On some Constructions of Shapeless Quasigroups


 Percival Norman
 3 years ago
 Views:
Transcription
1 Aleksandra Mileva 1 and Smile Markovski 2 1 Faculty of Computer Science, University Goce Delčev, Štip 2 Faculty of Computer Science and Computer Engineering, University Ss. Cyril and Methodius  Skopje Republic of Macedonia Loops 11 July 2527, Třešt, Czech Republic
2 Motivation Today, we can already speak about quasigroup based cryptography, because the number of new defined cryptographic primitives that use quasigroups is growing up: stream cipher EDON80 (Gligoroski et al. (estream 2008)), hash functions EDONR (Gligoroski et al. (SHA )) and NaSHA (Markovski and Mileva (SHA )), digital signature algorithm MQQDSA (Gligoroski et al. (ACAM 2008)), public key cryptosystem LQLPs (for s {104, 128, 160}) (Markovski et al. (SCC 2010)), etc.
3 Motivation Different authors seek quasigroups with different properties. Definition (Gligoroski et al (2006)) A quasigroup (Q, ) of order r is said to be shapeless iff it is nonidempotent, noncommutative, nonassociative, it does not have neither left nor right unit, it does not contain proper subquasigroups, and there is no k < 2r such that identities of the kinds x (x (x y)) = y, y = ((y x) x) x } {{ } } {{ } k k (1)
4 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
5 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
6 Complete mappings and orthomorphisms Definition (Mann (1949), Johnson et al (1961), Evans (1989)) A complete mapping of a quasigroup (group) (G, +) is a permutation φ : G G such that the mapping θ : G G defined by θ(x) = x + φ(x) (θ = I + φ, where I is the identity mapping) is again a permutation of G. The mapping θ is the orthomorphism associated to the complete mapping φ. A quasigroup (group) G is admissible if there is a complete mapping φ : G G.
7 Complete mappings and orthomorphisms If θ is the orthomorphism associated to the complete mapping φ, then φ is the orthomorphism associated to the complete mapping θ, The inverse of the complete mapping (orthomorphism) is also a complete mapping (orthomorphism) (Johnson et al (1961)), Two orthomorphisms θ 1 and θ 2 of G are said to be orthogonal if and only if θ 1 θ 1 2 is an orthomorphism of G too. Even more, every orthomorphism is orthogonal to I and θ 1 is orthogonal to θ if and only if θ 2 is an orthomorphism (Johnson et al (1961))
8 Sade s Diagonal method (Sade (1957)) Consider the group (Z n, +) and let θ be a permutation of the set Z n, such that φ(x) = x θ(x) is also a permutation. Define an operation on Z n by: x y = θ(x y) + y (2) where x, y Z n. Then (Z n, ) is a quasigroup (and we say that (Z n, ) is derived by θ).
9 Generalization Theorem Let φ be a complete mapping of the admissible group (G, +) and let θ be an orthomorphism associated to φ. Define operations and on G by x y = φ(y x) + y = θ(y x) + x, (3) x y = θ(x y) + y = φ(x y) + x, (4) where x, y G. Then (G, ) and (G, ) are quasigroups, opposite to each other, i.e., x y = y x for every x, y G.
10 Quasigroup conjugates Theorem Let φ : G G be a complete mapping of the abelian group (G, +) with associated orthomorphism θ : G G. Then all the conjugates of the quasigroup (G, ) can be obtain by the equation (4) and the following statements are true. a) The quasigroup (G, /) is derived by the orthomorphism θ 1 associated to the complete mapping φθ 1. b) The quasigroup (G, \) is derived by the orthomorphism θ( φ) 1 associated to the complete mapping ( φ) 1. c) The quasigroup (G, //) is derived by the orthomorphism φ( θ) 1 associated to the complete mapping ( θ) 1. d) The quasigroup (G, \\) is derived by the orthomorphism φ 1 associated to the complete mapping θφ 1. e) (G, ) = (G, ).
11 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
12 If θ(0) 0 then the quasigroup (G, ) has no idempotent elements. (G, ) does not have a left unit and if θ is different to the identity mapping it does not have a right unit either. (G, ) is nonassociative quasigroup. If θ(z) θ( z) z for some z G then (G, ) is noncommutative quasigroup.
13 The identity holds true in (G, ) iff θ l = I. The identity y = ((y x)... ) x } {{ } l x ( (x y)) = y } {{ } l holds true in (G, ) iff ( φ) l = I = (I θ) l.
14 If θ has a cycle containing 0 of length greater than G /2, then the quasigroup (G, ) cannot have a proper subquasigroup. The quasigroup (G, ) is without a proper subquasigroup if S G /2, where S = i=p+1 i=1 i=p+1 i=1 {θ i (0)} {λ i (0)} i=p+1 i=1 i=p+1 i=1 {θ(0) + θ i (0)} {λ(0) + λ i (0)} p = [ G /2] and λ = θ( φ) 1. i=p+1 i=1 i=p+1 i=1 {θ( θ i (0)) + θ i (0)} {λ( λ i (0)) + λ i (0)},
15 ((x x)... ) x = θ l (0) + x, } {{ } x ( (x x)) = ( φ) l (0) + x. } {{ } (5) l l ((x/ x)/... )/x = (θ 1 ) l (0) + x, } {{ } x/(... (x /x)) = (φθ 1 ) l (0) + x, } {{ } (6) l l ((x\x) \... ) \ x = ( θ( φ) 1 ) l (0)+x, x \ ( \ (x \x)) = (( φ) 1 ) l (0)+x, } {{ } } {{ } l l ((x// x)//... )//x } {{ } l ((x \\ x) \\... ) \\x } {{ } l (7) = ( φ( θ) 1 ) l (0)+x, x//(... //(x //x)) = ( ( θ) 1 ) l (0)+x, } {{ } l (8) = ( φ 1 ) l (0) + x, x \\( \\(x \\ x)) = (θφ 1 ) l (0) + x. } {{ } l (9)
16 (G, ) as an shapeless quasigroup Theorem Let θ be an orthomorphism of the abelian group (G, +), and let (G, ) be a quasigroup derived by θ by the equation (4). If θ is with the following properties: a) θ(0) 0 b) θ k I for all k < 2 G c) (I θ) k I for all k < 2 G d) θ(z) θ( Z) Z for some Z G e) S G /2, then (G, ) is a shapeless quasigroup.
17 Example of a shapeless quasigroup The abelian group (Z2 4, ) is given. x θ(x) φ(x) = x θ(x) We define the quasigroup operation as x y = θ(x y) y
18 Example of a shapeless quasigroup
19 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
20 Different generalizations of the Feistel Network Feistel Networks are introduced by H. Feistel (Scientific American, 1973). Parameterized Feistel Network (PFN) (Markovski and Mileva (QRS 2009)), Extended Feistel networks type1, type2 and type3 (Zheng et al (CRYPTO 1989)), Generalized FeistelNon Linear Feedback Shift Register (GFNLFSR) (Choy et al (ACISP 2009)). We will redefine the last two generalizations with parameters and over abelian groups.
21 Parameterized Feistel Network (PFN) Definition Let (G, +) be an abelian group, let f : G G be a mapping and let A, B, C G are constants. The Parameterized Feistel Network F A,B,C : G 2 G 2 created by f is defined for every l, r G by F A,B,C (l, r) = (r + A, l + B + f (r + C)).
22 Parameterized Feistel Network (PFN) In (Markovski and Mileva (QRS 2009)) is shown that if starting mapping f is bijection, the Parameterized Feistel Network F A,B,C and its square F 2 A,B,C are orthomorphisms of the group (G 2, +). More over, they are orthogonal orthomorphisms.
23 Parameterized Extended Feistel Network (PEFN) type1 Definition Let (G, +) be an abelian group, let f : G G be a mapping, let A 1, A 2,..., A n+1 G are constants and let n > 1 be an integer. The Parameterized Extended Feistel Network (PEFN) type1 F A1,A 2,...,A n+1 : G n G n created by f is defined for every (x 1, x 2,..., x n ) G n by F A1,A 2,...,A n+1 (x 1, x 2,..., x n ) = (x 2 + f (x 1 + A 1 ) + A 2, x 3 + A 3,..., x n + A n, x 1 + A n+1 ).
24 Parameterized Extended Feistel Network (PEFN) type1
25 Parameterized Extended Feistel Network (PEFN) type1 Theorem Let (G, +) be an abelian group, let n > 1 be an integer and A 1, A 2,..., A n+1 G. If F A1,A 2,...,A n+1 : G n G n is a PEFN type1 created by a bijection f : G G, then F A1,A 2,...,A n+1 is an orthomorphism of the group (G n, +).
26 Parameterized Generalized FeistelNon Linear Feedback Shift Register (PGFNLFSR) Definition Let (G, +) be an abelian group, let f : G G be a mapping, let A 1, A 2,..., A n+1 G are constants and let n > 1 be an integer. The Parameterized Generalized FeistelNon Linear Feedback Shift Register (PGFNLFSR) F A1,A 2,...,A n+1 : G n G n created by f is defined for every (x 1, x 2,..., x n ) G n by F A1,A 2,...,A n+1 (x 1, x 2,..., x n ) = (x 2 + A 1, x 3 + A 2,..., x n + A n 1, x x n + A n + f (x 1 + A n+1 )).
27 Parameterized Generalized FeistelNon Linear Feedback Shift Register (PGFNLFSR)
28 Parameterized Generalized FeistelNon Linear Feedback Shift Register (PGFNLFSR) Theorem Let we use the abelian group (Z m 2, ), let n = 2k be an integer and A 1, A 2,..., A n+1 Z m 2. If F A 1,A 2,...,A n+1 : (Z m 2 )n (Z m 2 )n is a PGFNLFSR created by a bijection f : Z m 2 Zm 2, then is an orthomorphism of the group ((Z m 2 )n, ). F A1,A 2,...,A n+1
29 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
30 Quasigroups derived by the PFN F A,B,C Proposition Let (G, +) be an abelian group and let f : G G be a bijection. Let F A,B,C be a Parameterized Feistel Network created by f, and let (G 2, ) be a quasigroup derived by F A,B,C by the equation (4). If F A,B,C is with the following properties: a)a 0 or B f (C) b) F k A,B,C I for all k < 2 G 2 c) (I F A,B,C ) k I for all k < 2 G 2 d) S G 2 /2, then (G 2, ) is a shapeless quasigroup.
31 Quasigroups derived by the PFN F A,B,C We made a mfile in MatLab that produce a starting bijection f : Z2 n Z 2 n and parameters A, B and C for the orthomorphism PFN F A,B,C which produce a shapeless quasigroup. The group operation is XOR. For n = 3, 4, 5, 6 execution time is less than a minute and for n = 7 is less than five minutes. These results correspond to shapeless quasigroups of order 2 6, 2 8, 2 10, 2 12 and 2 14.
32 Quasigroups derived by the PEFN type1 F A1,A 2,...,A n+1 Proposition Let (G, +) be an abelian group, let n > 1 be an integer, let A 1, A 2,..., A n+1 G and let f : G G be a bijection. Let F A1,A 2,...,A n+1 be a PEFN type1 created by f, and let (G n, ) be a quasigroup derived by F A1,A 2,...,A n+1 by the equation (4). If F A1,A 2,...,A n+1 is with the following properties: a)a i 0 for some i {3, 4,..., n + 1} or A 2 f (A 1) b) FA k 1,A 2,...,A n+1 I for all k < 2 G n c) (I F A1,A 2,...,A n+1 ) k I for all k < 2 G n d) S G n /2, then (G n, ) is a shapeless quasigroup.
33 Quasigroups derived by the PGFNLFSR F A1,A 2,...,A n+1 Proposition Let we use the abelian group (Z m 2, ), let n = 2k be an integer, let A 1, A 2,..., A n+1 Z m 2 and let f : Z m 2 Z m 2 be a bijection. Let F A1,A 2,...,A n+1 be a PGFNLFSR created by f, and let ((Z m 2 ) n, ) be a quasigroup derived by F A1,A 2,...,A n+1 by the equation (4). If F A1,A 2,...,A n+1 is with the following properties: a) A i 0 for some i {1, 2,..., n 1} or A n f (A n+1) b) F k A 1,A 2,...,A n+1 I for all k < 2 G n c) (I F A1,A 2,...,A n+1 ) k I for all k < 2 G n d) S G n /2, then ((Z m 2 ) n, ) is a shapeless quasigroup.
34 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5
35 We give some constructions of shapeless quasigroups of different orders by using quasigroups produced by diagonal method from orthomorphisms. We show that PEFN type1 produced by a bijection is an orthomorphism of the abelian group (G n, +) and that PGFNLFSR produced by a bijection is an orthomorphism of the ((Z m 2 )n, ). Also, we parameterized these orthomorphisms for the need of cryptography, so we can work with different quasigroups in every iterations of the future cryptographic primitives.
36 THANKS FOR YOUR ATTENTION!!!
arxiv:1010.3163v1 [cs.cr] 15 Oct 2010
The Digital Signature Scheme MQQSIG Intellectual Property Statement and Technical Description 10 October 2010 Danilo Gligoroski 1 and Svein Johan Knapskog 2 and Smile Markovski 3 and Rune Steinsmo Ødegård
More informationLecture 12. More algebra: Groups, semigroups, monoids, strings, concatenation.
V. Borschev and B. Partee, November 1, 2001 p. 1 Lecture 12. More algebra: Groups, semigroups, monoids, strings, concatenation. CONTENTS 1. Properties of operations and special elements...1 1.1. Properties
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More information(Q, ), (R, ), (C, ), where the star means without 0, (Q +, ), (R +, ), where the plussign means just positive numbers, and (U, ),
2 Examples of Groups 21 Some infinite abelian groups It is easy to see that the following are infinite abelian groups: Z, +), Q, +), R, +), C, +), where R is the set of real numbers and C is the set of
More informationCryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur
Cryptography and Network Security Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Module No. # 01 Lecture No. # 05 Classic Cryptosystems (Refer Slide Time: 00:42)
More informationCryptography and Network Security. Prof. D. Mukhopadhyay. Department of Computer Science and Engineering. Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 12 Block Cipher Standards
More informationGroups in Cryptography
Groups in Cryptography Çetin Kaya Koç http://cs.ucsb.edu/~koc/cs178 koc@cs.ucsb.edu Koç (http://cs.ucsb.edu/~koc) ucsb cs 178 intro to crypto winter 2013 1 / 13 Groups in Cryptography A set S and a binary
More informationCryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #10 Symmetric Key Ciphers (Refer
More informationCRYPTOGRAPHIC PRIMITIVES AN INTRODUCTION TO THE THEORY AND PRACTICE BEHIND MODERN CRYPTOGRAPHY
CRYPTOGRAPHIC PRIMITIVES AN INTRODUCTION TO THE THEORY AND PRACTICE BEHIND MODERN CRYPTOGRAPHY Robert Sosinski Founder & Engineering Fellow Known as "America's Cryptologic Wing", is the only Air Force
More informationLukasz Pater CMMS Administrator and Developer
Lukasz Pater CMMS Administrator and Developer EDMS 1373428 Agenda Introduction Why do we need asymmetric ciphers? Oneway functions RSA Cipher Message Integrity Examples Secure Socket Layer Single Sign
More informationTable of Contents. Bibliografische Informationen http://dnb.info/996514864. digitalisiert durch
1 Introduction to Cryptography and Data Security 1 1.1 Overview of Cryptology (and This Book) 2 1.2 Symmetric Cryptography 4 1.2.1 Basics 4 1.2.2 Simple Symmetric Encryption: The Substitution Cipher...
More informationCIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives
CIS 6930 Emerging Topics in Network Security Topic 2. Network Security Primitives 1 Outline Absolute basics Encryption/Decryption; Digital signatures; DH key exchange; Hash functions; Application of hash
More informationOverview of Cryptographic Tools for Data Security. Murat Kantarcioglu
UT DALLAS Erik Jonsson School of Engineering & Computer Science Overview of Cryptographic Tools for Data Security Murat Kantarcioglu Pag. 1 Purdue University Cryptographic Primitives We will discuss the
More informationT Cryptography and Data Security
Kaisa Nyberg Email: kaisa dot nyberg at aalto dot fi Department of Computer Science Aalto University School of Science Lecture 1: Arrangements Course contents Finite Groups and Rings Euclid s Algorithm
More informationNetwork Security Technology Network Management
COMPUTER NETWORKS Network Security Technology Network Management Source Encryption E(K,P) Decryption D(K,C) Destination The author of these slides is Dr. Mark Pullen of George Mason University. Permission
More informationCryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Cryptography and Network Security Prof. D. Mukhopadhyay Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture No. # 11 Block Cipher Standards (DES) (Refer Slide
More informationMathematics Notes for Class 12 chapter 3. Matrices
1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form
More informationA NOVEL STRATEGY TO PROVIDE SECURE CHANNEL OVER WIRELESS TO WIRE COMMUNICATION
A NOVEL STRATEGY TO PROVIDE SECURE CHANNEL OVER WIRELESS TO WIRE COMMUNICATION Prof. Dr. Alaa Hussain Al Hamami, Amman Arab University for Graduate Studies Alaa_hamami@yahoo.com Dr. Mohammad Alaa Al
More informationEXAM questions for the course TTM4135  Information Security May 2013. Part 1
EXAM questions for the course TTM4135  Information Security May 2013 Part 1 This part consists of 5 questions all from one common topic. The number of maximal points for every correctly answered question
More informationAsymmetric Cryptography. Mahalingam Ramkumar Department of CSE Mississippi State University
Asymmetric Cryptography Mahalingam Ramkumar Department of CSE Mississippi State University Mathematical Preliminaries CRT Chinese Remainder Theorem Euler Phi Function Fermat's Theorem Euler Fermat's Theorem
More informationOn the Influence of the Algebraic Degree of the Algebraic Degree of
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 1, JANUARY 2013 691 On the Influence of the Algebraic Degree of the Algebraic Degree of Christina Boura and Anne Canteaut on Abstract We present a
More informationCryptography and Network Security Chapter 3
Cryptography and Network Security Chapter 3 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 3 Block Ciphers and the Data Encryption Standard All the afternoon
More informationGeneral about the course. Course assignment. Outline. T Cryptosystems. Summary and review of lectures
General about the course T 110.5211 Cryptosystems Summary and review of lectures 4.12.2008 This is the fifth time the course was arranged We need a course covering practical cryptographic topics Security
More informationMATH 433 Applied Algebra Lecture 13: Examples of groups.
MATH 433 Applied Algebra Lecture 13: Examples of groups. Abstract groups Definition. A group is a set G, together with a binary operation, that satisfies the following axioms: (G1: closure) for all elements
More informationUnderstanding the division property
Understanding the division property Christina Boura (joint work with Anne Canteaut) ASK 2015, October 1, 2015 1 / 36 Introduction In Eurocrypt 2015, Yosuke Todo introduces a new property, called the division
More informationSecret File Sharing Techniques using AES algorithm. C. Navya Latha 200201066 Garima Agarwal 200305032 Anila Kumar GVN 200305002
Secret File Sharing Techniques using AES algorithm C. Navya Latha 200201066 Garima Agarwal 200305032 Anila Kumar GVN 200305002 1. Feature Overview The Advanced Encryption Standard (AES) feature adds support
More informationGalois Fields and Hardware Design
Galois Fields and Hardware Design Construction of Galois Fields, Basic Properties, Uniqueness, Containment, Closure, Polynomial Functions over Galois Fields Priyank Kalla Associate Professor Electrical
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationCS 758: Cryptography / Network Security
CS 758: Cryptography / Network Security offered in the Fall Semester, 2003, by Doug Stinson my office: DC 3122 my email address: dstinson@uwaterloo.ca my web page: http://cacr.math.uwaterloo.ca/~dstinson/index.html
More informationCryptography and Network Security
Cryptography and Network Security Spring 2012 http://users.abo.fi/ipetre/crypto/ Lecture 3: Block ciphers and DES Ion Petre Department of IT, Åbo Akademi University January 17, 2012 1 Data Encryption Standard
More informationApplied Cryptology. Ed Crowley
Applied Cryptology Ed Crowley 1 Basics Topics Basic Services and Operations Symmetric Cryptography Encryption and Symmetric Algorithms Asymmetric Cryptography Authentication, Nonrepudiation, and Asymmetric
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationStream Ciphers. Example of Stream Decryption. Example of Stream Encryption. Real Cipher Streams. Terminology. Introduction to Modern Cryptography
Introduction to Modern Cryptography Lecture 2 Symmetric Encryption: Stream & Block Ciphers Stream Ciphers Start with a secret key ( seed ) Generate a keying stream ith bit/byte of keying stream is a function
More informationLecture 9: Application of Cryptography
Lecture topics Cryptography basics Using SSL to secure communication links in J2EE programs Programmatic use of cryptography in Java Cryptography basics Encryption Transformation of data into a form that
More informationA NEW HASH ALGORITHM: Khichidi1
A NEW HASH ALGORITHM: Khichidi1 Abstract This is a technical document describing a new hash algorithm called Khichidi1 and has been written in response to a Hash competition (SHA3) called by National
More informationCrypto Basics. Ed Crowley. Spring 2010
Crypto Basics Ed Crowley Spring 2010 Kerckhoff s Principle Symmetric Crypto Overview Key management problem Attributes Modes Symmetric Key Algorithms DES Attributes Modes 3DES AES Other Symmetric Ciphers
More informationOutline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures
Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike
More informationA Factoring and Discrete Logarithm based Cryptosystem
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 11, 511517 HIKARI Ltd, www.mhikari.com A Factoring and Discrete Logarithm based Cryptosystem Abdoul Aziz Ciss and Ahmed Youssef Ecole doctorale de Mathematiques
More informationOn loops with universal antiautomorphic inverse property
On loops with universal antiautomorphic inverse property Department of Fundamental Mathematics Moldova State University Chisinau, Republic of Moldova The 5th International Mathematical Conference on Quasigroups
More informationCSC474/574  Information Systems Security: Homework1 Solutions Sketch
CSC474/574  Information Systems Security: Homework1 Solutions Sketch February 20, 2005 1. Consider slide 12 in the handout for topic 2.2. Prove that the decryption process of a oneround Feistel cipher
More informationSecurity usually depends on the secrecy of the key, not the secrecy of the algorithm (i.e., the open design model!)
1 A cryptosystem has (at least) five ingredients: 1. 2. 3. 4. 5. Plaintext Secret Key Ciphertext Encryption algorithm Decryption algorithm Security usually depends on the secrecy of the key, not the secrecy
More informationRemotely Keyed Encryption Using NonEncrypting Smart Cards
THE ADVANCED COMPUTING SYSTEMS ASSOCIATION The following paper was originally published in the USENIX Workshop on Smartcard Technology Chicago, Illinois, USA, May 10 11, 1999 Remotely Keyed Encryption
More informationRevision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic
More informationss. Cyril and Methodius University Faculty of Computer Science and Engineering Skopje, Macedonia
ss. Cyril and Methodius University Faculty of Computer Science and Engineering Skopje, Macedonia Prof. Dimitar Trajanov, THE UNIVERSITY Ss. Cyril and Methodius University in Skopje is oldest and the biggest
More informationAbstract Algebra Cheat Sheet
Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationDesign and Implementation of an Encryption Algorithm for use in RFID System
Design and Implementation of an Encryption Algorithm for use in RFID System M. B. Abdelhalim AASTMT Cairo, Egypt M. ElMahallawy, M. Ayyad AASTMT Cairo, Egypt A. Elhennawy ASU Cairo, Egypt Abstract The
More informationTextbooks: Matt Bishop, Introduction to Computer Security, AddisonWesley, November 5, 2004, ISBN 0321247442.
CSET 4850 Computer Network Security (4 semester credit hours) CSET Elective IT Elective Current Catalog Description: Theory and practice of network security. Topics include firewalls, Windows, UNIX and
More informationSymmetric and asymmetric cryptography overview
Symmetric and asymmetric cryptography overview Modern cryptographic methods use a key to control encryption and decryption Two classes of keybased encryption algorithms symmetric (secretkey) asymmetric
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationSTUDY GUIDE LINEAR ALGEBRA. David C. Lay University of Maryland College Park AND ITS APPLICATIONS THIRD EDITION UPDATE
STUDY GUIDE LINEAR ALGEBRA AND ITS APPLICATIONS THIRD EDITION UPDATE David C. Lay University of Maryland College Park Copyright 2006 Pearson AddisonWesley. All rights reserved. Reproduced by Pearson AddisonWesley
More informationThe Laws of Cryptography Cryptographers Favorite Algorithms
2 The Laws of Cryptography Cryptographers Favorite Algorithms 2.1 The Extended Euclidean Algorithm. The previous section introduced the field known as the integers mod p, denoted or. Most of the field
More informationInverses and powers: Rules of Matrix Arithmetic
Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3
More informationChapter 7: Products and quotients
Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products
More informationNetwork Security: Cryptography CS/SS G513 S.K. Sahay
Network Security: Cryptography CS/SS G513 S.K. Sahay BITSPilani, K.K. Birla Goa Campus, Goa S.K. Sahay Network Security: Cryptography 1 Introduction Network security: measure to protect data/information
More informationCIS433/533  Computer and Network Security Cryptography
CIS433/533  Computer and Network Security Cryptography Professor Kevin Butler Winter 2011 Computer and Information Science A historical moment Mary Queen of Scots is being held by Queen Elizabeth and
More informationNetwork Security. Gaurav Naik Gus Anderson. College of Engineering. Drexel University, Philadelphia, PA. Drexel University. College of Engineering
Network Security Gaurav Naik Gus Anderson, Philadelphia, PA Lectures on Network Security Feb 12 (Today!): Public Key Crypto, Hash Functions, Digital Signatures, and the Public Key Infrastructure Feb 14:
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More informationINFORMATION SECURITY A MULTIDISCIPLINARY. Stig F. Mjolsnes INTRODUCTION TO. Norwegian University ofscience & Technology. CRC Press
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN A MULTIDISCIPLINARY INTRODUCTION TO INFORMATION SECURITY Stig F. Mjolsnes Norwegian University ofscience & Technology Trondheim
More informationSolutions to Problem Set 1
YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467b: Cryptography and Computer Security Handout #8 Zheng Ma February 21, 2005 Solutions to Problem Set 1 Problem 1: Cracking the Hill cipher Suppose
More informationPythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures
More informationVALLIAMMAI ENGINEERING COLLEGE
VALLIAMMAI ENGINEERING COLLEGE (A member of SRM Institution) SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year and Semester : I / II Section : 1 Subject Code : NE7202
More informationProf. Jadranka Dabovic  Anastasovska, PhD Ass. Neda Zdraveva, MSci. Faculty of Law Iustinianus Primus Skopje
Teaching Intellectual Property at Law Schools in the Republic of Macedonia Prof. Jadranka Dabovic  Anastasovska, PhD Ass. Neda Zdraveva, MSci Faculty of Law Iustinianus Primus Skopje On the roots. Introduced
More informationEMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION
EMBEDDING DEGREE OF HYPERELLIPTIC CURVES WITH COMPLEX MULTIPLICATION CHRISTIAN ROBENHAGEN RAVNSHØJ Abstract. Consider the Jacobian of a genus two curve defined over a finite field and with complex multiplication.
More informationShared Secret = Trust
Trust The fabric of life! Holds civilizations together Develops by a natural process Advancement of technology results in faster evolution of societies Weakening the natural bonds of trust From time to
More informationPage 1 of 37 # $ $ " $# " #! " % $ &'(")"# *+,)#)./(0 1)2+)1*+, 3")#)./("*+4,%" )#)./(,,)#2+)( 3 %&'/ ",, &*)1*/252+))&1)1*'25 2+))6 " #% 6" " 6 " ",, ", #, 1+, 6 6,, " 3, )/, &'(" )/,6# 2+2(%,*(, 1, ")#).*+
More information!' /%!' / / 8 ' 2 % / L ' % M " 7'' / 75,G ; M
!' /% ' '/'' /%'' G " '' ''' M L 7%/% %% %% /!%,%%=>!' / / 5'=> 8 ' 2 % / ' 5'=> 7'/Q( /,%%=> L ' % M " 7'' / % /6 7%,5 6 7%,5=67> 7'E"/ % / L 2 % /,G 75,G ; M " ' /%,G. %/; , B 1 0.
More information(Notice also that this set is CLOSED, but does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)
Definition 3.1 Group Suppose the binary operation p is defined for elements of the set G. Then G is a group with respect to p provided the following four conditions hold: 1. G is closed under p. That is,
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationCLASSIFYING FINITE SUBGROUPS OF SO 3
CLASSIFYING FINITE SUBGROUPS OF SO 3 HANNAH MARK Abstract. The goal of this paper is to prove that all finite subgroups of SO 3 are isomorphic to either a cyclic group, a dihedral group, or the rotational
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationPublic Key Cryptography and RSA. Review: Number Theory Basics
Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and
More informationTalk announcement please consider attending!
Talk announcement please consider attending! Where: Maurer School of Law, Room 335 When: Thursday, Feb 5, 12PM 1:30PM Speaker: Rafael Pass, Associate Professor, Cornell University, Topic: Reasoning Cryptographically
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationComputer Security: Principles and Practice
Computer Security: Principles and Practice Chapter 20 PublicKey Cryptography and Message Authentication First Edition by William Stallings and Lawrie Brown Lecture slides by Lawrie Brown PublicKey Cryptography
More informationModern Block Cipher Standards (AES) Debdeep Mukhopadhyay
Modern Block Cipher Standards (AES) Debdeep Mukhopadhyay Assistant Professor Department of Computer Science and Engineering Indian Institute of Technology Kharagpur INDIA 721302 Objectives Introduction
More informationSoftware implementation of the geodatabase model of the Republic of Macedonia
Software implementation of the geodatabase model of the Republic of Macedonia Blagoj Delipetrov 1, Dragan Mihajlov 2, Marjan Delipetrov 3 1,3 University Goce Delcev, 1 Faculty of Informatics, 3 Faculty
More informationLecture 1: Introduction. CS 6903: Modern Cryptography Spring 2009. Nitesh Saxena Polytechnic University
Lecture 1: Introduction CS 6903: Modern Cryptography Spring 2009 Nitesh Saxena Polytechnic University Outline Administrative Stuff Introductory Technical Stuff Some Pointers Course Web Page http://isis.poly.edu/courses/cs6903s10
More informationPublicKey Cryptography. Oregon State University
PublicKey Cryptography Çetin Kaya Koç Oregon State University 1 Sender M Receiver Adversary Objective: Secure communication over an insecure channel 2 Solution: Secretkey cryptography Exchange the key
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationMATH36001 Background Material 2015
MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be
More information(x + a) n = x n + a Z n [x]. Proof. If n is prime then the map
22. A quick primality test Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find
More informationT Cryptology Spring 2009
T79.5501 Cryptology Spring 2009 Homework 2 Tutor : Joo Y. Cho joo.cho@tkk.fi 5th February 2009 Q1. Let us consider a cryptosystem where P = {a, b, c} and C = {1, 2, 3, 4}, K = {K 1, K 2, K 3 }, and the
More informationAdvanced Cryptography
Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.
More informationAsymmetric Encryption. With material from Jonathan Katz, David Brumley, and Dave Levin
Asymmetric Encryption With material from Jonathan Katz, David Brumley, and Dave Levin Warmup activity Overview of asymmetrickey crypto Intuition for El Gamal and RSA And intuition for attacks Digital
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLecture 9  Message Authentication Codes
Lecture 9  Message Authentication Codes Boaz Barak March 1, 2010 Reading: BonehShoup chapter 6, Sections 9.1 9.3. Data integrity Until now we ve only been interested in protecting secrecy of data. However,
More informationNetwork Security. A Quick Overview. Joshua Hill joshweb@untruth.org http://www.untruth.org
Network Security A Quick Overview Joshua Hill joshweb@untruth.org http://www.untruth.org Security Engineering What is Security Engineering? "Security Engineering is about building systems to remain dependable
More information3 1. Note that all cubes solve it; therefore, there are no more
Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationChapter 7. Permutation Groups
Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationLemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.
Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a
More informationCryptography and Network Security, PART IV: Reviews, Patches, and11.2012 Theory 1 / 53
Cryptography and Network Security, PART IV: Reviews, Patches, and Theory Timo Karvi 11.2012 Cryptography and Network Security, PART IV: Reviews, Patches, and11.2012 Theory 1 / 53 Key Lengths I The old
More informationLecture 4 Data Encryption Standard (DES)
Lecture 4 Data Encryption Standard (DES) 1 Block Ciphers Map nbit plaintext blocks to nbit ciphertext blocks (n = block length). For nbit plaintext and ciphertext blocks and a fixed key, the encryption
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More information