12-3-2020-1 CRT Chinese remainder theorem = - system of Simultaneous congruences x = a . Cms ) { tasked ; x a , Cme) all Sol 's form a residue class Icm ( mi ) * { E E 's f Hypnotism - def of km ¥ = 33 1- ④ ⇒ x E 33 Imod7I 10 I x - 3 i.e . lol x - 33 14 l x - 5 i. e - 14 ( x - 33 } # ⇐ 1cm ( 10,14 ) / x - 33 - 70
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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