UMass Lowell Computer Science 91.503

Analysis of Algorithms Prof. Karen Daniels

Fall, 2006

UMass Lowell Computer Science 91.503

Analysis of Algorithms Prof. Karen Daniels

Fall, 2006

Wednesday, 29 NovemberWednesday, 29 NovemberNumber-Theoretic AlgorithmsNumber-Theoretic Algorithms

Chapter 31 Chapter 31

Chapter DependenciesChapter Dependencies

Ch 31Number-Theoretic AlgorithmsRSA

Math: Number Theory

You’re responsible for material in this chapter that we discuss in lecture. (Note that this does not include sections 31.8 or 31.9.)

OverviewOverview

Motivation: RSAMotivation: RSA BasicsBasics Euclid’s GCD AlgorithmEuclid’s GCD Algorithm Chinese Remainder TheoremChinese Remainder Theorem Powers of an ElementPowers of an Element RSA DetailsRSA Details

Motivation: RSA

Motivation: RSA

RSA EncryptionRSA Encryption

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.531.5

MMSP AA ))(( MMPS AA ))((

RSA Digital SignatureRSA Digital Signature

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.631.6

assume Alice also sends her name so Bob knows whose public key to useassume Alice also sends her name so Bob knows whose public key to use

'))'(( MMSP AA

?

RSA CryptosystemRSA Cryptosystem

(31.19)(31.19)

(31.26)(31.26)

)(mod)( nMMP e )(mod)( nCCS d(31.35)(31.35) (31.36)(31.36)

encodeencode decodedecode

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

to be explained later….

need efficient ways to compute P(M), S(C)

RSA DependenceRSA Dependence

Correctness:Correctness: Euler’s Euler’s Function Function Fermat’s TheoremFermat’s Theorem Chinese Remainder TheoremChinese Remainder Theorem

Efficiency:Efficiency: Modular ExponentiationModular Exponentiation Primality TestingPrimality Testing

Security:Security: Difficulty of Factoring Large IntegersDifficulty of Factoring Large Integers

)(mod))(())(( nMMSPMPS ed

see chart of result dependencies on next slide (courtesy of Mark Micire)…see chart of result dependencies on next slide (courtesy of Mark Micire)…

Need to show:Need to show:

)(mod nMM ed

Basic Concepts

Basic Concepts

** Indicates that result is on chart of result dependenciesIndicates that result is on chart of result dependencies

Division & RemaindersDivision & Remainders

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.131.1

(3.8)(3.8) **

Equivalence Class Modulo nEquivalence Class Modulo n

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.1)(31.1)

(31.2)(31.2)

Common DivisorsCommon Divisors

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.3)(31.3)

(31.4)(31.4)

(31.5)(31.5)

**

**

Greatest Common DivisorGreatest Common Divisor

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.6)(31.6)

(31.7)(31.7)

(31.8)(31.8)

(31.9)(31.9)

(31.10)(31.10)

31.231.2

(3.8)(3.8)

(31.4)(31.4)

**

**

Greatest Common DivisorGreatest Common Divisor

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.331.3

(31.4)(31.4)

31.231.2

31.431.4

**

Relatively Prime IntegersRelatively Prime Integers

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.631.6

31.231.2

31.231.2

**

Relatively Prime IntegersRelatively Prime Integers

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.731.7

31.631.6

31.1-631.1-6 **

Greatest Common Divisor

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.931.9

(31.5)(31.5)

(3.8)(3.8)

(31.4)(31.4)

(31.3)(31.3)

(31.4)(31.4)

(31.3)(31.3)

(31.5)(31.5) (31.14)(31.14) (31.15)(31.15)

(31.14)(31.14)

(31.15)(31.15)

**

Euclid’s GCD Algorithm

Euclid’s GCD Algorithm

Euclid’s GCD AlgorithmEuclid’s GCD Algorithm

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

**

Also see Java code on course web Also see Java code on course web sitesite

Extended EuclidExtended Euclid

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.16)(31.16)

31.131.1

**

**

Chinese Remainder Theorem

Chinese Remainder Theorem

Modular ArithmeticModular Arithmetic

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

Finite GroupsFinite Groups

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

size of this group is 6size of this group is 6 size of this group is 8size of this group is 8

31.231.2

Additive group mod 6Additive group mod 6 Multiplicative group mod 15Multiplicative group mod 15

}1),gcd(:]{[* naZaZ nnn

elements relatively prime to nelements relatively prime to n

Finite GroupsFinite Groups

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.1231.12

Finite GroupsFinite Groups

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.1331.13

31.631.6

31.1231.12

31.2631.26

Euler’s Phi FunctionEuler’s Phi Function

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.19)(31.19) **

Lagrange’s TheoremLagrange’s Theorem

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.1531.15**

Finite GroupsFinite Groups31.1731.17 **

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.1831.18

31.1931.19

**

**

}1:{ )( kaa k

additive subgroup additive subgroup generated by generated by aa

wherewhere

aaaa k )(

kk

Solving Modular Linear EqSolving Modular Linear Eq

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.2031.20

(31.4)(31.4)

**

Solving Modular Linear EqSolving Modular Linear Eq

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.2231.22

31.1831.18

31.1831.18

31.2231.22

31.2431.24

**

**

Solving Modular Linear EqSolving Modular Linear Eq

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.2631.26

**

**

Chinese Remainder TheoremChinese Remainder Theorem

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.23)(31.23)

31.2731.27

(31.23)(31.23)

(31.24)(31.24)

(31.25)(31.25)

(31.26)(31.26)

**

Chinese Remainder TheoremChinese Remainder Theorem

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.2931.29 **

Powers of an Element

Powers of an Element

Theorems of Euler & FermatTheorems of Euler & Fermat

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.3031.30

31.3131.31

**

**

Modular ExponentiationModular Exponentiation

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

**

nab mod

Also see Java code on course web siteAlso see Java code on course web site

RSA Details RSA Details

RSA EncryptionRSA Encryption

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.531.5

MMSP AA ))(( MMPS AA ))((

RSA Digital SignatureRSA Digital Signature

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

31.631.6

assume Alice also sends her name so Bob knows whose public key to useassume Alice also sends her name so Bob knows whose public key to use

'))'(( MMSP AA

?

RSA CryptosystemRSA Cryptosystem

(31.19)(31.19)

(31.26)(31.26)

)(mod)( nMMP e )(mod)( nCCS d(31.35)(31.35) (31.36)(31.36)

encodeencode decodedecode

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

need efficient ways to compute P(M), S(C)

RSA CorrectnessRSA Correctness

source: 91.503 textbook Cormen et al.source: 91.503 textbook Cormen et al.

(31.35)(31.35) (31.36)(31.36)

31.3131.31

31.2931.29

by Thm 31.31 (Fermat)by Thm 31.31 (Fermat)