td-RSA-corrige

n= 319 e= 11
•319 = 11 ×29 1011 = 263 (mod 319) 2632= 216 ×319 + 265
•1333= 12 (mod 319) 13325 = 133 (mod 319)
•112= 121 (mod 280) 114= 81 (mod 280) 118= 121 (mod 280) 1116 = 81
(mod 280)
•95 = 64 + 31 81.11 = 51 (mod 280) 81.121 = 1 (mod 280)
M= 100
M0= 10011 (mod 319) = 265
d e
11 ∗d= 1 (mod 280) d= 51
•
•51 = (280/11) ∗2
•d= 11−1(mod 280) d= 11φ(280)−1(mod 280) =
11φ(7.5.8)−1(mod 280) = 11(6.4.4)−1(mod 280) = 1195 (mod 280) = 1164+16+8+4+2+1
(mod 280) = 81.81.121.81.121.11 = 81.11 = (mod 280) = 51 (mod 280)
M0= 133
13351 (mod 319) 13325 =
133 (mod 319) 133 ∗133 ∗133 (mod 319) = 12
625

φ(55) = 40 27.3 = 81 = 1 mod 40
φ(33) = 20 7.3 = 21 = 1 mod 20
m= 123mod 33 122= 12[33] m= 12
mod 33
(xeSmod nS)dPmod nP
y(yePmod nP)dSmod nS
(233mod55)7mod 33 = (127mod 33) = 12

(n, e)e= 3
n
e
(nW,3) (nJ,3) (nA,3)
x0≤x < nW, nJ, nA
x3mod nWx3mod nJx3
mod nAx
a k
ba1/kcO(log2a)
O(log2(nA+nJ+nW)) C=x3nW.nJ.nA
C=x3x < nW, nJ, nA
xRx=C1/3
x
c
c=meAmod nAc
0< r < nAx reAmod nA
y x.c mod nA
y u =ydA
mod nA
m
u.r−1mod nA
u.r−1mod nA=ydA.r−1mod nA=cdAxdAr−1mod nA=
cdA(xdAmod nA)r−1mod nA=cdArr−1mod nA=cdAmod nA=m
u.r−1mod nAO(log2nA)
m

n
(eA, dA) (eB, dB)eAeB
m
cA=meAmod nAcB=meBmod nB
m
m
eAeB
(eA, eB, n) (cA, cB)
n
r s reA+seB= 1
cr
A.cs Bmod n=mr.eA+s.eBmod n=
m
n

a b a 6= 0
b x x2≡amod b.
x a b
p q n =p.q
Z/nZ?
x2≡amod b(b−x)2≡amod b
a n a p
q
a6= 0 p x y =p−x
aZ/nZa p q
x1n−x1x2n−x2
(Z/pZ?,×)p−1
2
p
Z/nZ?
(b−x)2=b2−2bx +x2=x2=amod b
a=x2+kpq a ≡x2mod p p q
x y a =x2=y2mod p x2−y2= (x−y)(x+
y) = 0 mod pZ/pZp x −y6= 0
mod p x +y= 0 mod p y =p−x
amod nmod pmod q
a6= 0 mod p a u1=u u2=p−u
p v1=v v2=q−v q
a
n ui.q.q−1[p]+vj.p.p−1[q]mod n1≤i, j ≤2
x1n−x1x2
n−x2
g(Z/pZ?, .)x
g=x2gp−1= 1 mod p g
p g p−1
2=−1xp−1=−16= 1 p6= 2
xZ/pZg
p
g2ip−1
2
g2ix=gigi+p−1
2=−x=p−x