii About the Author Richard Anthony Mollin received his Ph.D. in mathematics (1975) from Queen’s University, Kingston, Ontario, Canada, where he was born. He is now a full professor in the Mathematics Department at the University of Calgary, Alberta, Canada.
Load more

Recommended publications
  • Mihir Bellare Curriculum Vitae Contents
    Mihir Bellare Curriculum vitae August 2018 Department of Computer Science & Engineering, Mail Code 0404 University of California at San Diego 9500 Gilman Drive, La Jolla, CA 92093-0404, USA. Phone: (858) 534-4544 ; E-mail: [email protected] Web Page: http://cseweb.ucsd.edu/~mihir Contents 1 Research areas 2 2 Education 2 3 Distinctions and Awards 2 4 Impact 3 5 Grants 4 6 Professional Activities 5 7 Industrial relations 5 8 Work Experience 5 9 Teaching 6 10 Publications 6 11 Mentoring 19 12 Personal Information 21 2 1 Research areas ∗ Cryptography and security: Provable security; authentication; key distribution; signatures; encryp- tion; protocols. ∗ Complexity theory: Interactive and probabilistically checkable proofs; approximability ; complexity of zero-knowledge; randomness in protocols and algorithms; computational learning theory. 2 Education ∗ Massachusetts Institute of Technology. Ph.D in Computer Science, September 1991. Thesis title: Randomness in Interactive Proofs. Thesis supervisor: Prof. S. Micali. ∗ Massachusetts Institute of Technology. Masters in Computer Science, September 1988. Thesis title: A Signature Scheme Based on Trapdoor Permutations. Thesis supervisor: Prof. S. Micali. ∗ California Institute of Technology. B.S. with honors, June 1986. Subject: Mathematics. GPA 4.0. Class rank 4 out of 227. Summer Undergraduate Research Fellow 1984 and 1985. ∗ Ecole Active Bilingue, Paris, France. Baccalauréat Série C, June 1981. 3 Distinctions and Awards ∗ PET (Privacy Enhancing Technologies) Award 2015 for publication [154]. ∗ Fellow of the ACM (Association for Computing Machinery), 2014. ∗ ACM Paris Kanellakis Theory and Practice Award 2009. ∗ RSA Conference Award in Mathematics, 2003. ∗ David and Lucille Packard Foundation Fellowship in Science and Engineering, 1996. (Twenty awarded annually in all of Science and Engineering.) ∗ Test of Time Award, ACM CCS 2011, given for [81] as best paper from ten years prior.
    [Show full text]
  • Stronger Security Notions for Trapdoor Functions and Applications
    STRONGER SECURITY NOTIONS FOR TRAPDOOR FUNCTIONS AND APPLICATIONS A Thesis Presented to The Academic Faculty by Adam O'Neill In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the College of Computing Georgia Institute of Technology December 2010 STRONGER SECURITY NOTIONS FOR TRAPDOOR FUNCTIONS AND APPLICATIONS Approved by: Professor Alexandra Boldyreva, Professor Chris Peikert Advisor College of Computing College of Computing Georgia Institute of Technology Georgia Institute of Technology Professor Mihir Bellare Professor Dana Randall Computer Science and Engineering College of Computing University of California, San Diego Georgia Institute of Technology Professor Richard Lipton Professor Patrick Traynor College of Computing College of Computing Georgia Institute of Technology Georgia Institute of Technology Date Approved: 9 August 2010 To my parents, John (Chuck) and Phyllis O'Neill, and my sister Katie, for their unconditional support. iii ACKNOWLEDGEMENTS My Ph.D. studies have been a significant undertaking, which would not have been possible without the help and guidance of many people (please forgive any omissions). First of all, I'd like to thank my parents, for encouraging me to pursue higher education and giving me the opportunity to do so. My intellectual development was greatly fostered during my time as an undergrad- uate at UCSD, and for that I have many friends and teaching assistants to thank. I would especially like to thank Daniel Bryant for helping me during my freshman year, and Derek Newland for inviting me to do an independent study with him. I would also like to thank Mihir Bellare for taking the time to help undergraduates find out about graduate school and encouraging them to take graduate-level courses.
    [Show full text]
  • Part V Public-Key Cryptosystems, I. Key Exchange, Knapsack
    Part V Public-key cryptosystems, I. Key exchange, knapsack, RSA CHAPTER 5: PUBLIC-KEY CRYPTOGRAPHY I. RSA The main problem of secret key (or symmetric) cryptography is that in order to send securely A secret message we need to send at first securely a secret key and therefore secret key cryptography is clearly not a sufficiently good tool for massive communication capable to protect secrecy, privacy and anonymity. prof. Jozef Gruska IV054 5. Public-key cryptosystems, I. Key exchange, knapsack, RSA 2/67 SECURE ENCRYPTION - a PRACTICAL POINT OF VIEW From practical point of view encryptions by a cryptosystem can be considered as secure if they cannot be broken by many (thousands) superomputeers with exaflop performance working for some years. prof. Jozef Gruska IV054 5. Public-key cryptosystems, I. Key exchange, knapsack, RSA 3/67 CONTENT In this chapter we describe the birth of public key cryptography, that can better manage key distribution problem, and three of its cryptosystems, especially RSA. The basic idea of a public key cryptography: In a public key cryptosystem not only the encryption and decryption algorithms are public, but for each user U also the key eU for encrypting messages (by anyone) for U is public. Moreover, each user U keeps secret another (decryption) key, dU , that can be used for decryption of messages that were addressed to him and encrypted with the help of the public encryption key eU . Encryption and decryption keys could (and should) be different - we can therefore speak also about asymmetric cryptography. Secret key cryptography, that has the same key for encryption and for decryption is also called symmetric cryptography.
    [Show full text]
  • Solving NTRU Challenges Using the New Progressive BKZ Library
    Solving NTRU Challenges Using the New Progressive BKZ Library Erik Mårtensson Department of Electrical and Information Technology Lund University Advisors: Professor Thomas Johansson and Qian Guo 2016-08-24 Printed in Sweden E-huset, Lund, 2016 Abstract NTRU is a public-key cryptosystem, where the underlying mathematical problem is currently safe against large-scale quantum computer attacks. The system is not as well investigated, as for example RSA and the company behind NTRU has created the NTRU Challenges, to remedy this. These challenges consist of 27 different public keys of increasing size, where the task in each challenge is to calculate (something similar to) the private key. The goal of this thesis was to examine different attacks against the NTRU Challenges and solve as many challenges as possible. By lattice reduction attacks, using a recently published new progressive BKZ algorithm, the first five challenges were solved, while the current biggest solved challenge by any researcher is challenge number seven. Keywords: NTRU Challenge, Progressive BKZ , BDD, Enumeration, SVP i ii Acknowledgements I would like to thank my main supervisor Professor Thomas Johansson for all his help and feedback during the project, for introducing me to the lattice-based cryptography area and last but not least for pointing out the necessity of solving a small problem correctly before tackling a big problem. I would like to thank my assistant supervisor Qian Guo for all his help and feed- back during the project and for introducing me to the progressive BKZ algorithm that turned out to become the focus of this thesis. I would like to thank all the researchers that patiently answered my many ques- tions during this project, with special thanks to Yoshinori Aono, Léo Ducas and Zhenfei Zhang for answering questions regarding the progressive BKZ algorithm and library, the BKZ plus BDD enumeration strategy for attacking NTRU and the NTRU Challenges respectively.
    [Show full text]
  • How to Strengthen Ntruencrypt to Chosen-Ciphertext Security in the Standard Model
    NTRUCCA: How to Strengthen NTRUEncrypt to Chosen-Ciphertext Security in the Standard Model Ron Steinfeld1?, San Ling2, Josef Pieprzyk3, Christophe Tartary4, and Huaxiong Wang2 1 Clayton School of Information Technology Monash University, Clayton VIC 3800, Australia [email protected] 2 Div. of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, 637371 lingsan,[email protected] 3 Centre for Advanced Computing - Algorithms and Cryptography, Dept. of Computing, Macquarie University, Sydney, NSW 2109, Australia [email protected] 4 Institute for Theoretical Computer Science, Tsinghua University, People's Republic of China [email protected] Abstract. NTRUEncrypt is a fast and practical lattice-based public-key encryption scheme, which has been standardized by IEEE, but until re- cently, its security analysis relied only on heuristic arguments. Recently, Stehlé and Steinfeld showed that a slight variant (that we call pNE) could be proven to be secure under chosen-plaintext attack (IND-CPA), assum- ing the hardness of worst-case problems in ideal lattices. We present a variant of pNE called NTRUCCA, that is IND-CCA2 secure in the standard model assuming the hardness of worst-case problems in ideal lattices, and only incurs a constant factor overhead in ciphertext and key length over the pNE scheme. To our knowledge, our result gives the rst IND- CCA2 secure variant of NTRUEncrypt in the standard model, based on standard cryptographic assumptions. As an intermediate step, we present a construction for an All-But-One (ABO) lossy trapdoor function from pNE, which may be of independent interest. Our scheme uses the lossy trapdoor function framework of Peik- ert and Waters, which we generalize to the case of (k −1)-of-k-correlated input distributions.
    [Show full text]
  • One-Time Trapdoor One-Way Functions
    One-Time Trapdoor One-Way Functions Julien Cathalo?1 and Christophe Petit??2 1 Smals Avenue Fonsny 20, 1060 Bruxelles, Belgium 2 UCL Crypto Group Universit´ecatholique de Louvain, Place du Levant 3, 1348 Louvain-la-Neuve, Belgium Abstract. Trapdoors are widely used in cryptography, in particular for digital signatures and public key encryption. In these classical applica- tions, it is highly desirable that trapdoors remain secret even after their use. In this paper, we consider positive applications of trapdoors that do not remain secret when they are used. We introduce and formally de- fine one-time trapdoor one-way functions (OTTOWF), a primitive simi- lar in spirit to classical trapdoor one-way functions, with the additional property that its trapdoor always becomes public after use. We provide three constructions of OTTOWF. Two of them are based on factoring assumptions and the third one on generic one-way functions. We then consider potential applications of our primitive, and in particular the fair exchange problem. We provide two fair exchange protocols using OTTOWF, where the trapdoor is used to provide some advantage to one of the parties, whereas any (abusive) use of this trapdoor will make the advantage available to the other party as well. We compare our proto- cols with well-established solutions for fair exchange and describe some scenarios where they have advantageous characteristics. These results demonstrate the interest of one-time trapdoor one-way functions, and suggest looking for further applications of them. Keywords Cryptographic primitive, trapdoor one-way function, fair exchange 1 Introduction In cryptography, a trapdoor is a secret piece of information that provides its holder with some special power or advantage.
    [Show full text]
  • Practical Lattice-Based Digital Signature Schemes J
    Practical Lattice-based Digital Signature Schemes J. Howe1, T. Pöppelmann2, M. O’Neill1, E. O’Sullivan1, T. Güneysu2 1 Centre for Secure Information Technologies (CSIT), Queen’s University Belfast, UK 2Horst Görtz Institute for IT-Security, Ruhr-University Bochum, Germany Abstract: Among the various post-quantum techniques that exist (such as multivariate, code or hash-based), the most promising is lattice-based cryptography, which has become a very viable alternative to number-theoretic cryptography. Its main advantage is that it allows for extended functionality and is, at the same time, more efficient for the basic primitives of public-key encryption and digital signatures. The focus of this presentation will be to survey recent developments in lattice-based digital signature schemes and in particular practical schemes that have been shown to improve upon the performance of equivalent RSA designs. Propositions for future research areas that are essential for the continued development of lattice-based cryptography will also be discussed. 1 Introduction With the onset of quantum computers ever looming, the computational power it could provide would cause instant insecurity to many of today’s universally used cryptographic schemes by virtue of Shor’s [1] algorithm. Specifically, schemes based on the discrete-logarithm problem or number-theoretic hard problems, which subsume almost all public-key encryption schemes used on the Internet, including elliptic-curve cryptography (ECC), RSA and DSA would be vulnerable. Accordingly, this has motivated the post-quantum era of cryptography, which refers to the construction of cryptographic algorithms to withstand quantum reductions. Amongst many important areas in post-quantum research (such as multivariate, code or hash-based) lattice-based cryptography is disputably the most auspicious.
    [Show full text]
  • 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).
    [Show full text]
  • Construction of Efficient Authentication Schemes Using Trapdoor Hash Functions
    University of Kentucky UKnowledge University of Kentucky Doctoral Dissertations Graduate School 2011 CONSTRUCTION OF EFFICIENT AUTHENTICATION SCHEMES USING TRAPDOOR HASH FUNCTIONS Santosh Chandrasekhar University of Kentucky, [email protected] Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Chandrasekhar, Santosh, "CONSTRUCTION OF EFFICIENT AUTHENTICATION SCHEMES USING TRAPDOOR HASH FUNCTIONS" (2011). University of Kentucky Doctoral Dissertations. 162. https://uknowledge.uky.edu/gradschool_diss/162 This Dissertation is brought to you for free and open access by the Graduate School at UKnowledge. It has been accepted for inclusion in University of Kentucky Doctoral Dissertations by an authorized administrator of UKnowledge. For more information, please contact [email protected]. ABSTRACT OF DISSERTATION Santosh Chandrasekhar The Graduate School University of Kentucky 2011 CONSTRUCTION OF EFFICIENT AUTHENTICATION SCHEMES USING TRAPDOOR HASH FUNCTIONS ABSTRACT OF DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Engineering at the University of Kentucky By Santosh Chandrasekhar Lexington, Kentucky Director: Dr. Mukesh Singhal Lexington, Kentucky 2011 Copyright c Santosh Chandrasekhar 2011 ABSTRACT OF DISSERTATION CONSTRUCTION OF EFFICIENT AUTHENTICATION SCHEMES USING TRAPDOOR HASH FUNCTIONS In large-scale distributed systems, where adversarial attacks can have widespread impact, authentication provides protection from threats involving impersonation of entities and tampering of data. Practical solutions to authentication problems in distributed systems must meet specific constraints of the target system, and provide a reasonable balance between security and cost. The goal of this dissertation is to address the problem of building practical and efficient authentication mechanisms to secure distributed applications.
    [Show full text]
  • The Factoring Dead: Preparing for the Cryptopocalypse
    THE FACTORING DEAD: PREPARING FOR THE CRYPTOPOCALYPSE Javed Samuel — javed[at]isecpartners[dot]com iSEC Partners, Inc 123 Mission Street, Suite 1020 San Francisco, CA 94105 https://www.isecpartners.com March 20, 2014 Abstract This paper will explain the latest breakthroughs in the academic cryptography community and look ahead at what practical issues could arise for popular cryptosystems. Specifically, we will focus on the recent major devel- opments in discrete mathematics and their potential ability to undermine our trust in the most basic asymmetric primitives, including RSA. We will explain the basic theories behind RSA and the state-of-the-art in large number- ing factoring, and how several recent papers may point the way to massive improvements in this area. The paper will then switch to the practical aspects of the doomsday scenario, and will answer the question “What happens the day after RSA is broken?” We will point out the many obvious and hidden uses of RSA and related algorithms and outline how software engineers and security teams can operate in a post-RSA world. We will also discuss the results of our survey of popular products and software, and point out the ways in which individuals can prepare for the “zombie cryptopocalypse”. 1 INTRODUCTION Over the past few years, there have been numerous attacks on the current SSL infrastructure. These have ranged from BEAST [97], CRIME [88], Lucky 13 [2][86], RC4 bias attacks [1][91] and BREACH [42]. These attacks all show the fragility of the current SSL architecture as vulnerabilities have been found in a variety of features ranging from compression, timing and padding [90].
    [Show full text]
  • Trapdoor Functions from the Computational Diffie-Hellman Assumption
    Trapdoor Functions from the Computational Diffie-Hellman Assumption∗ Sanjam Gargy Mohammad Hajiabadiz June 29, 2018 Abstract Trapdoor functions (TDFs) are a fundamental primitive in cryptography. Yet, the current set of assumptions known to imply TDFs is surprisingly limited, when compared to public-key encryption. We present a new general approach for constructing TDFs. Specifically, we give a generic construction of TDFs from any Hash Encryption (D¨ottlingand Garg [CRYPTO '17]) satisfying a novel property which we call recyclability. By showing how to adapt current Com- putational Diffie-Hellman (CDH) based constructions of hash encryption to yield recyclability, we obtain the first construction of TDFs with security proved under the CDH assumption. While TDFs from the Decisional Diffie-Hellman (DDH) assumption were previously known, the possibility of basing them on CDH had remained open for more than 30 years. 1 Introduction Trapdoor functions (TDFs) are a fundamental primitive in cryptography, historically pre-dating the complexity-based development of public key encryption (PKE) [DH76, RSA78]. Informally, TDFs are a family of functions, where each function in the family is easy to compute given the function's index key, and also easy to invert given an associated trapdoor key. The security requirement is that a randomly chosen function from the family should be hard to invert without knowledge of a trapdoor key. A salient difference between TDFs and PKE lies in their inversion (decryption) algorithms: while the inversion algorithm of a TDF recovers the entire pre-image in full, the decryption algorithm of a PKE only recovers the corresponding plaintext, and not necessarily the randomness.
    [Show full text]
  • Section 5 Public Key Crypto Topics: RSA, ECC, Cas Administrivia
    Section 5 Public Key Crypto Topics: RSA, ECC, CAs Administrivia ● Homework 2 due next Wednesday (02-10) ○ Individual assignment ○ Hands-on cryptography ● Final Project checkpoint #1 due next next Wednesday (02-17) ○ Group members’ names and UWNetIDs ○ Presentation topic RSA: Review, Practice, and Future Public Key Cryptography Review Alice wants to send Bob an encrypted message ● Goal: Confidentiality ● Problem: Eve can intercept key Public Key Cryptography Review Solution: public key cryptography (aka asymmetric cryptography) ● Public-private keypair ● Alice encrypts using Bob’s public key ● Bob decrypts using Bob’s private key RSA Cryptosystem Review Key generation: ● Generate large primes p, q ● Compute N=pq and φ(N)=(p-1)(q-1) ● Choose e coprime to φ(N) ○ Typically e=3 or e=216+1=65537 ● Find (unique) d such that ed ≡ 1 (mod φ(N)) Adi Shamir, Ron Rivest, Len Adleman [Photo from Dan Wright] Public key = (e, N); Private key = (d, N) Encryption of m: c = me mod N Decryption of c: cd mod N = (me mod N)d mod N = m1 mod N = m RSA Practice Public key: N = 33, e = 7 Cryptograms: Step 1: Find φ(N) Step 2: Find the decryption key, d - ed ≡ 1 (mod φ(N)) Step 3: Decrypt the cryptogram - cd mod N = m - ‘A’ = 1, ‘B’ = 2, ... Cowabunga! RSA-2048: RSA Strength “RSA problem”: decrypt only using the public key ● Factoring N is hard ● No known efficient algorithm ● Trapdoor function: easy to go forward, hard to go back RSA Factoring Challenge (1991-2007) ● Cash prizes for factoring large N values (up to $200,000 (!)) ● Only the smallest 23 of 54 factored so far..
    [Show full text]