Generic attacks and index calculus. D. J. Bernstein University of Illinois at Chicago


 Arthur Martin
 2 years ago
 Views:
Transcription
1 Generic attacks and index calculus D. J. Bernstein University of Illinois at Chicago
2 The discretelogarithm problem Define Ô = Easy to prove: Ô is prime. Can we find an integer Ò ¾ Ô 1 such that 5Ò mod Ô = ? Easy to prove: Ò 5Ò mod Ô permutes Ô 1. So there exists an Ò such that 5Ò mod Ô = Could find Ò by brute force. Is there a faster way?
3 Typical cryptanalytic application: Ô Imagine standard = in the DiffieHellman protocol. User chooses secret key Ò, publishes 5Ò mod Ô = Can attacker quickly solve the discretelogarithm problem? 5Ò Given public key mod Ô, quickly find secret key Ò? (Warning: This is one way to attack the protocol. Maybe there are better ways.)
4 Relations to ECC: 1. Some DL techniques also apply to ellipticcurve DL problems. Use in evaluating security of an elliptic curve. 2. Some techniques don t apply. Use in evaluating advantages of elliptic curves compared to multiplication. 3. Tricky: Some techniques have extra applications to some curves. See Tanja Lange s talk on Weil descent etc.
5 Understanding brute force Can compute successively 5 1 mod Ô = 5, 5 2 mod Ô = 25, 5 3 mod Ô = 125,, 5 8 mod Ô = , 5 9 mod Ô = ,, mod Ô = 1. At some point we ll find Ò with 5Ò mod Ô = Maximum cost of computation: Ô 1 mults by 5 mod Ô; Ô 1 nanoseconds on a CPU that does 1 mult/nanosecond.
6 This is negligible work for Ô But users can standardize a larger Ô, making the attack slower. Attack cost scales linearly: 2 50 mults for Ô 2 50, mults for Ô 2 100, etc. (Not exactly linearly: cost of mults grows with Ô. But this is a minor effect.)
7 Computation has a good chance of finishing earlier. Chance scales linearly: 1 2 chance of 1 2 cost; 1 10 chance of 1 10 cost; etc. So users should choose large Ò. That s pointless. We can apply random selfreduction : choose random Ö, say ; 5Ö compute mod Ô = ; 5Ö Ò compute 5 mod Ô as (5Ò ( mod Ô)) mod Ô; compute discrete log; subtract Ö mod Ô 1; obtain Ò.
8 Computation can be parallelized. One lowcost chip can run many parallel searches. Example, 2 6 e: one chip, 2 10 cores on the chip, each 2 30 mults/second? Maybe; see SHARCS workshops for detailed cost analyses. Attacker can run many parallel chips. Example, 2 30 e: 2 24 chips, so 2 34 cores, so 2 64 mults/second, so 2 89 mults/year.
9 Multiple targets and giant steps Computation can be applied to many targets at once. Given 100 DL targets 5Ò 1 mod Ô, 5Ò 2 mod Ô,, 5Ò 100 mod Ô: Can find all of Ò 1 Ò 2 Ò 100 with Ô 1 mults mod Ô. Simplest approach: First build a sorted table containing 5Ò 1 mod Ô,, 5Ò 100 mod Ô. Then check table for 5 1 mod Ô, 5 2 mod Ô, etc.
10 Interesting consequence #1: Solving all 100 DL problems isn t much harder than solving one DL problem. Interesting consequence #2: Solving at least one out of 100 DL problems is much easier than solving one DL problem. When did this computation find its first Ò? Typically (Ô 1) 100 mults.
11 Can use random selfreduction to turn a single target into multiple targets. Given 5Ò mod Ô: Choose random Ö 1 Ö 2 Ö 100. Compute 5Ö 1 5Ò mod Ô, 5Ö 2 5Ò mod Ô, etc. Solve these 100 DL problems. Typically (Ô 1) 100 mults to find at least one Ö + Ò mod Ô 1, immediately revealing Ò.
12 Also spent some mults to compute each 5Ö mod Ô: lgô mults for each. Faster: Choose Ö = Ö 1 with Ö 1 (Ô 1) 100. Compute 5Ö 1 mod Ô; 5Ö 1 5Ò mod Ô; 5 2Ö 15Ò mod Ô; 5 3Ö 15Ò mod Ô; etc. Just 1 mult for each new lgô + (Ô 1) 100 mults to find Ò given 5Ò mod Ô.
13 Faster: Increase 100 to Ô Ô. Only 2 Ô Ô mults to solve one DL problem! Shanks babystepgiantstep discretelogarithm algorithm. Example: Ô = , 5Ò mod Ô = Compute mod Ô = Then compute 1000 targets: Ò mod Ô = , Ò mod Ô = , Ò mod Ô = ,, Ò mod Ô =
14 Build a sorted table of targets: 2573 = Ò mod Ô, 3371 = Ò mod Ô, 3593 = Ò mod Ô, 4960 = Ò mod Ô, 5218 = Ò mod Ô,, = Ò mod Ô. Look up 5 1 mod Ô, 5 2 mod Ô, 5 3 mod Ô, etc. in this table mod Ô = ; find = Ò mod Ô in the table of targets; so 755 = Ò mod Ô 1; deduce Ò =
15 Eliminating storage Improved method: Define Ü 0 = 1; Ü +1 = 5Ü mod Ô if Ü ¾ 3Z; Ü +1 = Ü 2 mod Ô if Ü ¾ 2 + 3Z; Ü +1 = 5ÒÜ mod Ô otherwise. Then Ü = 5 Ò+ mod Ô where ( 0 0 ) = (0 0) and ( +1 +1) = ( + 1), or ( +1 +1) = (2 2 ), or ( +1 +1) = ( + 1 ). Search for a collision in Ü : Ü 1 = Ü 2? Ü 2 = Ü 4? Ü 3 = Ü 6? Ü 4 = Ü 8? Ü 5 = Ü 10? etc. Deduce linear equation for Ò.
16 The Ü s enter a cycle, typically within Ô Ô steps. Example: , Modulo : Ü 1 = 5Ò = Ü 2 = 5 2Ò = = Ü 3 = 5 2Ò+1 = = Ü 4 = 5 2Ò+2 = = Ü 5 = 5 2Ò+3 = = Ü 6 = 5 2Ò+4 = = Ü 7 = 5 4Ò+8 = = Ü 8 = 5 4Ò+9 = = etc.
17 Ü 1785 = Ò = Ü 3570 = Ò = (Cycle length is 357.) Conclude that Ò Ò (mod Ô 1), so Ò (mod (Ô 1) 6). Only 6 possible Ò s. Try each of them. Find that 5Ò mod Ô = for Ò = (Ô 1) 6, i.e., for Ò =
18 This is Pollard s rho method. Optimized: Ô Ô mults. Another method, similar speed: Pollard s kangaroo method. Can parallelize both methods. van Oorschot/Wiener parallel DL using distinguished points. Bottom line: With mults, distributed across many cores, have chance 2 Ô of finding Ò from 5Ò mod Ô. With 2 90 mults (a few years?), have chance Ô. Negligible if, e.g., Ô
19 Factors of the group order Assume 5 has order. Given Ü, a power of 5: 5 has order, and Ü is a power of 5. Compute = log 5 Ü. 5 has order, and Ü 5 is a power of 5. Compute Ñ = log 5 (Ü 5 ). Then Ü = 5 +Ñ.
20 This PohligHellman method converts an order DL into an order DL, an order DL, and a few exponentiations. e.g. Ô = , Ü = : Ô 1 = 6 where = Compute log 5 6(Ü 6 ) = Compute Ü = Compute log = 3. Then Ü = = Use rho: Ô + Ô mults. Better if factors further: apply PohligHellman recursively.
21 All of the techniques so far apply to elliptic curves. An elliptic curve over FÕ has Õ + 1 points so can compute ECDL using Ô Õ ellipticcurve adds. Need quite large Õ. If largest prime divisor of number of points is much smaller than Õ then PohligHellman method computes ECDL more quickly. Need larger Õ; or change choice of curve.
22 Index calculus Have generated many group elements 5 Ò+ mod Ô. Deduced equations for Ò from random collisions. Index calculus obtains discretelogarithm equations in a different way. Example for Ô = : Can completely factor 3 (Ô 3) as in Q so (mod Ô) so log 5 ( 1) + log log log 5 5 (mod Ô 1).
23 Can completely factor 62 (Ô + 62) as so log log 5 31 log log log log log 5 29 (mod Ô 1). Try to completely factor 1 (Ô + 1), 2 (Ô + 2), etc. Find factorization of (Ô + ) as product of powers of for each of the following s: 5100, 4675, 3128, 403, 368, 147, 3, 62, 957, 2912, 3857, 6877.
24 Each complete factorization produces a log equation. Now have 12 linear equations for log 5 2 log 5 3 log Free equations: log 5 5 = 1, (Ô log 5 ( 1) = 1) 2. By linear algebra compute log 5 2 log 5 3 log (If this hadn t been enough, could have searched more s.) By similar technique obtain discrete log of any target.
25 For Ô ½, index calculus scales surprisingly well: cost Ô where 0. Compare to rho: Ô 1 2. Specifically: searching ¾ 1 2 Ý 2, with lgý ¾ Ç( Ô lgôlg lgô), finds Ý complete factorizations into primes Ý, and computes discrete logs. (Assuming standard conjectures. Have extensive evidence.)
26 Latest indexcalculus variants use the numberfield sieve and the functionfield sieve. To compute discrete logs in FÕ: lg cost ¾ Ç((lgÕ) 1 3 (lg lgõ) 2 3 ). For security: Õ to stop rho; Õ to stop NFS. We don t know any indexcalculus methods for ECDL! except for some curves.
Arithmetic algorithms for cryptology 5 October 2015, Paris. Sieves. Razvan Barbulescu CNRS and IMJPRG. R. Barbulescu Sieves 0 / 28
Arithmetic algorithms for cryptology 5 October 2015, Paris Sieves Razvan Barbulescu CNRS and IMJPRG R. Barbulescu Sieves 0 / 28 Starting point Notations q prime g a generator of (F q ) X a (secret) integer
More informationDigital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document?
Cryptography Digital Signatures Professor: Marius Zimand Digital signatures are meant to realize authentication of the sender nonrepudiation (Note that authentication of sender is also achieved by MACs.)
More informationMATH 168: FINAL PROJECT Troels Eriksen. 1 Introduction
MATH 168: FINAL PROJECT Troels Eriksen 1 Introduction In the later years cryptosystems using elliptic curves have shown up and are claimed to be just as secure as a system like RSA with much smaller key
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 informationAn Overview of Integer Factoring Algorithms. The Problem
An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm
More informationElliptic Curve Hash (and Sign)
Elliptic Curve Hash (and Sign) (and the 1up problem for ECDSA) Daniel R. L. Brown Certicom Research ECC 2008, Utrecht, Sep 2224 2008 Dan Brown (Certicom) Elliptic Curve Hash (and Sign) ECC 2008 1 / 43
More informationStudy of algorithms for factoring integers and computing discrete logarithms
Study of algorithms for factoring integers and computing discrete logarithms First IndoFrench Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department
More informationElements of Applied Cryptography Public key encryption
Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let
More informationCryptographic Algorithms and Key Size Issues. Çetin Kaya Koç Oregon State University, Professor http://islab.oregonstate.edu/koc koc@ece.orst.
Cryptographic Algorithms and Key Size Issues Çetin Kaya Koç Oregon State University, Professor http://islab.oregonstate.edu/koc koc@ece.orst.edu Overview Cryptanalysis Challenge Encryption: DES AES Message
More informationPrimality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More informationRSA Question 2. Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p1)(q1) = φ(n). Is this true?
RSA Question 2 Bob thinks that p and q are primes but p isn t. Then, Bob thinks Φ Bob :=(p1)(q1) = φ(n). Is this true? Bob chooses a random e (1 < e < Φ Bob ) such that gcd(e,φ Bob )=1. Then, d = e 1
More informationPublicKey Cryptanalysis 1: Introduction and Factoring
PublicKey Cryptanalysis 1: Introduction and Factoring Nadia Heninger University of Pennsylvania July 21, 2013 Adventures in Cryptanalysis Part 1: Introduction and Factoring. What is publickey crypto
More informationALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION
ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More informationFACTORING. n = 2 25 + 1. fall in the arithmetic sequence
FACTORING The claim that factorization is harder than primality testing (or primality certification) is not currently substantiated rigorously. As some sort of backward evidence that factoring is hard,
More informationFinal Exam. IT 4823 Information Security Administration. Rescheduling Final Exams. Kerberos. Idea. Ticket
IT 4823 Information Security Administration Public Key Encryption Revisited April 5 Notice: This session is being recorded. Lecture slides prepared by Dr Lawrie Brown for Computer Security: Principles
More informationFactoring Algorithms
Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors
More informationFactoring & Primality
Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount
More informationELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS. Carl Pomerance
ELEMENTARY THOUGHTS ON DISCRETE LOGARITHMS Carl Pomerance Given a cyclic group G with generator g, and given an element t in G, the discrete logarithm problem is that of computing an integer l with g l
More informationFactorization Methods: Very Quick Overview
Factorization Methods: Very Quick Overview Yuval Filmus October 17, 2012 1 Introduction In this lecture we introduce modern factorization methods. We will assume several facts from analytic number theory.
More informationFactoring and Discrete Log
Factoring and Discrete Log Nadia Heninger University of Pennsylvania June 1, 2015 Textbook RSA [Rivest Shamir Adleman 1977] Public Key N = pq modulus e encryption exponent Private Key p, q primes d decryption
More informationWhat output size resists collisions in a xor of independent expansions?
What output size resists collisions in a xor of independent expansions? Daniel J. Bernstein Department of Mathematics, Statistics, and Computer Science (MC 249) University of Illinois at Chicago, Chicago,
More informationPublic Key Cryptography. Performance Comparison and Benchmarking
Public Key Cryptography Performance Comparison and Benchmarking Tanja Lange Department of Mathematics Technical University of Denmark tanja@hyperelliptic.org 28.08.2006 Tanja Lange Benchmarking p. 1 What
More informationCryptography and Network Security Chapter 10
Cryptography and Network Security Chapter 10 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 10 Other Public Key Cryptosystems Amongst the tribes of Central
More informationPrinciples of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms
Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport
More informationLecture 3: OneWay Encryption, RSA Example
ICS 180: Introduction to Cryptography April 13, 2004 Lecturer: Stanislaw Jarecki Lecture 3: OneWay Encryption, RSA Example 1 LECTURE SUMMARY We look at a different security property one might require
More informationIs n a Prime Number? Manindra Agrawal. March 27, 2006, Delft. IIT Kanpur
Is n a Prime Number? Manindra Agrawal IIT Kanpur March 27, 2006, Delft Manindra Agrawal (IIT Kanpur) Is n a Prime Number? March 27, 2006, Delft 1 / 47 Overview 1 The Problem 2 Two Simple, and Slow, Methods
More informationECE 842 Report Implementation of Elliptic Curve Cryptography
ECE 842 Report Implementation of Elliptic Curve Cryptography WeiYang Lin December 15, 2004 Abstract The aim of this report is to illustrate the issues in implementing a practical elliptic curve cryptographic
More informationCryptography: RSA and the discrete logarithm problem
Cryptography: and the discrete logarithm problem R. Hayden Advanced Maths Lectures Department of Computing Imperial College London February 2010 Public key cryptography Assymmetric cryptography two keys:
More informationSECURITY IMPROVMENTS TO THE DIFFIEHELLMAN SCHEMES
www.arpapress.com/volumes/vol8issue1/ijrras_8_1_10.pdf SECURITY IMPROVMENTS TO THE DIFFIEHELLMAN SCHEMES Malek Jakob Kakish Amman Arab University, Department of Computer Information Systems, P.O.Box 2234,
More informationCIS 5371 Cryptography. 8. Encryption 
CIS 5371 Cryptography p y 8. Encryption  Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: Allornothing secrecy.
More informationMathematics of Cryptography Part I
CHAPTER 2 Mathematics of Cryptography Part I (Solution to OddNumbered Problems) Review Questions 1. The set of integers is Z. It contains all integral numbers from negative infinity to positive infinity.
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 informationMathematics of Cryptography
CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter
More informationFACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY
FACTORING LARGE NUMBERS, A GREAT WAY TO SPEND A BIRTHDAY LINDSEY R. BOSKO I would like to acknowledge the assistance of Dr. Michael Singer. His guidance and feedback were instrumental in completing this
More informationFaster Cryptographic Key Exchange on Hyperelliptic Curves
Faster Cryptographic Key Exchange on Hyperelliptic Curves No Author Given No Institute Given Abstract. We present a key exchange procedure based on divisor arithmetic for the real model of a hyperelliptic
More informationShort Programs for functions on Curves
Short Programs for functions on Curves Victor S. Miller Exploratory Computer Science IBM, Thomas J. Watson Research Center Yorktown Heights, NY 10598 May 6, 1986 Abstract The problem of deducing a function
More informationFactoring. Factoring 1
Factoring Factoring 1 Factoring Security of RSA algorithm depends on (presumed) difficulty of factoring o Given N = pq, find p or q and RSA is broken o Rabin cipher also based on factoring Factoring like
More informationFactHacks: RSA factorization in the real world
FactHacks: RSA factorization in the real world Daniel J. Bernstein University of Illinois at Chicago Technische Universiteit Eindhoven Nadia Heninger Microsoft Research New England Tanja Lange Technische
More informationCryptography and Network Security
Cryptography and Network Security Fifth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA PrivateKey Cryptography traditional private/secret/single key cryptography uses one key shared
More informationOutline. Cryptography. Bret Benesh. Math 331
Outline 1 College of St. Benedict/St. John s University Department of Mathematics Math 331 2 3 The internet is a lawless place, and people have access to all sorts of information. What is keeping people
More informationCurve25519: new DiffieHellman speed records
Curve25519: new DiffieHellman speed records Daniel J. Bernstein djb@cr.yp.to Abstract. This paper explains the design and implementation of a highsecurity ellipticcurvediffiehellman function achieving
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationHighspeed cryptography and DNSCurve. D. J. Bernstein University of Illinois at Chicago
Highspeed cryptography and DNSCurve D. J. Bernstein University of Illinois at Chicago Stealing Internet mail: easy! Given a mail message: Your mail software sends a DNS request, receives a server address,
More informationA New Generic Digital Signature Algorithm
Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationChaCha, a variant of Salsa20
ChaCha, a variant of Salsa20 Daniel J. Bernstein Department of Mathematics, Statistics, and Computer Science (M/C 249) The University of Illinois at Chicago Chicago, IL 60607 7045 snuffle6@box.cr.yp.to
More informationRuntime and Implementation of Factoring Algorithms: A Comparison
Runtime and Implementation of Factoring Algorithms: A Comparison Justin Moore CSC290 Cryptology December 20, 2003 Abstract Factoring composite numbers is not an easy task. It is classified as a hard algorithm,
More informationOverview of PublicKey Cryptography
CS 361S Overview of PublicKey Cryptography Vitaly Shmatikov slide 1 Reading Assignment Kaufman 6.16 slide 2 PublicKey Cryptography public key public key? private key Alice Bob Given: Everybody knows
More informationLecture Note 5 PUBLICKEY CRYPTOGRAPHY. Sourav Mukhopadhyay
Lecture Note 5 PUBLICKEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security  MA61027 Modern/Publickey cryptography started in 1976 with the publication of the following paper. W. Diffie
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationHomework 5 Solutions
Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which
More informationThe RSA Algorithm. Evgeny Milanov. 3 June 2009
The RSA Algorithm Evgeny Milanov 3 June 2009 In 1978, Ron Rivest, Adi Shamir, and Leonard Adleman introduced a cryptographic algorithm, which was essentially to replace the less secure National Bureau
More informationFaster deterministic integer factorisation
David Harvey (joint work with Edgar Costa, NYU) University of New South Wales 25th October 2011 The obvious mathematical breakthrough would be the development of an easy way to factor large prime numbers
More informationAn Approach to Shorten Digital Signature Length
Computer Science Journal of Moldova, vol.14, no.342, 2006 An Approach to Shorten Digital Signature Length Nikolay A. Moldovyan Abstract A new method is proposed to design short signature schemes based
More informationCryptography and Network Security Chapter 8
Cryptography and Network Security Chapter 8 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 8 Introduction to Number Theory The Devil said to Daniel Webster:
More informationCryptanalysis with a costoptimized FPGA cluster
Cryptanalysis with a costoptimized FPGA cluster Jan Pelzl, Horst Görtz Institute for ITSecurity, Germany UCLA IPAM Workshop IV Special Purpose Hardware for Cryptography: Attacks and Applications December
More informationTechniques of Asymmetric File Encryption. Alvin Li Thomas Jefferson High School For Science and Technology Computer Systems Lab
Techniques of Asymmetric File Encryption Alvin Li Thomas Jefferson High School For Science and Technology Computer Systems Lab Abstract As more and more people are linking to the Internet, threats to 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 informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More informationCUNSHENG DING HKUST, Hong Kong. Computer Security. Computer Security. Cunsheng DING, HKUST COMP4631
Cunsheng DING, HKUST Lecture 08: Key Management for Onekey Ciphers Topics of this Lecture 1. The generation and distribution of secret keys. 2. A key distribution protocol with a key distribution center.
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 informationA SOFTWARE COMPARISON OF RSA AND ECC
International Journal Of Computer Science And Applications Vol. 2, No. 1, April / May 29 ISSN: 97413 A SOFTWARE COMPARISON OF RSA AND ECC Vivek B. Kute Lecturer. CSE Department, SVPCET, Nagpur 9975549138
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More information3. Computational Complexity.
3. Computational Complexity. (A) Introduction. As we will see, most cryptographic systems derive their supposed security from the presumed inability of any adversary to crack certain (number theoretic)
More informationELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM
ELLIPTIC CURVES AND LENSTRA S FACTORIZATION ALGORITHM DANIEL PARKER Abstract. This paper provides a foundation for understanding Lenstra s Elliptic Curve Algorithm for factoring large numbers. We give
More informationIMPLEMENTATION OF ELLIPTIC CURVE CRYPTOGRAPHY ON TEXT AND IMAGE
IMPLEMENTATION OF ELLIPTIC CURVE CRYPTOGRAPHY ON TEXT AND IMAGE Mrs. Megha Kolhekar Assistant Professor, Department of Electronics and Telecommunication Engineering Fr. C. Rodrigues Institute of Technology,
More informationTHE ADVANTAGES OF ELLIPTIC CURVE CRYPTOGRAPHY FOR WIRELESS SECURITY KRISTIN LAUTER, MICROSOFT CORPORATION
T OPICS IN WIRELESS SECURITY THE ADVANTAGES OF ELLIPTIC CURVE CRYPTOGRAPHY FOR WIRELESS SECURITY KRISTIN LAUTER, MICROSOFT CORPORATION Q 2 = R 1 Q 2 R 1 R 1 As the wireless industry explodes, it faces
More informationThe Mathematics of the RSA PublicKey Cryptosystem
The Mathematics of the RSA PublicKey Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through
More informationDigital Signature. Raj Jain. Washington University in St. Louis
Digital Signature Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse57111/
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationPublic Key Cryptography Overview
Ch.20 PublicKey Cryptography and Message Authentication I will talk about it later in this class Final: Wen (5/13) 16301830 HOLM 248» give you a sample exam» Mostly similar to homeworks» no electronic
More informationElliptic Curve Cryptography
Elliptic Curve Cryptography Elaine Brow, December 2010 Math 189A: Algebraic Geometry 1. Introduction to Public Key Cryptography To understand the motivation for elliptic curve cryptography, we must first
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationDetermining the Optimal Combination of Trial Division and Fermat s Factorization Method
Determining the Optimal Combination of Trial Division and Fermat s Factorization Method Joseph C. Woodson Home School P. O. Box 55005 Tulsa, OK 74155 Abstract The process of finding the prime factorization
More informationSpeeding up XTR. P.O.Box 513, 5600 MB Eindhoven, The Netherlands stam@win.tue.nl
Speeding up XTR Martijn Stam 1, and Arjen K. Lenstra 2 1 Technische Universiteit Eindhoven P.O.Box 513, 5600 MB Eindhoven, The Netherlands stam@win.tue.nl 2 Citibank, N.A. and Technische Universiteit Eindhoven
More informationShor s algorithm and secret sharing
Shor s algorithm and secret sharing Libor Nentvich: QC 23 April 2007: Shor s algorithm and secret sharing 1/41 Goals: 1 To explain why the factoring is important. 2 To describe the oldest and most successful
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
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 informationSHARK A Realizable Special Hardware Sieving Device for Factoring 1024bit Integers
SHARK A Realizable Special Hardware Sieving Device for Factoring 1024bit Integers Jens Franke 1, Thorsten Kleinjung 1, Christof Paar 2, Jan Pelzl 2, Christine Priplata 3, Colin Stahlke 3 1 University
More informationAuthentication requirement Authentication function MAC Hash function Security of
UNIT 3 AUTHENTICATION Authentication requirement Authentication function MAC Hash function Security of hash function and MAC SHA HMAC CMAC Digital signature and authentication protocols DSS Slides Courtesy
More informationCSCE 465 Computer & Network Security
CSCE 465 Computer & Network Security Instructor: Dr. Guofei Gu http://courses.cse.tamu.edu/guofei/csce465/ Public Key Cryptogrophy 1 Roadmap Introduction RSA DiffieHellman Key Exchange Public key and
More informationMathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR
Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGrawHill The McGrawHill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,
More informationParallel Collision Search with Cryptanalytic Applications
Parallel Collision Search with Cryptanalytic Applications Paul C. van Oorschot and Michael J. Wiener Nortel, P.O. Box 3511 Station C, Ottawa, Ontario, K1Y 4H7, Canada 1996 September 23 Abstract. A simple
More informationThe Future of Digital Signatures. Johannes Buchmann
The Future of Digital Signatures Johannes Buchmann Digital Signatures Digital signatures document sign signature verify valid / invalid secret public No ITSecurity without digital signatures Software
More informationA simple and fast algorithm for computing exponentials of power series
A simple and fast algorithm for computing exponentials of power series Alin Bostan Algorithms Project, INRIA ParisRocquencourt 7815 Le Chesnay Cedex France and Éric Schost ORCCA and Computer Science Department,
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 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 informationPrimality Testing and Factorization Methods
Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,
More informationThe Factoring Dead Preparing for the Cryptopocalypse
The Factoring Dead Preparing for the Cryptopocalypse Thomas Ptacek, Matasano Tom Ritter, isec Partners Javed Samuel, isec Partners Alex Stamos, Artemis Internet Agenda Introduction The Math New Advances
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 informationFactoring pq 2 with Quadratic Forms: Nice Cryptanalyses
Factoring pq 2 with Quadratic Forms: Nice Cryptanalyses Phong Nguyễn http://www.di.ens.fr/~pnguyen & ASIACRYPT 2009 Joint work with G. Castagnos, A. Joux and F. Laguillaumie Summary Factoring A New Factoring
More informationCapture Resilient ElGamal Signature Protocols
Capture Resilient ElGamal Signature Protocols Hüseyin Acan 1, Kamer Kaya 2,, and Ali Aydın Selçuk 2 1 Bilkent University, Department of Mathematics acan@fen.bilkent.edu.tr 2 Bilkent University, Department
More informationSignature Schemes. CSG 252 Fall 2006. Riccardo Pucella
Signature Schemes CSG 252 Fall 2006 Riccardo Pucella Signatures Signatures in real life have a number of properties They specify the person responsible for a document E.g. that it has been produced by
More informationImplementation of Elliptic Curve Digital Signature Algorithm
Implementation of Elliptic Curve Digital Signature Algorithm Aqeel Khalique Kuldip Singh Sandeep Sood Department of Electronics & Computer Engineering, Indian Institute of Technology Roorkee Roorkee, India
More informationElementary factoring algorithms
Math 5330 Spring 013 Elementary factoring algorithms The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat s Little Theorem and some related ideas, one can
More informationRecommendation for PairWise Key Establishment Schemes Using Discrete Logarithm Cryptography (Revised)
NIST Special Publication 80056A Recommendation for PairWise Key Establishment Schemes Using Discrete Logarithm Cryptography (Revised) Elaine Barker, Don Johnson, and Miles Smid C O M P U T E R S E C
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationD. J. Bernstein University of Illinois at Chicago. See online version of paper, particularly for bibliography: http://cr.yp.to /papers.
The tangent FFT D. J. Bernstein University of Illinois at Chicago See online version of paper, particularly for bibliography: http://cr.yp.to /papers.html#tangentfft Algebraic algorithms f 0 f 1 g 0 g
More informationNotes on Network Security Prof. Hemant K. Soni
Chapter 9 Public Key Cryptography and RSA PrivateKey Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications
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 information