Premier Exercice : Combinaisons linéaires

Z(a, b)∈Z2L(a, b)
{ax +by (x, y)∈Z2}a b
π(a, b)≥0L(a, b)
π(a, b)
L(a, b) = π(a, b)Z.
(a, b)∈Z2b6= 0 amod b a b
r∈Z0≤r < |b|b a −r
π(a, b)
a b
a≤b
a b
b6= 0
r←− amod b
a←− b
b←− r
a
π(a, b)
n n a b
0≤a, b < 2n
x y
ax +by =π(a, b)
(a, b, n, m)∈Z4π(a, b) = 1 α α −n
a α −m b

s
` n a1, . . . , an`(x1, . . . , xn)∈ {0,1}n
s=Pn
i=1 xiai(x1, . . . , xn)∈ {0,1}n
m ` O(m` log m)
poly(n, `)n `
n ` c1c2c3
c1×nc2`c3n ` poly(n, `)
2npoly(n, `)
n
2n/2poly(n, `) 2n/2poly(n, `)
n
2n/2poly(n, `) 2n/4poly(n, `)

Z(a, b)∈Z2L(a, b){ax +
by (x, y)∈Z2}a b
π(a, b)≥0L(a, b)
π(a, b)
L(a, b) = π(a, b)Z.
(a, b)∈Z2b6= 0 amod b a b
r∈Z0≤r < |b|b a −r
π(a, b)
a b
a≤b
a b
b6= 0
r←− amod b
a←− b
b←− r
a
π(a, b)
n n a b
0≤a, b < 2n
x y
ax +by =π(a, b)
(a, b, n, m)∈Z4π(a, b) =
1α α −n a α −m b

s ` n
a1, . . . , an`(x1, . . . , xn)∈ {0,1}n
s=Pn
i=1 xiai(x1, . . . , xn)∈ {0,1}n
m ` O(m` log m)
poly(n, `)n ` n `
c1c2c3c1×nc2`c3
n ` poly(n, `)
2npoly(n, `)
n
2n/2poly(n, `) 2n/2poly(n, `)
n
2n/2poly(n, `) 2n/4poly(n, `)