Enoncé

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
d e
M0= 133
625

(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)
c
c=meAmod nAc
0< r < nAx reAmod nA
y x.c mod nA
y u =ydA
mod nA
m
u.r−1mod nA
n
(eA, dA) (eB, dB)eAeB
m
cA=meAmod nAcB=meBmod nB
m
m
eAeB

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?
a < n
x a 6= 0 n
n
x1x2
(n−x1) (n−x2)a n
n
u=x1−x2mod n v =x1+x2mod n u.v ≡0 mod n
1≤u, v < n
p q n u v
n
Z/nZ(x) = x2mod n

n=pq
p q
xA< n a =x2
A
mod n n a
xA
n
r
y=r2mod n z =xA.r mod n
y z
a.y −z2= 0 mod n
r xA
z y =z2/a
mod n
r y =r2mod n y
b∈ {0,1}b
z=r y n
z=xA.r mod n y.m
mod n
y.ab−z2= 0 mod n
k