UMass Lowell Computer Science 91.503 Analysis of Algorithms Prof. Karen Daniels Spring, 2010

UMass Lowell Computer Science 91.503

Analysis of Algorithms Prof. Karen Daniels

Spring, 2010

UMass Lowell Computer Science 91.503

Analysis of Algorithms Prof. Karen Daniels

Spring, 2010

Tuesday, 27 AprilTuesday, 27 AprilNumber-Theoretic AlgorithmsNumber-Theoretic Algorithms

Chapter 31Chapter 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., 3source: 91.503 textbook Cormen et al., 3 rdrd edition edition

to be explained later….

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

Assume M < n

(31.20)(31.20)

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

20022002

(Eqn. 31.20)(Eqn. 31.20)

with thanks to Mark Micire

EUCLID GCDEUCLID GCD EXTENDED-EUCLIDEXTENDED-EUCLID

Notes on Primality TestingNotes on Primality Testing

• Efficient primality testing has been goal for > 2,000 Efficient primality testing has been goal for > 2,000 years.years.

• Early attempts required exponential time.Early attempts required exponential time.• Miller-Rabin (Section 31.8) primality test is a Miller-Rabin (Section 31.8) primality test is a

randomized polynomial-time algorithm (1980’s).randomized polynomial-time algorithm (1980’s).• Agrawal, Kayal, Saxena provided a deterministic Agrawal, Kayal, Saxena provided a deterministic

polynomial-time algorithm (2002).polynomial-time algorithm (2002).

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 DivisorGreatest 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 elements relatively prime to nn

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

**

**

'' where,modsolution a as has mod then If :31.23 . 0 nyaxd nx'(b/d) x n)b (axd|bThm

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 **

Corollary 31.28Corollary 31.28. If . If nn11, , nn22, …, , …, nnkk are pairwise relatively prime and are pairwise relatively prime and n = n =

nn11nn22…n…nkk, then, for any integers , then, for any integers aa11, , aa22, …, , …, aakk, the set of simultaneous , the set of simultaneous

equations for equations for i = 1, 2, …, ki = 1, 2, …, k, has a unique solution , has a unique solution

modulo modulo nn for the unknown for the unknown xx..

x ai mod ni ,

04/19/2304/19/23 3636

NumTheoryNumTheory

a 2 mod 5 , a 3 mod 13 ,ExampleExample. .

Given the Given the twotwo equations equations what is what is aa mod 65 mod 65? Note that ? Note that 65 = 5•1365 = 5•13..

The table of moduli wrt 5 and 13 for all integers in The table of moduli wrt 5 and 13 for all integers in ZZ6565..

source: 91.503 textbook Cormen et al. & Prof. Pecellisource: 91.503 textbook Cormen et al. & Prof. Pecelli

Table can be generated diagonally.

04/19/2304/19/23 3737

NumTheoryNumTheory

Knowing that find Knowing that find aa mod 65 mod 65..

We have We have

aa11 = 2, n = 2, n11 = 5 , m = 5 , m11 = n/n = n/n11 = 13, = 13,

aa22 = 3, n = 3, n22 = 13, m = 13, m22 = n/n = n/n22 = 5 = 5..

We can compute: We can compute:

m1 1 13 1 2 mod 5 ; m2

1 5 1 8 mod13 .c1 13 2 mod 5 26; c2 5 8 mod13 40;

a 226 340 mod 65 52 120 mod 65 42 mod 65 .

a 2 mod 5 and a 3 mod13

source: 91.503 textbook Cormen et al. & Prof. Pecellisource: 91.503 textbook Cormen et al. & Prof. Pecelli

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

**

**

31.2031.20

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., 3source: 91.503 textbook Cormen et al., 3 rdrd edition edition

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

(31.20)(31.20)

RSA CorrectnessRSA Correctness

source: 91.503 textbook Cormen et al. 3source: 91.503 textbook Cormen et al. 3 rdrd edition edition

(31.37)(31.37) (31.38)(31.38)

31.31)31.31)

31.2931.29

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

)(mod0 :Case pM

)(mod0 :Case pM pp

qq