12-3-2020-1CRT Chinese remainder theorem=-

system ofSimultaneous congruences

x = a. Cms ) {tasked;x a, Cme)

all Sol 's forma residue class

Icm(mi )

* { E E 's f Hypnotism-def of km¥ = 331-

④ ⇒ x E 33 Imod7I10 I x - 3 i.e . lol x- 3314 l x -5 i.e - 14 ( x - 33 }#

⇐ 1cm ( 10,14) / x-33-70

Sufficient conditionCRT of solubilityIf msn.imuasepairwisetd.pn.me#then F solution(ti) ( x = a , mod mi)-NOT a necessary condition-

X 0 mod 17

×=0w ) EEEf-m , - -- ma ) (aa . . . - an)x I ⑨ 0 (m .)

÷ :

x - ④o Chul=

T

Proof for= gcdcm.ms- IG) - x Ea . Cm ,)(2) X=azCm#

X = U,m

, tuzmz find u , uz

gFa

(1)⇐ uzmz Ea , (m ,) Faz

G) ⇐ Tim ,= an Cma) / za ,- b/c gcdlm.mil/a,fi:D#¥394 -* I if V

DX# § ⇐ '⇒ do) ]xeuc.IO tuz. 7

Uzi ? I 3 (co ) uz = - lU,-10 I 5 (7) u ,

= 4

x - 40-7=32 9

proof of CRT1<=2 ✓k 23 induction DLP-

RSA public - keycryptosystem-

X . plaintex messageE-(X) - encode : ciphertext

← DIE = Xdecryption

Encryption key

← Decryption key-

9 . --

- - - → 9

try page El

a.at#oComputational complexity

xg

pines 315.7

ReE: public key E algorithmprivate key D "

¥?-

given E , for most YD CY) hand to compute-

D domain : set of possibleplaintexts{ ciphertexts

E :D → DD : D →D permutations

- lE = D DIFFIE

MERKLE .HELLMANN1978

Rivest,Shamir

, Adlerian

RSAP# g prime number ( laye)'II 'The.es/PYiuYekye:g-cd(eiM)='

private key.: f=(e- ' and M) fef =L (M)-

D - {0,1 . . . . ,N - I }XED ECX) :=(XemodN)

1) ( Y) : - (YtmodN)-

the #K×et=× mod N )Fet

of =L wed M M-- Kalp-I ,q - 1)Then Atx) ( xef=x mod N)-* N=pg

NTS ×et=× ( p)xef=× Ce) }*-

ef ⇒ ( M)

case I plx xef=O=p)case 2 ptx XP-1=-1 Cp) TITp- II Mlef - I

×ef - I =#P -Nk = Ik =/ (p)x -- -

yet = x ( p)✓

FACTORING IS HARDON AVERAGE :

P , g random primes→ pg is hard to factor-

e if ⇒ CM)t

=pq= ? M - km (p-I. s- l )

=

in fact , knowing M ← factoring N-

Only known attacks on RSArequire factoring ppg

-

a -digit beep Grin )" we believe "

movie

SNEAKERSthriller

-

The (Peter Shor 1990s )

Factory is solvable in

quantum polynomial time-

Post - quantum cryptographylattice-based4¥→

.--- -

p-

DCECXD - AEID CD) -Y-

Digital signature

DIM)Es MP-

N, Nz . - - N

••ooo

p = god (Ni .Nj) f- I-

pick random 100 -digit number

Plein =a÷÷aioo