14 Applications of Number Theory

8/9/2019 14 Applications of Number Theory

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Applications of Number

Theory

CS 202

Epp section 10.4Aaron Bloomfield


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About this lecture set

! "ant to introduce #SA$ The most commonly used crypto%raphic al%orithm

today

&uch of the underlyin% theory "e "ill not be ableto %et to$ !t's beyond the scope of this course

&uch of "hy this all "or(s "on't be tau%ht$ !t's )ust an introduction to ho" it "or(s


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*ublic (ey crypto%raphy

E+erybody has a (ey that encrypts and a

separate (ey that decrypts

$ They are not interchan%able

The encryption (ey is made public

The decryption (ey is (ept pri+ate


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*ublic (ey crypto%raphy %oals

ey %eneration should be relati+ely easy

Encryption should be easy

3ecryption should be easy

$ ith the ri%ht (ey

Crac(in% should be veryhard


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!s that number prime

5se the 6ermat primality test

7i+en8

$ n8 the number to test for primality

$ k8 the number of times to test -the certainty/

The al%orithm is8repeat ktimes:

pick arandomly in the range [1, n1]

if an1mod n 1 then return composite

return probably prime


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9et n: 10;

$ !teration 18 a:


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9et n: 101

$ !teration 18 a: ;;8 ;;100mod 101 : 1

$ !teration 28 a: ?08 ?0100

mod 101 : 1$ !teration @8 a: 148 14100mod 101 : 1

$ !teration 48 a: @8 @100mod 101 : 1

$At this point> 101 has a -/4 : 1,1? chance of still

bein% composite


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&ore on the 6ermat primality test

Each iteration hal+es the probability that the number is a

composite

$ *robability : -/k

$ !f k: 100> probability it's a composite is -/100: 1 in 1.2 10@0

that the number is composite 7reater chance of ha+in% a hard"are error

$ Thus> k: 100 is a %ood +alue

o"e+er> this is not certain$ There are (no"n numbers that are composite but "ill al"ays

report prime by this test

Source8 http8,,en."i(ipedia.or%,"i(i,6ermatprimalitytest


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Googles recruitment campaignGoogles recruitment campaign


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#SA

Stands for the in+entors8 #on #i+est> Adi Shamir

and 9en Adleman

Three parts8$ ey %eneration

$ Encryptin% a messa%e

$ 3ecryptin% a messa%e


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ey %eneration steps

1. Choose t"o random lar%e prime numbers p D q> and

n:pq

2. Choose an inte%er 1 F eF n"hich is relati+ely prime to-pG1/-qG1/

@. Compute dsuch that d * eH 1 -mod -pG1/-qG1//

$ #ephrased8 de mod -pG1/-qG1/ : 1

4. 3estroy all records ofpand q


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ey %eneration> step 1

Choose t"o randomlar%e prime numberspD q

$ !n reality> 204= bit numbers are recommended

That's ?1 di%its

$ 6rom last lecture8 chance of a random odd 204= bitnumber bein% prime is about 1,10

e can compute if a number is prime relati+ely Iuic(ly +ia

the 6ermat primality test

e choosep: 10 and q:


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Ja+a code to find a bi% prime number8

BigInteger prime = ne BigInteger!numBits, certainty, random"#

The number ofbits of the prime

Certainty that the

number is a prime

The random number

%enerator


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Ja+a code to find a bi% prime number8

import $a%a&math&'#import $a%a&util&'#

class Big(rime )

static int num*igits = +1#static int certainty = 1--#

static final double ./02 = 3ath&log!1-"43ath&log!2"#

static int numBits = !int" !num*igits ' ./02"#

public static %oid main !5tring args[]" )6andom random = ne 6andom!"#BigInteger prime = ne BigInteger !numBits, certainty,

random"#5ystem&out&println !prime"#

77


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o" lon% does this ta(e

$ eep in mind this is Ja+a

$ These tests done on a =;0 &hK *entium machine

$A+era%e of 100 trials -certainty : 100/

$ 200 di%its -??4 bits/8 about 1.; seconds

$ ?1 di%its -204= bits/8 about ; seconds


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*ractical considerations

$ pand qshould not be too close to%ether

$ -pG1/ and -qG1/ should not ha+e small prime factors

$ 5se a %ood random number %enerator


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Choose an inte%er 1 F eF n"hich is relati+ely

prime to -pG1/-qG1/

There are al%orithms to do this efficiently$ e aren't %oin% o+er them in this course

Easy "ay to do this8 ma(e ebe a prime number$ !t only has to be relati+ely prime to -pG1/-qG1/> but can

be fully prime


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ey %eneration> step @

Compute dsuch that8

d * eH 1 -mod -pG1/-qG1//$ #ephrased8 de mod -pG1/-qG1/ : 1

There are al%orithms to do this efficiently$ e aren't %oin% o+er them in this course

e choose d: 4?? step @

Ja+a code to find d8

import $a%a&math&'#

class 8ind* )

public static %oid main !5tring args[]" )

BigInteger p9 = ne BigInteger!1-1+"#BigInteger e = ne BigInteger !;


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3estroy all records ofpand q

!f "e (no" p and q> then "e can compute the

pri+ate encryption (ey from the public decryption(ey

d * eH 1 -mod -pG1/-qG1//


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The (eys

e ha+e n:pq: 10@ e: =;> and d: 4??e/ : -10@ =;/

The pri+ate (ey is -n>d/ : -10@ 4?? nis not pri+ate

$ Lnly dis pri+ate

!n reality>d and eare ?00 -or so/ di%it numbers

$ Thus nis a 1200 -or so/ di%it number


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Encryptin% messa%es

To encode a messa%e81. Encode the messa%e minto a number

2. Split the number into smaller numbers mF n

@. 5se the formula c = me

mod n cis the cipherteMt> and mis the messa%e

Ja+a code to do the last step8

$ m.mod*o" -e> n/$ here the ob)ect m is the Bi%!nte%er to encrypt


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Encryptin% messa%es eMample

1. Encode the messa%e into a number$ Strin% is 7o Ca+aliersO$ &odified ASC!! codes8

41 =1 02 @ ? == ? = ; 1 =4 =; 0@ 0@

2. Split the number into numbers F n$ #ecall that n: 10@

14 Applications of Number Theory

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