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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1.2 Prime

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The primes less than 200:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199

1.2 Prime (Continued)

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

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3 Congruences

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4 The Chinese Remainder Theorem

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numbers.

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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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19 Example

19 Example

OracleRoot Square

9 Groups, Rings, Fields9.1 Groups

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n

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

Rbaabba

R

cbaacabacbcaba

cba

Ra aaa

Rcba

cbacba

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)

.operations usual under the fields

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!

Lecture 2 Basic Number Theory and Algebra. In modern cryptographic systems,the messages are...

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Transcript of Lecture 2 Basic Number Theory and Algebra. In modern cryptographic systems,the messages are...