Feuille 5 : Retour sur RSA et factorisation

k B1, . . . , Bke= 3
ni1≤i≤k
ni1≤i≤k
m Bi
m3P:= Qk
i=1 nim P > m3
k
n eBeC
eBeC
m
m
(221,11) (221,7)
210 58
m
n p q p > q
p q  := p−q t =p+q
2
s=p−q
2
n=t2−s2
s t √n
n
11598781

p n
p−√n < 4
√4n
N= 1829
B={2,5,7}43,49,52,53, . . .
ρ
n p
n
x1, x2, . . . , xi, . . . Z/nZxi=xjmod p i < j
npgcd(xi−xj, n)
N
1/2√N
n
xix1
xi+1 =P(xi)P∈Z[X]
xi=xjmod p=⇒xi+1 =xj+1 mod p
xi=xjmod p i < j xu=x2umod p
u u < j
(xi+1, x2(i+1)) (xi, x2i)
(xi)
Z/nZ√p
≤n n
n= 7171 x1= 39 P(x) = x2+ 1
d p q
n p q
e ϕ(n)n p
q d
d n
e d n B ϕ(n)
λ= ppcm(p−1, q −1) a∈(Z/nZ)×, aλ= 1
aλ/2n

H={a∈(Z/nZ)×, aλ/2≡ ±1}H
(Z/nZ)×
b∈(Z/nZ)×b p −1p
(q−1)/2q
p−1=2vpp0q−1=2vqq0p0, q0
vp≥vqλ/2vpp0, q0
b H x (Z/nZ)×
x H 1/2
x∈(Z/nZ)×λ B k
xλ/2=xB/2k+1
n
n e d
n= 77 e= 7 d= 43