lecture 2 basic number theory and algebra. in modern cryptographic systems,the messages are...

Lecture 2 Basic Number Theory and Algebra

In modern cryptographic systems,the messages are represented by numerical values prior to being encrypted and transmitted. The encryption processes are mathematical operations that turn the input numerical value into output numerical values. Building, analyzing, and attacking these cryptosystem requires mathematical tools. The most important of these is number theory, especially the theory of congruences.

Outline Basic Notions Solving ax+by=d=gcd(a,b) Congruence The Chinese Remainder Theorem Fermat’s Little Theorem and Euler’s Theorem Primitive Root Inverting Matrices Mod n Square Roots Mod n Groups Rings Fields

1 Basic Notions1.1 Divisibility

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The primes less than 200:

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1.2 Prime (Continued)

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1.3 Greatest Common Divisor (Continued)

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1.3 Greatest Common Divisor (Continued)

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2 Solving ax+by=d=gcd(a,b)

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3 Congruences

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4 Example

3.2 Division (Continued)

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3.2 Division (Continued)

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4 The Chinese Remainder Theorem

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4 The Chinese Remainder Theorem (Continued)

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4 The Chinese Remainder Theorem (Continued)

numbers.

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9 Example

4 The Chinese Remainder Theorem (Continued)

).35(mod34)7(mod1),5(mod1

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),35(mod6)7(mod1),5(mod1

),35(mod1)7(mod1),5(mod1

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4 The Chinese Remainder Theorem (Continued)

solutions.2 has )(mod1 then primes, odddistinct

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5 Fermat’s Little Theorem and Euler’s Theorem

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5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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14 Example

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5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

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6 Primitive Root

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7 Inverting Matrices Mod n

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7 Inverting Matrices Mod n (Continued)

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16 Example

15 Example

7 Inverting Matrices Mod n (Continued)

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8 Square Roots Mod n

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8 Square Roots Mod n (Continued)

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19 Example

18 Example

17 Example

8 Square Roots Mod n (Continued)

. factoring toequivalentnally computatio is )(mod

to, solutionsfour thefindingThen . modroot squere a has

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2

2

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2 Principle Basic

19 Example

19 Example

OracleRoot Square

9 Groups, Rings, Fields9.1 Groups

. , allfor (4)

e,furthermor if, e)commutativ(or abelian is groupA

.1 such that , of inverse

thecalled ,element an exists G thereeach For (3)

. allfor 11that

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11

1

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G

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6 Definition

9.1 Groups (Continued)

.,element identity

withgroup, a form XOR, ofoperation with the},,{set The (3)

1.element identity with , modulo

tionmultiplica ofoperation under the group a is set theHowever,

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1

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20 Example

9.2 Rings

. , allfor if ring ecommutativ a is ring The

.

, , allfor )()()( and )()(

)( is,That .over vedistributi is operation The (4)

. allfor 1 1that

such 0, 1 with 1, denotedidentity tivemultiplica a is There (3)

. , , allfor

)( )( is,That e.associativ is operation The (2)

0. denotedidentity with groupabelian an is ) (R, (1)

axioms. following thesatisfying ,on

ation)(multiplic and (addition) denotedy arbitraril operations

binary with twoset a of consists ),,( ringA

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R

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cba

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R

RR

7 Definition

9.2 Rings (Continued)

ring. ecommutativ a is modulo performed

tion multiplica andaddition with set The (2)

ring.

ecommutativ a istion multiplica andaddition of

operations usual with the Zintegers ofset The (1)

n

Zn

21 Example

9.3 Fields

order. its called is elements ofnumber The

finite. is elements ofnumber theif finite is structure algebraA #

prime. is

If number. prime a is ifonly and if ) modulotion multiplica

andaddition of operations usual (under the field a is (2)

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form numberscomplex theand , numbers real the, numbers

rational theHowever, 1. and 1 are inverses tivemultiplicawith

integers zero-nononly thesince field, anot istion multiplica and

addition of operations usual under the integers ofset The (1)

inverses. tivemultiplica have elements

zero-non allin which ring ecommutativ a is fieldA

nnn

Zn

CRQ

22 Example

8 Definition

Thank you!