1 Philosophie des tests de primalité 2 Témoins de Fermat

∼
n∈N∗
n
O(log(n)c)c > 0
n
n
npgcd n
n
P(n, x)
n=⇒ ∀x∈Z,P(n, x).
xP(n, x)n
n
n
n n
p x ∈Zp
xp−1≡1 (mod p)
n>2x∈Z
n n xn−16≡ 1 mod n
n n
n n
n

n1<x<n
n
n
x x n
561
n
ϕ(n)
2
n
n>1
n p n p −1n−1
x n xn−1≡1 (mod n)
p p −1=2αq q ∈Nx∈Z
p
xq≡1 mod p∃06i6α−1, x2iq≡ −1 mod p.
n>2n−1 = 2αm m ∈N
x∈Z
xm6≡ 1 mod n∀i= 0,1, . . . , α −1, x2im6≡ −1 mod n,
n

x
n
n
n= 561
n x 1<
x < n n
n > 9n−1 = 2αm m
M
M=nx∈Z/nZ|xm= 1 x2im=−1i∈ {0,1, . . . , α −1}o.
#M6ϕ(n)
4ϕ
Z/nZ∗n−1
Z/nZ∗
p ε > 0Z/pεZ∗Z/pεZ
pε−1(p−1)
G g= 1
pgcd(#G, )
Z/abZa
mba ma
0−→ Z/aZ−→ Z/abZ−→ aZ/abZ−→ 0
xmod b7−→ bx mod ab
ymod ab 7−→ ay mod ab
ay mod ab 7→ ymod a aZ/abZ→Z/bZ
n=Qp|npεpp n p −1 =
2αpmpmpβ= min{αp, p |n}2β2
(p−1) p n
N+,N−N
N+=nx∈Z/nZ|x2β−1m= 1o,N−=nx∈Z/nZ|x2β−1m=−1o,N=N+∪ N−.
M⊂N #N6ϕ(n)
4
• M ⊂ N
Z/nZ∗
Z/nZ∗'Y
p|n
Z/pεpZ∗.

ϕ(pεp) = pεp−1(p−1) x∈Z/nZ∗xm= 1 mod n x ∈ N
x2im=−1i∈ {0,1, . . . , α −1}(xm)2i=−1 mod n(xm)2i=
−1 mod pεpp n xm2i+1
pεpp n 2i+1 |ϕ(pεp)p i+1 6β
x∈ N
• N+x2β−1m= 1 pεp
p n
#N+=Y
p|n
pgcd 2β−1m, pεp−1(p−1)=Y
p|n
2β−1pgcd (m, p −1) .
N−p n
x2β−1m=−1pεp
{x∈Z/pεpZ|x2β−1m=−1}={x∈Z/pεpZ|x2βm= 1}\{x∈Z/pεpZ|x2β−1m= 1}
#{x∈Z/pεpZ|x2β−1m=−1}= pgcd 2βm, pεp−1(p−1)−pgcd 2β−1m, pεp−1(p−1))
= 2βpgcd(m, p −1) −2β−1pgcd(m, p −1)
= 2β−1pgcd(m, p −1).
#N+= #N−
#N= 2 Y
p|n
2β−1pgcd (m, p −1) .
#N
ϕ(n)= 2 Y
p|n
2β−1pgcd (p−1, m)
(p−1)pεp−1
?
61
4⇐⇒ Y
p|n
(p−1)
2β−1pgcd (p−1, m)×pεp−1?
>8.
(p−1)
2β−1pgcd(p−1,m)>2n
(2) n p q
εp>2 (2)
>4p>8n=pq (2)
(p−1)
2β−1pgcd (p−1, m)=(q−1) 2β−1pgcd (q−1, m)= 2,
(p−1=2βpgcd(p−1, m)=2αpmp
q−1=2βpgcd(q−1, m)=2αqmq
=⇒
αp=αq=β
pgcd(p−1, m) = mp
pgcd(q−1, m) = mq.
mpm(p−1) mp
2βmq=q−1≡pq −1≡n−1≡2αm≡0 (mod mp)⇒mp|mq.
mq|mpmp=mqp=q n
n=pεpn > 9p>5
εp>2p>3εp>3 (2) 

x[2, n −2] n
x n log(n)Z/nZ
O(log(n)3)
log(n)
O(log(n)4)
n
n
log(n)n61
4log(n)61
n
log(n)2
O(log(n)c)
n∈Z
62 log(n)2
p x ∈Np x 62 log(p)2
p, p0p6=p0x∈Np
p0x62 log(pp0)2
p x ∈Z/pZ∗
x⇐⇒ xp−1
2= 1, x ⇐⇒ xp−1 2=−1.
p x ∈Z
x
p
x
p=
0p x,
1x p p,
−1x p p.
x
p≡xp−1
2(mod p)
16x6
2 log(n)2
xm≡1 (mod n)∃06i6α−1, x2im≡ −1 (mod n)
n−1=2αm m n
•n