The Paillier Cryptosystem

Review of Elementary Cryptography. For more material, see my notes of CSE 5351, available on my webpage

Review of Elemetary Cryptography For more material, see my otes of CSE 5351, available o my webpage Outlie Security (CPA, CCA, sematic security, idistiguishability) RSA ElGamal Homomorphic ecryptio 2 Two

Factoring Algorithms and Other Attacks on the RSA 1/12

Factorig Algorithms ad Other Attacks o the RSA T-79550 Cryptology Lecture 8 April 8, 008 Kaisa Nyberg Factorig Algorithms ad Other Attacks o the RSA / The Pollard p Algorithm Let B be a positive iteger

Message Authentication Codes. Reading: Chapter 4 of Katz & Lindell

Message Autheticatio Codes Readig: Chapter 4 of Katz & Lidell 1 Message autheticatio Bob receives a message m from Alice, he wats to ow (Data origi autheticatio) whether the message was really set by Alice.

A Generalization of Paillier s Public-Key System with Applications to Electronic Voting

A Generalization of Paillier s Public-Key System with Applications to Electronic Voting Ivan Damgård, Mads Jurik and Jesper Buus Nielsen Aarhus University, Dept. of Computer Science, BRICS Abstract. We

1 Summary: Binary and Logic

1 Summary: Biary ad Logic Biary Usiged Represetatio : each 1-bit is a power of two, the right-most is for 2 0 : 0110101 2 = 2 5 + 2 4 + 2 2 + 2 0 = 32 + 16 + 4 + 1 = 53 10 Usiged Rage o bits is [0...2

An Overview of Homomorphic Encryption

An Overview of Homomorphic Encryption Alexander Lange Department of Computer Science Rochester Institute of Technology Rochester, NY 14623 May 9, 2011 Alexander Lange (RIT) Homomorphic Encryption May 9,

Lecture 11: Pseudorandom functions

COM S 6830 Cryptography Oct 1, 2009 Istructor: Rafael Pass 1 Recap Lecture 11: Pseudoradom fuctios Scribe: Stefao Ermo Defiitio 1 (Ge, Ec, Dec) is a sigle message secure ecryptio scheme if for all uppt

Estimating the φ(n) of Upper/Lower Bound in its RSA Cryptosystem

Estimatig the φ() of Upper/Lower Boud i its RSA Cryptosystem Cheglia Liu 1 ad Ziwei Ye 2 1 Departmet of Electrical Egieerig, Natioal Tsig-Hua Uiversity, Taiwa cheglia.liu@gmail.com 2 Departmet of Computer

7. Modern Techniques. Data Encryption Standard (DES)

7. Moder Techiques. Data Ecryptio Stadard (DES) The objective of this chapter is to illustrate the priciples of moder covetioal ecryptio. For this purpose, we focus o the most widely used covetioal ecryptio

International Journal of Engineering Science Invention Research & Development; Vol. II Issue VIII February e-issn:

Iteratioal Joural of Egieerig Sciece Ivetio Research & Developmet; Vol. II Issue VIII February 216 www.ijesird.com eissn: 236 MULTIPLE ENCRYPTIONS OF VARIOUS CIPHERS A. ChadraSekhar*,1, B.Ravi Kumar 2,

Lossy Trapdoor Functions from Smooth Homomorphic Hash Proof Systems

Electroic Colloquium o Computatioal Complexity, Revisio 1 of Report No 127 (2009) Lossy Trapdoor Fuctios from Smooth Homomorphic Hash Proof Systems July 4, 2010 Abstract I STOC 08, Peikert ad Waters itroduced

Lecture Notes 15 : Voting, Homomorphic Encryption

6.857 Computer and Network Security October 29, 2002 Lecture Notes 15 : Voting, Homomorphic Encryption Lecturer: Ron Rivest Scribe: Ledlie/Ortiz/Paskalev/Zhao 1 Introduction The big picture and where we

Non-Interactive Zero-Knowledge from Homomorphic Encryption. Ivan Damgård (Aarhus Universitet) Nelly Fazio, Antonio Nicolosi (NYU)

Non-Interactive Zero-Knowledge from Homomorphic Encryption Ivan Damgård (Aarhus Universitet) Nelly Fazio, Antonio Nicolosi (NYU) January 27th, 2006 NYU Crypto Reading Group Zero-Knowledge and Interaction

Simon Blackburn. Sean Murphy. Jacques Stern. Laboratoire d'informatique, Ecole Normale Superieure, Abstract

The Cryptaalysis of a Public Key Implemetatio of Fiite Group Mappigs Simo Blackbur Sea Murphy Iformatio Security Group, Royal Holloway ad Bedford New College, Uiversity of Lodo, Egham, Surrey TW20 0EX,

Solutions to Math 347 Practice Problems for the final

Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is

Two-Input Functional Encryption for Inner Products from Bilinear Maps

Two-Iput Fuctioal Ecryptio for Ier Products from Biliear Maps Kwagsu Lee Dog Hoo Lee Abstract Fuctioal ecryptio is a ew paradigm of public-key ecryptio that allows a user to compute f x o ecrypted data

Extracting Witnesses from Proofs of Knowledge in the Random Oracle Model

Extracting Witnesses from Proofs of Knowledge in the Random Oracle Model Jens Groth Cryptomathic and BRICS, Aarhus University Abstract We prove that a 3-move interactive proof system with the special soundness

Injections, Surjections, and the Pigeonhole Principle

Ijectios, Surjectios, ad the Pigeohole Priciple 1 (10 poits Here we will come up with a sloppy boud o the umber of parethesisestigs (a (5 poits Describe a ijectio from the set of possible ways to est pairs

Efficient Multiplicative Homomorphic E-Voting

Efficient Multiplicative Homomorphic E-Voting Kun Peng and Feng Bao Institute for Infocomm Research, Singapore dr.kun.peng@gmail.com Abstract. Multiplicative homomorphic e-voting is proposed by Peng et

Threshold Undeniable RSA Signature Scheme

Threshold Undeniable RSA Signature Scheme Guilin Wang 1, Sihan Qing 1, Mingsheng Wang 1, and Zhanfei Zhou 2 1 Engineering Research Center for Information Security Technology; State Key Laboratory of Information

The structure of finite rings. The multiplicative residues. Modular exponentiation. and finite exponentiation

The structure of fiite rigs ad fiite expoetiatio The multiplicative residues We have see that the fiite rig Z p is a field, that is, every o-zero elemet of Z p has a multiplicative iverse It is a covetio

ORTHOGONAL MATRIX IN CRYPTOGRAPHY

Orthogoal Matrix i Cryptography ORTHOGONAL MATRIX IN CRYPTOGRAPHY Yeray Cachó Sataa Member of CriptoRed (U.P.M.) ABSTRACT I this work is proposed a method usig orthogoal matrix trasform properties to ecrypt

Towards a DL-based Additively Homomorphic Encryption Scheme

Towards a DL-based Additively Homomorphic Encryption Scheme Guilhem Castagnos 1 and Benoît Chevallier-Mames 2 1 DMI-XLIM, Université de Limoges, 123, Avenue Albert-Thomas 87060 Limoges Cedex, France guilhem.castagnos@unilim.fr

An extension of the RSA trapdoor in a KEM/DEM framework

A extesio of the RSA trapdoor i a KEM/DEM framework Bogda Groza Politehica Uiversity of Timisoara Faculty of Automatics ad Computers Bd. Vasile Parva r. 2, 300223 Timisoara, Romaia mail: bogda.groza@aut.upt.ro

Oblivious Transfer using Elliptic Curves

Oblivious Trasfer usig Elliptic Curves bhishek Parakh Louisiaa State Uiversity, ato Rouge, L May 4, 006 bstract: This paper proposes a algorithm for oblivious trasfer usig elliptic curves lso, we preset

Winter 2011 Josh Benaloh Brian LaMacchia

Winter 2011 Josh Benaloh Brian LaMacchia Fun with Public-Key Tonight we ll Introduce some basic tools of public-key crypto Combine the tools to create more powerful tools Lay the ground work for substantial

Last time, we talked about how Equation (1) can simulate Equation (2). We asserted that Equation (2) can also simulate Equation (1).

6896 Quatum Complexity Theory Sept 23, 2008 Lecturer: Scott Aaroso Lecture 6 Last Time: Quatum Error-Correctio Quatum Query Model Deutsch-Jozsa Algorithm (Computes x y i oe query) Today: Berstei-Vazirii

Homomorphic Encryption. Liam Morris

Homomorphic Encryption Liam Morris Topics What Is Homomorphic Encryption? Partially Homomorphic Cryptosystems Fully Homomorphic Cryptosystems Benefits of Homomorphism Drawbacks of Homomorphism What Is

Minimum-Knowledge Schemes for low-power, low-memory Devices

Miimum-Kowledge Schemes for low-power, low-memory Deices T. Newe & T. Coffey Data Commuicatios Security Group, Departmet of lectroic & Computer gieerig, Uiersity of imerick, relad. Fax: 353-61-338176 Abstract

Definition: For a positive integer n, if 0<a<n and gcd(a,n)=1, a is relatively prime to n. Ahmet Burak Can Hacettepe University

Number Theory, Public Key Cryptography, RSA Ahmet Burak Can Hacettepe University abc@hacettepe.edu.tr The Euler Phi Function For a positive integer n, if 0

Gentry s ideal-lattice based encryption scheme. Gentry s STOC 09 paper - Part III

Getry s ideal-lattice based ecryptio scheme Getry s STOC 09 paper - Part 1 From Micciacio's paper Why ideal lattices --- as opposed to just ideals or lattices? We described a ideal-based ecryptio scheme

CPE 776:DATA SECURITY & CRYPTOGRAPHY. Some Number Theory and Classical Crypto Systems

CPE 776:DATA SECURITY & CRYPTOGRAPHY Some Number Theory and Classical Crypto Systems Dr. Lo ai Tawalbeh Computer Engineering Department Jordan University of Science and Technology Jordan Some Number Theory

Improving Helios with Everlasting Privacy Towards the Public Denise Demirel, Jeroen van de Graaf, Roberto Araújo

Improving Helios with Everlasting Privacy Towards the Public Denise Demirel, Jeroen van de Graaf, Roberto Araúo 15.08.2012 Fachbereich 20 CDC Denise Demirel 1 Helios Introduced 2008 by Ben Adida Web application

Threshold Cryptography

Threshold Cryptography Cloud Security Mechanisms Björn Groneberg - Summer Term 2013 09.07.2013 Threshold Cryptography 1 ? 09.07.2013 Threshold Cryptography 2 Threshold Cryptography Sharing Secrets Treasure

Structure of a Typical Digital System Data Inputs

ecture RT Desig Methodology Trasitio from the & Iterface to a Correspodig Block Diagram Structure of a Typical Digital System Data Iputs Datapath (Executio Uit) Data Outputs Cotrol Sigals Status Sigals

International Journal of Advanced Research in Computer Science and Software Engineering

Volume 2, Issue 11, November 2012 ISSN: 2277 128X Iteratioal Joural of Advaced Research i Computer Sciece ad Software Egieerig Research Paper Available olie at: www.ijarcsse.com A Digital Sigature Algorim

Identity-Based Cryptography on Hidden-Order Groups

vailable olie at www.sciecedirect.com Procedia Egieerig 9 (0) 067 07 0 Iteratioal Worshop o Iformatio ad Electroics Egieerig (IWIEE) Idetity-Based Cryptography o Hidde-Order Groups Chalgu Li a a Key Laboratory

Advanced Cryptography 03/06/2007. Lecture 8

Advanced Cryptography 03/06/007 Lecture 8 Lecturer: Victor Shoup Scribe: Prashant Puniya Overview In this lecture, we will introduce the notion of Public-Key Encryption. We will define the basic notion

A Block Cipher Using Linear Congruences

Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

EulerPhi Notatios Traditioal ame Euler totiet fuctio Traditioal otatio Φ Mathematica StadardForm otatio EulerPhi Primary defiitio 3.06.02.000.0 Φ gcd,k, ; For oegative iteger, the Euler totiet fuctio Φ

Cryptographic Voting Systems (Ben Adida)

Cryptographic Voting Systems (Ben Adida) Click to edit Master subtitle style Jimin Park Carleton University COMP 4109 Seminar 15 February 2011 If you think cryptography is the solution to your problem.

Primality Test. Rong-Jaye Chen

Primality Test Rog-Jaye Che OUTLINE [1] Modular Arithmetic Algorithms [2] Quadratic Residues [3] Primality Testig p2. [1] Modular Arithmetic Algorithms 1. The itegers a divides b a b a{ 1, b} If b has

Wrap of Number Theory & Midterm Review. Recall: Fundamental Theorem of Arithmetic

Wrap of Number Theory & Midterm Review F Primes, GCD, ad LCM (Sectio 3.5 i text) F Midterm Review Sectios.-.7 Propositioal logic Predicate logic Rules of iferece ad proofs Sectios.-.3 Sets ad Set operatios

BRICS. A Generalisation, a Simplification and some Applications of Paillier s Probabilistic Public-Key System

BRICS Basic Research in Computer Science BRICS RS-00-45 Damgård & Jurik: Generalisation and Applications of Paillier s Probabilistic Public-Key System A Generalisation, a Simplification and some Applications

during transmission safeguard information Cryptography: used to CRYPTOGRAPHY BACKGROUND OF THE MATHEMATICAL

THE MATHEMATICAL BACKGROUND OF CRYPTOGRAPHY Cryptography: used to safeguard information during transmission (e.g., credit card number for internet shopping) as opposed to Coding Theory: used to transmit

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

Some ZK security proofs for Belenios

Some ZK security proofs for Belenios Pierrick Gaudry CNRS, INRIA, Université de Lorraine January 30, 2017 The purpose of this document is to justify the use of ZK proofs in Belenios. Most of them are exactly

Practical Verifiable Encryption and Decryption of Discrete Logarithms

Practical Verifiable Encryption and Decryption of Discrete Logarithms Jan Camenisch IBM Zurich Research Lab Victor Shoup New York University p.1/27 Verifiable encryption of discrete logs Three players:

Further More on Key Wrapping. 2011/2/17 SKEW2011 Lyngby Nagoya University Yasushi Osaki, Tetsu Iwata

Further More o Key Wrappig 011//17 SKEW011 Lygby Nagoya Uiversity Yasushi Osaki, Tetsu Iwata 1 What is key wrappig? Used to ecrypt specialized data, such as cryptographic keys A key wrappig that also esures

Question: Total Points: Score:

University of California, Irvine COMPSCI 134: Elements of Cryptography and Computer and Network Security Midterm Exam (Fall 2016) Duration: 90 minutes November 2, 2016, 7pm-8:30pm Name (First, Last): Please

Dr George Danezis University College London, UK

Dr George Danezis University College London, UK Identity as a proxy to check credentials Username decides access in Access Control Matrix Sometime it leaks too much information Real world examples Tickets

Chapter 8 Public-key Cryptography and Digital Signatures

Chapter 8 Public-key Cryptography and Digital Signatures v 1. Introduction to Public-key Cryptography 2. Example of Public-key Algorithm: Diffie- Hellman Key Exchange Scheme 3. RSA Encryption and Digital

The multiplicative structure of finite field and a construction of LRC

IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio

3.2.4 Integer and Number Theoretical Functions

418 3. Advaced Mathematics i Mathematica 3.2.4 Iteger ad Number Theoretical Fuctios Mod[k, ] k modulo (positive remaider from dividig k by ) Quotiet[m, ] the quotiet of m ad (iteger part of m/) GCD[ 1,

Homework 3 Solutions

5233/IOC5063 Theory of Cryptology, Fall 205 Instructor Prof. Wen-Guey Tzeng Homework 3 Solutions 7-Dec-205 Scribe Amir Rezapour. Consider an unfair coin with head probability 0.5. Assume that the coin

Identity-based Encryption from Codes with Rank Metric

Idetity-based Ecryptio from Codes with Rak Metric Philippe Gaborit 1, Adrie Hauteville 1,2, Duog Hieu Pha 1, ad Jea-Pierre Tillich 2 1 Uiversité de Limoges, XLIM-DMI, 123, Av. Albert Thomas, 87060 Limoges

Elliptic Curve Cryptography

Elliptic Curve Cryptography Elliptic Curves An elliptic curve is a cubic equation of the form: y + axy + by = x 3 + cx + dx + e where a, b, c, d and e are real numbers. A special addition operation is

Number Theory & Modern Cryptography

Number Theory & Modern Cryptography Week 12 Stallings: Ch 4, 8, 9, 10 CNT-4403: 2.April.2015 1 Introduction Increasing importance in cryptography Public Key Crypto and Signatures Concern operations on

Cryptographic Protocols. Steve Lai

Cryptographic Protocols Steve Lai This course: APPLICATIONS (security) Encryption Schemes Crypto Protocols Sign/MAC Schemes Pseudorandom Generators And Functions Zero-Knowledge Proof Systems Computational

Efficient RSA Key Generation and Threshold Paillier in the Two-Party Setting

An extended abstract of this paper was published in the proceedings of CT-RSA 2012. Efficient RSA Key Generation and Threshold Paillier in the Two-Party Setting Carmit Hazay Gert Læssøe Mikkelsen Tal Rabin

( ) = p and P( i = b) = q.

MATH 540 Radom Walks Part 1 A radom walk X is special stochastic process that measures the height (or value) of a particle that radomly moves upward or dowward certai fixed amouts o each uit icremet of

Computational Intelligence Winter Term 2018/19

Computatioal Itelligece Witer Term 28/9 Prof. Dr. Güter Rudolph Lehrstuhl für Algorithm Egieerig (LS ) Fakultät für Iformatik TU Dortmud Pla for Today Lecture Evolutioary Algorithms (EA) Optimizatio Basics

Design and Analysis of ALGORITHM (Topic 2)

DR. Gatot F. Hertoo, MSc. Desig ad Aalysis of ALGORITHM (Topic 2) Algorithms + Data Structures = Programs Lessos Leared 1 Our Machie Model: Assumptios Geeric Radom Access Machie (RAM) Executes operatios

Introduction to Cybersecurity Cryptography (Part 4)

Introduction to Cybersecurity Cryptography (Part 4) Review of Last Lecture Blockciphers Review of DES Attacks on Blockciphers Advanced Encryption Standard (AES) Modes of Operation MACs and Hashes Message

Econ 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara

Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio

How to Shuffle in Public

How to Shuffle in Public Ben Adida Harvard (work done at MIT) Douglas Wikström ETH Zürich TCC 27 February 24th, 27 How to Shuffle in Public Ben Adida Harvard (work done at MIT) Douglas Wikström ETH Zürich

Independence of the Miller-Rabin and Lucas Probable Prime Tests

Idepedece of the Miller-Rabi ad Lucas Probable Prime Tests Alec Leg Metor: avid Corwi March 30, 017 1 Abstract I the moder age, public-key cryptography has become a vital compoet for secure olie commuicatio.

Introduction to Cybersecurity Cryptography (Part 4)

Introduction to Cybersecurity Cryptography (Part 4) Review of Last Lecture Blockciphers Review of DES Attacks on Blockciphers Advanced Encryption Standard (AES) Modes of Operation MACs and Hashes Message

Estimation of a population proportion March 23,

1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes

Math778P Homework 2 Solution

Math778P Homework Solutio Choose ay 5 problems to solve. 1. Let S = X i where X 1,..., X are idepedet uiform { 1, 1} radom variables. Prove that E( S = 1 ( 1 1 Proof by Day Rorabaugh: Let S = X i where

Network Security Technology Spring, 2018 Tutorial 3, Week 4 (March 23) Due Date: March 30

Network Security Technology Spring, 2018 Tutorial 3, Week 4 (March 23) LIU Zhen Due Date: March 30 Questions: 1. RSA (20 Points) Assume that we use RSA with the prime numbers p = 17 and q = 23. (a) Calculate

Basic Sets. Functions. MTH299 - Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.

Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the

Public-Key Cryptosystems CHAPTER 4

Public-Key Cryptosystems CHAPTER 4 Introduction How to distribute the cryptographic keys? Naïve Solution Naïve Solution Give every user P i a separate random key K ij to communicate with every P j. Disadvantage:

Introduction to Cryptography. Lecture 8

Introduction to Cryptography Lecture 8 Benny Pinkas page 1 1 Groups we will use Multiplication modulo a prime number p (G, ) = ({1,2,,p-1}, ) E.g., Z 7* = ( {1,2,3,4,5,6}, ) Z p * Z N * Multiplication

Iterated Encryption and Wiener s attack on RSA

Iterated Encryption Euler s function Euler s function: φ(n) = {1 x n : gcd(x, n) = 1} Theorem (Euler) If n is a positive integer and m is a positive integer coprime to n then m φ(n) mod n = 1. Iterated

Dirichlet s Theorem on Arithmetic Progressions

Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty

RSA Algorithm. Factoring, EulerPhi, Breaking RSA. Çetin Kaya Koç Spring / 14

RSA Algorithm http://koclab.org Çetin Kaya Koç Spring 2018 1 / 14 Well-Known One-Way Functions Discrete Logarithm: Given p, g, and x, computing y in y = g x (mod p) is EASY Given p, g, y, computing x in

Polynomial reduction. Outline Lecture. Non deterministic polynomial time. Example 1 : discrete log. Lecture: Polynomial reduction.

Outlie Lecture Part 1 : Asymmetric cryptography, oe way fuctio, complexity Part 2 : arithmetic complexity ad lower bouds : expoetiatio Part 3 : Provable security ad polyomial time reductio : P, NP classes.

CryptoComputing with rationals

CryptoComputing with rationals Pierre-Alain Fouque 1,2, Jacques Stern 2, and Geert-Jan Wackers 3 1 D.C.S.S.I. Crypto Lab 51, bd Latour-Maubourg, F-75007 Paris, France 2 École Normale Supérieure, Département

University of Tokyo: Advanced Algorithms Summer Lecture 6 27 May. Let s keep in mind definitions from the previous lecture:

University of Tokyo: Advanced Algorithms Summer 2010 Lecture 6 27 May Lecturer: François Le Gall Scribe: Baljak Valentina As opposed to prime factorization, primality testing is determining whether a given

Basics of Probability Theory (for Theory of Computation courses)

Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.

A brief introduction to linear algebra

CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad

NBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?

NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily

DIFFERENTIAL CRYPTANALYSIS FOR A 3-ROUND SPN

IRNTIL RYPTNLYSIS OR -ROUN SPN M. Tolga Sakallı rca uluş daç Şahi atma üyüksaraçoğlu e-mail: tolga@trakya.edu.tr. e-mail: ercab@trakya.edu.tr e-mail: adacs@trakya.edu.tr e-mail: fbuyuksaracoglu@trakya.edu.tr

CIS 551 / TCOM 401 Computer and Network Security

CIS 551 / TCOM 401 Computer and Network Security Spring 2008 Lecture 15 3/20/08 CIS/TCOM 551 1 Announcements Project 3 available on the web. Get the handout in class today. Project 3 is due April 4th It

CPSC 467b: Cryptography and Computer Security

CPSC 467b: Cryptography and Computer Security Michael J. Fischer Lecture 11 February 21, 2013 CPSC 467b, Lecture 11 1/27 Discrete Logarithm Diffie-Hellman Key Exchange ElGamal Key Agreement Primitive Roots

ECEN 655: Advanced Channel Coding Spring Lecture 7 02/04/14. Belief propagation is exact on tree-structured factor graphs.

ECEN 655: Advaced Chael Codig Sprig 014 Prof. Hery Pfister Lecture 7 0/04/14 Scribe: Megke Lia 1 4-Cycles i Gallager s Esemble What we already kow: Belief propagatio is exact o tree-structured factor graphs.

Optimal Design of Accelerated Life Tests with Multiple Stresses

Optimal Desig of Accelerated Life Tests with Multiple Stresses Y. Zhu ad E. A. Elsayed Departmet of Idustrial ad Systems Egieerig Rutgers Uiversity 2009 Quality & Productivity Research Coferece IBM T.

Efficient RSA Key Generation and Threshold Paillier in the Two-Party Setting

Efficient RSA Key Generation and Threshold Paillier in the Two-Party Setting Carmit Hazay Gert Læssøe Mikkelsen Tal Rabin Tomas Toft Abstract The problem of generating an RSA composite in a distributed

End-of-Year Contest. ERHS Math Club. May 5, 2009

Ed-of-Year Cotest ERHS Math Club May 5, 009 Problem 1: There are 9 cois. Oe is fake ad weighs a little less tha the others. Fid the fake coi by weighigs. Solutio: Separate the 9 cois ito 3 groups (A, B,

Information Theory and Coding

Sol. Iformatio Theory ad Codig. The capacity of a bad-limited additive white Gaussia (AWGN) chael is give by C = Wlog 2 ( + σ 2 W ) bits per secod(bps), where W is the chael badwidth, is the average power

Merkle-Hellman Knapsack Cryptosystem in Undergraduate Computer Science Curriculum

Merkle-Hellma Kapsack Cryptosystem i Udergraduate Computer Sciece Curriculum Y. Kortsarts, Y. Kemper 2 Computer Sciece Departmet, Wideer Uiversity, Chester, PA, USA 2 Computer Sciece Departmet, Holo Istitute

Randomized Algorithms I, Spring 2018, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 25, 2018)

Radomized Algorithms I, Sprig 08, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 5, 08). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black

1 Hash tables. 1.1 Implementation

Lecture 8 Hash Tables, Uiversal Hash Fuctios, Balls ad Bis Scribes: Luke Johsto, Moses Charikar, G. Valiat Date: Oct 18, 2017 Adapted From Virgiia Williams lecture otes 1 Hash tables A hash table is a

Chapter 6: Mining Frequent Patterns, Association and Correlations

Chapter 6: Miig Frequet Patters, Associatio ad Correlatios Basic cocepts Frequet itemset miig methods Costrait-based frequet patter miig (ch7) Associatio rules 1 What Is Frequet Patter Aalysis? Frequet

Report on Private Information Retrieval over Unsynchronized Databases

Report o Private Iformatio Retrieval over Usychroized Databases Lembit Valgma Supervised by Vitaly Skachek May 25, 217 1 Problem Statemet There are may challeges cocerig olie privacy. Private iformatio

Resampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.

Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator

MTAT Cryptology II. Zero-knowledge Proofs. Sven Laur University of Tartu

MTAT.07.003 Cryptology II Zero-knowledge Proofs Sven Laur University of Tartu Formal Syntax Zero-knowledge proofs pk (pk, sk) Gen α 1 β 1 β i V pk (α 1,...,α i 1 ) α i P sk (β 1,...,β i 1 ) (pk,sk)? R

Algorithm Analysis. Algorithms that are equally correct can vary in their utilization of computational resources

Algorithm Aalysis Algorithms that are equally correct ca vary i their utilizatio of computatioal resources time ad memory a slow program it is likely ot to be used a program that demads too much memory

YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE

YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE CPSC 467a: Crytograhy ad Comuter Security Notes 16 (rev. 1 Professor M. J. Fischer November 3, 2008 68 Legedre Symbol Lecture Notes 16 ( Let be a odd rime,