Calcul de racines carrées dans les corps nis 0. Introduction

m KPA
p q
x=m2KPAx p q
m
x p q
KPA
KPA=pq x KPA±u, ±v KPA
u2−v2= (u−v)(u+v)KPA
x
q2q= 2mFqa
F∗
qa a2m−1
O(log2(q)3)
q3 4 aF∗2
q
a a(q+1)/4O(log2(q)3)
q−1q2st t
H t
h=G0⊂G1⊂ ··· ⊂ Gs−1⊂Gs=F∗
q,
i{0, ..., s −1}Gi+1/Gi2
gF∗
qF∗ qe≤s
h H a =geh

e h a
h
h u
e a F∗
qe
uge/2a
gF∗
q
1/2a0(slog2(q)3)
σ σ(x) = x(q−1)/2xF∗
q
q x2−tx +a t Fq(q+ 1)/2
Fq
tFqx2−tx +a
1
2−1 2q
tFq
∆ = t2−4aFq
∆ = t2−4aFqx(q+1)/2x2−tx +a
aFq
1
2−1 2qa0(log2(q)3)
N=pq N +y2=x2p=x−y, q =x+y
B LB(N)p p 6B N
p b B rbb2N
LB(N)b(¯vp(rb))p∈LB(N)Fl
2
¯vp(r)p2r p l
LB(N)
•l b1, ..., blB√N+ 1
•(ci) (bi)Pi(ci)=0
•s=Qcit=Qp∈LB(N)p(Pvp(rci))/2s2≡t2N
B(ci)

•
m
K
K K
•a q = 2m
•a q ≡3 4
•h e
q= 17
•σ
∆