Structures algébriques et algorithmes Table des matières

n
n
n
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≤Nbi·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·Bi
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≤Mbi·Bi
c=ab
i= 0 . . . N
ci←0
j= 0 . . . M
p←ai×bjp p =p1·B+p00≤pi< B
ci←ci+p0·Bi+j+p1·Bi+j+1
(−1)s+tP0≤i≤Nci·Bi
N M O(NM)
N
Z

π(n)n n
π(n)∼n
ln n
n≥59
n
ln(n)1 + 1 2 ln(n)< π(n)<n ln(n)1 + 3 2 ln(n).
t πtt
πt=π(2t)−π(2t−1)>2t−2
(t−1) ln(2).
n
n√n
O(√n)
n
n
k←2
k n k ≤√n
k←k+ 1
k n
t O(2t/2)
n > 1n−1 = 2su u 1< a < n
au6= 1 (mod n)a2iu6=−1 (mod n)i∈[[0, s −1]] n a
n
n a
n n
n
n < 1
4
s