Efficient Montgomery Modular Multiplication Algorithm Using Complement and Partition Techniques Speaker: Te-Jen Chang.

Slides:

Advertisements

Similar presentations

On Karatsuba Multiplication Algorithm

Advertisements

Asymmetric-Key Cryptography

CSE331: Introduction to Networks and Security Lecture 19 Fall 2002.

Public Key Encryption Algorithm

Notation Intro. Number Theory Online Cryptography Course Dan Boneh

Elliptic curve arithmetic and applications to cryptography By Uros Abaz Supervised by Dr. Shaun Cooper and Dr. Andre Barczak.

Abdullah Sheneamer CS591-F2010 Project of semester Presentation University of Colorado, Colorado Springs Dr. Edward RSA Problem and Inside PK Cryptography.

OOP/Java1 Public Key Crytography From: Introduction to Algorithms Cormen, Leiserson and Rivest.

RSA ( Rivest, Shamir, Adleman) Public Key Cryptosystem

Public Key Crytography1 From: Introduction to Algorithms Cormen, Leiserson and Rivest.

WS Algorithmentheorie 03 – Randomized Algorithms (Public Key Cryptosystems) Prof. Dr. Th. Ottmann.

Cryptography1 CPSC 3730 Cryptography Chapter 9 Public Key Cryptography and RSA.

Theory I Algorithm Design and Analysis (9 – Randomized algorithms) Prof. Dr. Th. Ottmann.

Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications.

WS Algorithmentheorie 03 – Randomized Algorithms (Public Key Cryptosystems) Prof. Dr. Th. Ottmann.

Fall 2010/Lecture 311 CS 426 (Fall 2010) Public Key Encryption and Digital Signatures.

Dr.Saleem Al_Zoubi1 Cryptography and Network Security Third Edition by William Stallings Public Key Cryptography and RSA.

Public Key Algorithms 4/17/2017 M. Chatterjee.

CSE 246: Computer Arithmetic Algorithms and Hardware Design Numbers: RNS, DBNS, Montgomory Prof Chung-Kuan Cheng Lecture 3.

ASYMMETRIC CIPHERS.

 Introduction  Requirements for RSA  Ingredients for RSA  RSA Algorithm  RSA Example  Problems on RSA.

Andreas Steffen, , 4-PublicKey.pptx 1 Internet Security 1 (IntSi1) Prof. Dr. Andreas Steffen Institute for Internet Technologies and Applications.

Montgomery multiplication Algorithm Mohammad Farmani Under supervision of : Dr. S. Bayat-sarmadi 2 nd. Semister, Sharif University of Technology.

CS5204 – Fall Cryptographic Security Presenter: Hamid Al-Hamadi October 13, 2009.

Chi-Cheng Lin, Winona State University CS 313 Introduction to Computer Networking & Telecommunication Network Security (A Very Brief Introduction)

Network and Communications Network Security Department of Computer Science Virginia Commonwealth University.

RSA Ramki Thurimella.

Cryptography: RSA & DES Marcia Noel Ken Roe Jaime Buccheri.

10/1/2015 9:38:06 AM1AIIS. OUTLINE Introduction Goals In Cryptography Secrete Key Cryptography Public Key Cryptograpgy Digital Signatures 2 10/1/2015.

CS 627 Elliptic Curves and Cryptography Paper by: Aleksandar Jurisic, Alfred J. Menezes Published: January 1998 Presented by: Sagar Chivate.

Networks Management and Security Lecture 3.

Midterm Review Cryptography & Network Security

Public Key Encryption CS432 – Security in Computing Copyright © 2005, 2008 by Scott Orr and the Trustees of Indiana University.

Modular Arithmetic with Applications to Cryptography Lecture 47 Section 10.4 Wed, Apr 13, 2005.

Public-Key Encryption

Public Key Cryptography. symmetric key crypto requires sender, receiver know shared secret key Q: how to agree on key in first place (particularly if.

1 Public-Key Cryptography and Message Authentication.

Cryptography and Network Security Chapter 9 - Public-Key Cryptography

CS461/ECE422 Spring 2012 Nikita Borisov — UIUC1.  Text Chapters 2 and 21  Handbook of Applied Cryptography, Chapter 8 

Some Number Theory Modulo Operation: Question: What is 12 mod 9?

PUBLIC-KEY CRYPTOGRAPH IT 352 : Lecture 2- part3 Najwa AlGhamdi, MSc – 2012 /1433.

Cryptography and Network Security Public Key Cryptography and RSA.

Chapter 3 – Public Key Cryptography and RSA (A). Private-Key Cryptography traditional private/secret/single-key cryptography uses one key shared by both.

Scott CH Huang COM 5336 Cryptography Lecture 6 Public Key Cryptography & RSA Scott CH Huang COM 5336 Cryptography Lecture 6.

Elliptic Curve Cryptography

Chapter 9 Public Key Cryptography and RSA. Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender.

Public Key Algorithms Lesson Introduction ●Modular arithmetic ●RSA ●Diffie-Hellman.

Faster Implementation of Modular Exponentiation in JavaScript

Elliptic Curve Cryptography Celia Li Computer Science and Engineering November 10, 2005.

Cryptography issues – elliptic curves Presented by Tom Nykiel.

Lecture 9 Elliptic Curves. In 1984, Hendrik Lenstra described an ingenious algorithm for factoring integers that relies on properties of elliptic curves.

Computer Security Lecture 5 Ch.9 Public-Key Cryptography And RSA Prepared by Dr. Lamiaa Elshenawy.

EPFL-IC-IIF-LACAL Marcelo E. Kaihara April 27 th, 2007 Algorithms for public-key cryptology Montgomery Arithmetic.

1 An Ordered Multi-Proxy Multi-Signature Scheme Authors: Min-Shiang Hwang, Shiang-Feng Tzeng, Shu-Fen Chiou Speaker: Shu-Fen Chiou.

Introduction to Number Theory

Introduction to Elliptic Curves CSCI 5857: Encoding and Encryption.

Lecture 3 (Chapter 9) Public-Key Cryptography and RSA Prepared by Dr. Lamiaa M. Elshenawy 1.

Introduction to Elliptic Curve Cryptography CSCI 5857: Encoding and Encryption.

RSA Pubic Key Encryption CSCI 5857: Encoding and Encryption.

Chapter 9 – Public Key Cryptography and RSA Every Egyptian received two names, which were known respectively as the true name and the good name, or the.

CSCE 715: Network Systems Security Chin-Tser Huang University of South Carolina.

Motivation Basis of modern cryptosystems

RSA Algorithm Date: 96/10/17 Wun-Long Yang. Outline Introduction to RSA algorithm RSA efficient implementation & profiling.

Public Key Cryptography. Asymmetric encryption is a form of cryptosystem in which Encryption and decryption are performed using the different keys—one.

Lecture 5 Asymmetric Cryptography. Private-Key Cryptography Traditional private/secret/single key cryptography uses one key Shared by both sender and.

Public Key Encryption.

ICS 353: Design and Analysis of Algorithms

The Application of Elliptic Curves Cryptography in Embedded Systems

Introduction to Cryptography

The RSA Public-Key Encryption Algorithm

Presentation transcript:

Efficient Montgomery Modular Multiplication Algorithm Using Complement and Partition Techniques Speaker: Te-Jen Chang

2006/11/2 TANET2006 台灣網際網路研討會 2 Outline Introduction Modular Arithmetic Complement Technique General Integer Multiplication Montgomery Algorithm Jeong-Burleson Algorithm

2006/11/2 TANET2006 台灣網際網路研討會 3 Outline (Cont.) Lee-Jeong-Kwon Algorithm Proposed Algorithm Computational Complexity Conclusions and Future Works

2006/11/2 TANET2006 台灣網際網路研討會 4 Introduction The computation of the modular exponentiation is a fundamental and important arithmetic operation in many scientific investigations, especially in the area of cryptology. High-speed, area-efficient, and low-power Montgomery modular multipliers for RSA algorithm have been developed for digital signature and user authentication in high-speed network systems and smart cards.

2006/11/2 TANET2006 台灣網際網路研討會 5 Introduction (Cont.) The RSA (Rivest Shamir Adleman) is one of the most widely used public-key cryptosystems. The encryption function is based on modular exponentiation with key size of 1024 or 2048 bits, and the modular multiplication is one of the major computation methods in cryptography systems.

2006/11/2 TANET2006 台灣網際網路研討會 6 Modular Arithmetic Equation l Equation 2 Equation 3 Equation 4

2006/11/2 TANET2006 台灣網際網路研討會 7 Modular Arithmetic (Cont.) The processes of addition, subtraction, and multiplication using Equation (1), Equation (2), Equation (3), and Equation (4) are called residue arithmetic or modular arithmetic.

2006/11/2 TANET2006 台灣網際網路研討會 8 Complement Technique where and if ; ; for i = 0, 1, …, k-1. Equation 5 Equation 6

2006/11/2 TANET2006 台灣網際網路研討會 9 General Integer Multiplication Figure 1. General integer multiplication (A*B). Equation 7 Equation 8

2006/11/2 TANET2006 台灣網際網路研討會 10 General Integer Multiplication (Cont.) Example 1: Let A = ( ) 2 = (512) 10, B = ( ) 2 = (692) 10, N = 7, k = 10, Evaluate. Here, B H = (10101) 2 = (21) 10 and B L = (10100) 2 = (20) 10 from Equation (7). Then, as in Figure 1, = 354,304.

2006/11/2 TANET2006 台灣網際網路研討會 11 Montgomery Algorithm Equation 9 Figure 2. Montgomery algorithm (A*B* mod N).

2006/11/2 TANET2006 台灣網際網路研討會 12 Montgomery Algorithm (Cont.) Example 2 Let A = ( ) 2 = (512) 10, B = ( ) 2 = (692) 10, N = 7, k = 10, Evaluate. Here, B H = (10101) 2 = (21) 10 and B L = (10100) 2 = (20) 10 from Equation (9). Then, as in Figure 2, k is the bit length of the exponent. = 3.

2006/11/2 TANET2006 台灣網際網路研討會 13 Jeong-Burleson Algorithm Figure 3. Jeong- Burleson algorithm (A*B mod N). Equation 10

2006/11/2 TANET2006 台灣網際網路研討會 14 Example 3 Let A = ( ) 2 = (512) 10, B = ( ) 2 = (692) 10, N = 7, k = 10, Evaluate. Here, B H = (10101) 2 = (21) 10 and B L = (10100) 2 = (20) 10 from Equation (7). Then, as in Figure 3, = 6. Jeong-Burleson Algorithm (Cont.)

2006/11/2 TANET2006 台灣網際網路研討會 15 Lee-Jeong-Kwon Algorithm Figure 4. Lee- Jeong-Kwon algorithm (A*B*mod N). Equation 11

2006/11/2 TANET2006 台灣網際網路研討會 16 We adopt the complement technique to Lee- Jeong-Kwon algorithm to efficiently evaluate modular multiplications. Case 1: Ham (A) >, and Case 2: Ham (A) ≦. Proposed Algorithm

2006/11/2 TANET2006 台灣網際網路研討會 17 The number of iterations for the modular multiplication “ ” = * (the number of iterations for modular multiplication in Case 1) + * (the number of iterations for modular multiplication in Case 2). Proposed Algorithm (Cont.) Equation 12

2006/11/2 TANET2006 台灣網際網路研討會 18 Case 1: Ham( Ａ ) > The number of iterations for a modular multiplication in Case 1 is Proposed Algorithm (Cont.)

2006/11/2 TANET2006 台灣網際網路研討會 19 Case 2: Ham( Ａ )  The number of iterations for a modular multiplication in Case 2 is Proposed Algorithm (Cont.)

2006/11/2 TANET2006 台灣網際網路研討會 20 The overall computational complexity of the modular arithmetic: Speedup ratio: (Comparing Lee-Jeong-Kwon algorithm with the proposed algorithm) Proposed Algorithm (Cont.) Equation 14

2006/11/2 TANET2006 台灣網際網路研討會 21 Conclusions and Future Works The computational complexity of the proposed algorithm is +, which is less than in Lee-Jeong-Kwon algorithm, where k is the bit- length of the multiplicand and the multiplier. The proposed modular multiplication algorithm has speedup ratio of 2.

2006/11/2 TANET2006 台灣網際網路研討會 22 Conclusions and Future Works (Cont.) Some new techniques are studied and proposed to fast evaluate modular multiplications such as the unified-multipliers method, elliptic curve encrypt method, and improved Montgomery algorithm.

2006/11/2 TANET2006 台灣網際網路研討會 23 Conclusions and Future Works (Cont.) In the future, we will try to properly study the combination of techniques mentioned in binary multiplication to further decrease the computational complexity for cryptographic applications such as the smart card and key exchange schemes.

2006/11/2 TANET2006 台灣網際網路研討會 24 Thanks for your attention.