Advanced Encryption Standard
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Public Key Cryptography And
PublicPublic KeyKey CryptographyCryptography andand RSARSA Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 [email protected] Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-11/ Washington University in St. Louis CSE571S ©2011 Raj Jain 9-1 OverviewOverview 1. Public Key Encryption 2. Symmetric vs. Public-Key 3. RSA Public Key Encryption 4. RSA Key Construction 5. Optimizing Private Key Operations 6. RSA Security These slides are based partly on Lawrie Brown’s slides supplied with William Stallings’s book “Cryptography and Network Security: Principles and Practice,” 5th Ed, 2011. Washington University in St. Louis CSE571S ©2011 Raj Jain 9-2 PublicPublic KeyKey EncryptionEncryption Invented in 1975 by Diffie and Hellman at Stanford Encrypted_Message = Encrypt(Key1, Message) Message = Decrypt(Key2, Encrypted_Message) Key1 Key2 Text Ciphertext Text Keys are interchangeable: Key2 Key1 Text Ciphertext Text One key is made public while the other is kept private Sender knows only public key of the receiver Asymmetric Washington University in St. Louis CSE571S ©2011 Raj Jain 9-3 PublicPublic KeyKey EncryptionEncryption ExampleExample Rivest, Shamir, and Adleman at MIT RSA: Encrypted_Message = m3 mod 187 Message = Encrypted_Message107 mod 187 Key1 = <3,187>, Key2 = <107,187> Message = 5 Encrypted Message = 53 = 125 Message = 125107 mod 187 = 5 = 125(64+32+8+2+1) mod 187 = {(12564 mod 187)(12532 mod 187)... (1252 mod 187)(125 mod 187)} mod 187 Washington University in -
On the NIST Lightweight Cryptography Standardization
On the NIST Lightweight Cryptography Standardization Meltem S¨onmez Turan NIST Lightweight Cryptography Team ECC 2019: 23rd Workshop on Elliptic Curve Cryptography December 2, 2019 Outline • NIST's Cryptography Standards • Overview - Lightweight Cryptography • NIST Lightweight Cryptography Standardization Process • Announcements 1 NIST's Cryptography Standards National Institute of Standards and Technology • Non-regulatory federal agency within U.S. Department of Commerce. • Founded in 1901, known as the National Bureau of Standards (NBS) prior to 1988. • Headquarters in Gaithersburg, Maryland, and laboratories in Boulder, Colorado. • Employs around 6,000 employees and associates. NIST's Mission to promote U.S. innovation and industrial competitiveness by advancing measurement science, standards, and technology in ways that enhance economic security and improve our quality of life. 2 NIST Organization Chart Laboratory Programs Computer Security Division • Center for Nanoscale Science and • Cryptographic Technology Technology • Secure Systems and Applications • Communications Technology Lab. • Security Outreach and Integration • Engineering Lab. • Security Components and Mechanisms • Information Technology Lab. • Security Test, Validation and • Material Measurement Lab. Measurements • NIST Center for Neutron Research • Physical Measurement Lab. Information Technology Lab. • Advanced Network Technologies • Applied and Computational Mathematics • Applied Cybersecurity • Computer Security • Information Access • Software and Systems • Statistical -
Introduction to Public Key Cryptography and Clock Arithmetic Lecture Notes for Access 2010, by Erin Chamberlain and Nick Korevaar
1 Introduction to Public Key Cryptography and Clock Arithmetic Lecture notes for Access 2010, by Erin Chamberlain and Nick Korevaar We’ve discussed Caesar Shifts and other mono-alphabetic substitution ciphers, and we’ve seen how easy it can be to break these ciphers by using frequency analysis. If Mary Queen of Scots had known this, perhaps she would not have been executed. It was a long time from Mary Queen of Scots and substitution ciphers until the end of the 1900’s. Cryptography underwent the evolutionary and revolutionary changes which Si- mon Singh chronicles in The Code Book. If you are so inclined and have appropriate leisure time, you might enjoy reading Chapters 2-5, to learn some of these historical cryptography highlights: People came up with more complicated substitution ciphers, for example the Vigen`ere square. This Great Cipher of France baffled people for quite a while but a de- termined cryptographer Etienne Bazeries had a Eureka moment after three years of work, cracked the code, and possibly found the true identity of the Man in the Iron Mask, one of the great mysteries of the seventeeth century. Edgar Allan Poe and Sir Arthur Conan Doyle even dabbled in cryptanalysis. Secrecy was still a problem though because the key needed to be sent, and with the key anyone could encrypt and decrypt the messages. Frequency analysis was used to break all of these codes. In 1918 Scherbius invented his Enigma machine, but Alan Turing’s machine (the first computer?) helped in figuring out that supposedly unbreakable code, and hastened the end of the second World War. -
Choosing Key Sizes for Cryptography
information security technical report 15 (2010) 21e27 available at www.sciencedirect.com www.compseconline.com/publications/prodinf.htm Choosing key sizes for cryptography Alexander W. Dent Information Security Group, University Of London, Royal Holloway, UK abstract After making the decision to use public-key cryptography, an organisation still has to make many important decisions before a practical system can be implemented. One of the more difficult challenges is to decide the length of the keys which are to be used within the system: longer keys provide more security but mean that the cryptographic operation will take more time to complete. The most common solution is to take advice from information security standards. This article will investigate the methodology that is used produce these standards and their meaning for an organisation who wishes to implement public-key cryptography. ª 2010 Elsevier Ltd. All rights reserved. 1. Introduction being compromised by an attacker). It also typically means a slower scheme. Most symmetric cryptographic schemes do The power of public-key cryptography is undeniable. It is not allow the use of keys of different lengths. If a designer astounding in its simplicity and its ability to provide solutions wishes to offer a symmetric scheme which provides different to many seemingly insurmountable organisational problems. security levels depending on the key size, then the designer However, the use of public-key cryptography in practice is has to construct distinct variants of a central design which rarely as simple as the concept first appears. First one has to make use of different pre-specified key lengths. -
Quantum Computing Threat: How to Keep Ahead
Think openly, build securely White paper: Quantum Computing Threat: How to Keep Ahead Ë PQShield Ƕ February 2021 © PQShield Ltd | www.pqshield.com | PQShield Ltd, Oxford, OX2 7HT, UK Cryptographic agility and a clear roadmap to the upcoming NIST standards are key to a smooth and secure transition. 1 Background 1.1 New Cryptography Standards The NIST (U.S. National Institute of Standards and Technology) Post‐Quantum Cryptography (PQC) Project has been running since 2016 and is now in its third, final evaluation round. This project standardizes new key establishment and digital signature algorithms that have been designed to be resistant against attacks by quantum computers. These new algorithms are intended to replace current classical‐security RSA and Elliptic Cryptography (ECDH, ECDSA) standards in applications. The particular mathematical problems that RSA and Elliptic Cryptography are based onareeasy (polynomial‐time solvable) for quantum computers. Unless complemented with quantum‐safe cryptography, this will allow for the forgery of digital signatures (integrity compromise) and de‐ cryption of previously encrypted data (confidentiality compromise) in the future. This is aheight‐ ened risk for organizations that need to secure data for long periods of time; government orga‐ nizations that handle and secure classified information have largely been the drivers ofthepost‐ quantum standardization and its adoption in the field, but banks, financial services, healthcare providers, those developing intellectual property and many others increasingly feel the need to ensure that they can protect their customers and IP – now and in the future. 1.2 Quantum Threat and Post-Quantum Cryptography The current position of NCSC (UK’s National Cyber Security Centre) and NSA (USA’s National Se‐ curity Agency) is that the best mitigation against the threat of quantum computers is quantum‐ safe cryptography, also known as post‐quantum cryptography (PQC). -
The Mathemathics of Secrets.Pdf
THE MATHEMATICS OF SECRETS THE MATHEMATICS OF SECRETS CRYPTOGRAPHY FROM CAESAR CIPHERS TO DIGITAL ENCRYPTION JOSHUA HOLDEN PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Copyright c 2017 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TR press.princeton.edu Jacket image courtesy of Shutterstock; design by Lorraine Betz Doneker All Rights Reserved Library of Congress Cataloging-in-Publication Data Names: Holden, Joshua, 1970– author. Title: The mathematics of secrets : cryptography from Caesar ciphers to digital encryption / Joshua Holden. Description: Princeton : Princeton University Press, [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016014840 | ISBN 9780691141756 (hardcover : alk. paper) Subjects: LCSH: Cryptography—Mathematics. | Ciphers. | Computer security. Classification: LCC Z103 .H664 2017 | DDC 005.8/2—dc23 LC record available at https://lccn.loc.gov/2016014840 British Library Cataloging-in-Publication Data is available This book has been composed in Linux Libertine Printed on acid-free paper. ∞ Printed in the United States of America 13579108642 To Lana and Richard for their love and support CONTENTS Preface xi Acknowledgments xiii Introduction to Ciphers and Substitution 1 1.1 Alice and Bob and Carl and Julius: Terminology and Caesar Cipher 1 1.2 The Key to the Matter: Generalizing the Caesar Cipher 4 1.3 Multiplicative Ciphers 6 -
Cryptographic Algorithms and Key Sizes for Personal Identity Verification
Archived NIST Technical Series Publication The attached publication has been archived (withdrawn), and is provided solely for historical purposes. It may have been superseded by another publication (indicated below). Archived Publication Series/Number: NIST Special Publication 800-78-2 Title: Cryptographic Algorithms and Key Sizes for Personal Identity Verification Publication Date(s): February 2010 Withdrawal Date: December 2010 Withdrawal Note: SP 800-78-2 is superseded in its entirety by the publication of SP 800-78-3 (December 2010). Superseding Publication(s) The attached publication has been superseded by the following publication(s): Series/Number: NIST Special Publication 800-78-3 Title: Cryptographic Algorithms and Key Sizes for Personal Identity Verification Author(s): W. Timothy Polk, Donna F. Dodson, William E. Burr, Hildegard Ferraiolo, David Cooper Publication Date(s): December 2010 URL/DOI: http://dx.doi.org/10.6028/NIST.SP.800-78-3 Additional Information (if applicable) Contact: Computer Security Division (Information Technology Lab) Latest revision of the SP 800-78-4 (as of August 7, 2015) attached publication: Related information: http://csrc.nist.gov/groups/SNS/piv/ Withdrawal N/A announcement (link): Date updated: ƵŐƵƐƚϳ, 2015 NIST Special Publication 800-78-2 Cryptographic Algorithms and Key Sizes for Personal Identity Verification W. Timothy Polk Donna F. Dodson William. E. Burr I N F O R M A T I O N S E C U R I T Y Computer Security Division Information Technology Laboratory National Institute of Standards and Technology Gaithersburg, MD, 20899-8930 February 2010 U.S. Department of Commerce Gary Locke, Secretary National Institute of Standards and Technology Dr. -
Outline Block Ciphers
Block Ciphers: DES, AES Guevara Noubir http://www.ccs.neu.edu/home/noubir/Courses/CSG252/F04 Textbook: —Cryptography: Theory and Applications“, Douglas Stinson, Chapman & Hall/CRC Press, 2002 Reading: Chapter 3 Outline n Substitution-Permutation Networks n Linear Cryptanalysis n Differential Cryptanalysis n DES n AES n Modes of Operation CSG252 Classical Cryptography 2 Block Ciphers n Typical design approach: n Product cipher: substitutions and permutations n Leading to a non-idempotent cipher n Iteration: n Nr: number of rounds → 1 2 Nr n Key schedule: k k , k , …, k , n Subkeys derived according to publicly known algorithm i n w : state n Round function r r-1 r n w = g(w , k ) 0 n w : plaintext x n Required property of g: ? n Encryption and Decryption sequence CSG252 Classical Cryptography 3 1 SPN: Substitution Permutation Networks n SPN: special type of iterated cipher (w/ small change) n Block length: l x m n x = x(1) || x(2) || … || x(m) n x(i) = (x(i-1)l+1, …, xil) n Components: π l → l n Substitution cipher s: {0, 1} {0, 1} π → n Permutation cipher (S-box) P: {1, …, lm} {1, …, lm} n Outline: n Iterate Nr times: m substitutions; 1 permutation; ⊕ sub-key; n Definition of SPN cryptosytems: n P = ?; C = ?; K ⊆ ?; n Algorithm: n Designed to allow decryption using the same algorithm n What are the parameters of the decryption algorithm? CSG252 Classical Cryptography 4 SPN: Example n l = m = 4; Nr = 4; n Key schedule: n k: (k1, …, k32) 32 bits r n k : (k4r-3, …, k4r+12) z 0 1 2 3 4 5 6 7 8 9 A B C D E F π S(z) E 4 D 1 2 F B 8 3 A 6 C -
The Whirlpool Secure Hash Function
Cryptologia, 30:55–67, 2006 Copyright Taylor & Francis Group, LLC ISSN: 0161-1194 print DOI: 10.1080/01611190500380090 The Whirlpool Secure Hash Function WILLIAM STALLINGS Abstract In this paper, we describe Whirlpool, which is a block-cipher-based secure hash function. Whirlpool produces a hash code of 512 bits for an input message of maximum length less than 2256 bits. The underlying block cipher, based on the Advanced Encryption Standard (AES), takes a 512-bit key and oper- ates on 512-bit blocks of plaintext. Whirlpool has been endorsed by NESSIE (New European Schemes for Signatures, Integrity, and Encryption), which is a European Union-sponsored effort to put forward a portfolio of strong crypto- graphic primitives of various types. Keywords advanced encryption standard, block cipher, hash function, sym- metric cipher, Whirlpool Introduction In this paper, we examine the hash function Whirlpool [1]. Whirlpool was developed by Vincent Rijmen, a Belgian who is co-inventor of Rijndael, adopted as the Advanced Encryption Standard (AES); and by Paulo Barreto, a Brazilian crypto- grapher. Whirlpool is one of only two hash functions endorsed by NESSIE (New European Schemes for Signatures, Integrity, and Encryption) [13].1 The NESSIE project is a European Union-sponsored effort to put forward a portfolio of strong cryptographic primitives of various types, including block ciphers, symmetric ciphers, hash functions, and message authentication codes. Background An essential element of most digital signature and message authentication schemes is a hash function. A hash function accepts a variable-size message M as input and pro- duces a fixed-size hash code HðMÞ, sometimes called a message digest, as output. -
Selecting Cryptographic Key Sizes
J. Cryptology (2001) 14: 255–293 DOI: 10.1007/s00145-001-0009-4 © 2001 International Association for Cryptologic Research Selecting Cryptographic Key Sizes Arjen K. Lenstra Citibank, N.A., 1 North Gate Road, Mendham, NJ 07945-3104, U.S.A. [email protected] and Technische Universiteit Eindhoven Eric R. Verheul PricewaterhouseCoopers, GRMS Crypto Group, Goudsbloemstraat 14, 5644 KE Eindhoven, The Netherlands eric.verheul@[nl.pwcglobal.com, pobox.com] Communicated by Andrew Odlyzko Received September 1999 and revised February 2001 Online publication 14 August 2001 Abstract. In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm-based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems. Key words. Symmetric key length, Public key length, RSA, ElGamal, Elliptic curve cryptography, Moore’s law. 1. Introduction 1.1. The Purpose of This Paper Cryptography is one of the most important tools that enable e-commerce because cryp- tography makes it possible to protect electronic information. The effectiveness of this protection depends on a variety of mostly unrelated issues such as cryptographic key size, protocol design, and password selection. Each of these issues is equally important: if a key is too small, or if a protocol is badly designed or incorrectly used, or if a pass- word is poorly selected or protected, then the protection fails and improper access can be gained. In this article we give some guidelines for the determination of cryptographic key sizes. -
Block Cipher and Data Encryption Standard (DES)
Block Cipher and Data Encryption Standard (DES) 2021.03.09 Presented by: Mikail Mohammed Salim Professor 박종혁 Cryptography and Information Security 1 Block Cipher and Data Encryption Standard (DES) Contents • What is Block Cipher? • Padding in Block Cipher • Ideal Block Cipher • What is DES? • DES- Key Discarding Process • Des- 16 rounds of Encryption • How secure is DES? 2 Block Cipher and Data Encryption Standard (DES) What is Block Cipher? • An encryption technique that applies an algorithm with parameters to encrypt blocks of text. • Each plaintext block has an equal length of ciphertext block. • Each output block is the same size as the input block, the block being transformed by the key. • Block size range from 64 -128 bits and process the plaintext in blocks of 64 or 128 bits. • Several bits of information is encrypted with each block. Longer messages are encoded by invoking the cipher repeatedly. 3 Block Cipher and Data Encryption Standard (DES) What is Block Cipher? • Each message (p) grouped in blocks is encrypted (enc) using a key (k) into a Ciphertext (c). Therefore, 푐 = 푒푛푐푘(푝) • The recipient requires the same k to decrypt (dec) the p. Therefore, 푝 = 푑푒푐푘(푐) 4 Block Cipher and Data Encryption Standard (DES) Padding in Block Cipher • Block ciphers process blocks of fixed sizes, such as 64 or 128 bits. The length of plaintexts is mostly not a multiple of the block size. • A 150-bit plaintext provides two blocks of 64 bits each with third block of remaining 22 bits. • The last block of bits needs to be padded up with redundant information so that the length of the final block equal to block size of the scheme. -
The Enigma History and Mathematics
The Enigma History and Mathematics by Stephanie Faint A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master of Mathematics m Pure Mathematics Waterloo, Ontario, Canada, 1999 @Stephanie Faint 1999 I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by pho tocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. 11 The University of Waterloo requires the signatures of all persons using or pho tocopying this thesis. Please sign below, and give address and date. ill Abstract In this thesis we look at 'the solution to the German code machine, the Enigma machine. This solution was originally found by Polish cryptologists. We look at the solution from a historical perspective, but most importantly, from a mathematical point of view. Although there are no complete records of the Polish solution, we try to reconstruct what was done, sometimes filling in blanks, and sometimes finding a more mathematical way than was originally found. We also look at whether the solution would have been possible without the help of information obtained from a German spy. IV Acknowledgements I would like to thank all of the people who helped me write this thesis, and who encouraged me to keep going with it. In particular, I would like to thank my friends and fellow grad students for their support, especially Nico Spronk and Philippe Larocque for their help with latex.