EUCLID 's 'fElements

( ca . 350 BCE

F-hans primes --

t¥)(Fp)(p primeT xp > x)

threshold

HeftyTprime

Aoi . as I-

Proof by contradictionsuppose q Ex

: . g l I

P E O ( q )P = - I Cs )

-

.

.o'ql I →e

Hwy II (6)F - primes

= - I (6)-

CANCELLATION LAW

cx=cy (m)Hc ITIassuming gcdH

x Ey (m)-

DO CFOcx=gq}⇒x=y (m')n'= -m-

→gcdlc ,m)

if god (44=1 we getCANC .

LAW

FERMAT LITTLE THEOREM

Fl T FLIp

(Fx) (x" = x (p))-

heart of public- key

cryptographye - cokehence

p=x(2)- 21 x? x = x (x - D •p =3 x

'sEx (3)→

- 31 x'-x =x(x2-Heyxlxtl)-

51×5-x = x (x"- D=⇒ (x'tf (x'- l )

= (x - 1) xlxti) . (x't l)urge

we would like

E:?? GI?

It I = Cx -2)Gtz)=xna

mod 5

XIX IK- 2)(x- 1) xlxtDCxt2)=Dhad 5

Fl Tv .1 . p prime

¥ ⇒ Cp)) ¥ Fxx"'

II CplTx

not enoughtf Hitsmancanc . ✓DOlaw version 2

(G)( ptx ⇒ x''' ' Ii Cps)-

S EZS closed under additionmeans

(tag)( x ,yes→ xty ES)-

DEI M E Z

M is a noctule if① M¥0② M is closed under

subtraction

d-x.g)(x, y EM ⇒x - y ⇐M)

DEF M E Z is a moduleif ① M -40② M is closed under

subtraction

xiy EM → x -y EM

z?t.ee - lo - o :c :• 14=22 h -

€61 221+1 1) Odd - odd-

=even

Tf = 21 ✓ h= ,5,3 5-3 X

M = 321 rn=3

fM=nZI← cyclicAc

modules

Ith All modules arecyclic-Bonus ( use Euclid 's lance)-

① If M is a modulethen 0 EM-

on EM ⇒ O - m-m EMa-7- b/c M¥0-

Do ¢ is closedunder subtraction

② If MEN ⇒ - m EMTootle

-m= O - m EM ✓

③ M module → closedunder addition

a.BE/7--7atbEM-

tf at b- a - f-b)'EMin

C-M-

part of HW

nEM⇒nZ. . .⑦

Mtn H module is cyclic-

Coney: F god for anyBezout listFlam of mutes-

Iee.S E Zd is a greatest commondivisor of Sif-

① d is a common dir.i. e . #s es) ( d l s)② of e is a common dir.then eld

Proof that god (a.b) existsand can be written as-

a Lin . comb Ext by

7-x.ykxtbylx.y.CZ/3=-----Ta2tbZLEMMAa2/tbZis a module

HD ed:@d)(a¥I=d¥Chain This I. is a go . c.d .① dla = a. It b. o dlb② ela.ee/b=zg!yaxt-byihEd=d - I

DI g. c.d . exists forany set

and can be written

as linear comb .-

Sf S int .v

a tin . comb meansof S

bin comb of afinite subset of S-

t.c.m.eu#hEEa7zLEMMA M , Mz : modules

⇒ M,n Mz module-

PI Q M , nMz # 0b/c both contain 0

② M,nMz closed under

④sublet

-

set of common hueltiplesof E. be : a 21 n BI-

module y''Ff

d-

TI : m 1. c.n.

TEX