Notes de cours

q=pn
http://www.cacr.math.uwaterloo.ca/hac/

657
657 = 6 ×102+ 5 ×101+ 7 ×100,
0
9 = 10 −1
B
657 = 1 ×29+ 0 ×28+ 1 ×27+ 0 ×26+ 0 ×25+ 1 ×24+ 0 ×23+ 0 ×22+ 0 ×21+ 1 ×20,
1010010001
0k657 0657 00657
0
n > 0B≥2|n|B
n B
|657|10 = 3 |657|2= 10
|n|B=blogBnc+ 1.
bxcx
xlogBx B x
logBx=ln x
ln B
|n|B≈logBn,
n
|n|B
|n|B0
≈ln B0
ln B= logBB0.
B= 10 B0= 2
log10 2≈0.3010299957,

n
[[0,2n−1]] [[−2n−1,2n−1−1]]
n= 32 232 −1≈4×109−231 ≈ −2×109231 −1≈2×109
n= 64 264 −1≈18 ×1018 −263 ≈ −9×1018 263 −1≈9×1018
B= 2nn= 32 64 N
s
a= (−1)sX
0≤i≤N
ai·Bi
s N a0a1aN
a= (−1)sP0≤i≤Nai·Bib= (−1)sP0≤i≤Nai·Bis
c= (−1)sP0≤i≤N+1 ci·Bic=a+b
γ0←0
i= 0 . . . N
ci←ai+bi+γiγi+1 ←0
ci≥B
ci←ci−b γi+1 ←1
cN+1 ←γN+1
(−1)sP0≤i≤N+1 ci
NO(N)
A(X) =
P0≤i≤NaiXiB(X) = P0≤i≤NbiXi
http://gmplib.org/

A(X) = P0≤i≤NaiXiB(X) = P0≤i≤NbiXi≤N
C(X) = P0≤i≤NciXiC=A+B
i= 0 . . . N
ci←ai+bi
P0≤i≤NciXi
a= (−1)sP0≤i≤Nai·Bib= (−1)sP0≤i≤Mai·Bi
c=ab
i= 0 . . . N
di←0
j= 0 . . . M
p←ai×bjp p =p1·B+p00≤pi< B
di←di+p0·Bi+j+p1·Bi+j+1
(−1)s+tP0≤i≤Ndi
N M O(NM)
G ?
?
∀a, b, c ∈G(a?b)? c =a ? (b?c) ;
?e
∃e∈G∀a∈G a ? e=e? a =a;
∀a∈G∃b∈G a ? b =b?a=e.
?
∀a, b ∈G a ? b =b ? a.
Z Q R C
+
Z/nZn
K\ {0}K=Q,R,C
×

R[X]R R
Snn◦
N
A∗A
×
Z\ {0}
1−1
Q\ {0}R\ {0}C\ {0}
Z/nZ\{0}n
Z/nZ∗n
ϕ(n)ϕ n ≥2
ϕ(n) = nY
p|n
(1 −1
p)
p n ϕ
n=Qr
i=1 psi
i
ϕ r
Y
i=1
psi
i!=
r
Y
i=1
(psi
i−psi−1 i).
+×
·
a, b ∈G a +b a ×b a ·b ab
m a ∈G m ∈N\ {0}ma =a+. . . +a
| {z }
m a
am=a·. . . ·a
| {z }
m a
m= 0 0a=0a0=1
m=−1−a a−1
−m a ∈G m ∈N(−m)a=m(−a)a−m= (a−1)m
m, p ∈Za∈G(m+p)a=ma +pa am+p=am·ap
m, p ∈Za∈G(mp)a=m(pa)amp = (am)p