Chapitre 2 Exemples d`algorithmes itératifs et récursifs - IMJ-PRG

a∈Zb∈Zm∈Z
P GCD(a, b) = P GCD(b, a −mb).
m=q
a b
a, b
a b
(a, b)
r←amod b
r
b
(b, r)
a, b
a b
(a, b)
b
r←amod b
a←b b ←r
a

a, b b ≥3a > 0
a b
(a, b)
a6=b
a
a←a/2
a>b
a←a−b
2
r←b−a
2
b←a a ←r
a
b > 0
a∈Zb∈Zd=P GCD(a, b)u v
d=ua +vb
(u, v, d)a.u +v.b =d
(a, b)
A←(1,0, a)Aii A
B←(0,1, b)Bii B
reste ←b
6=
q←A3reste
temp ←A−q×B
A←B
B←temp
reste ←B3
A
∗ ∗

− ∗
∗ ∗ −
∗ ∗
− ∗
∗ ∗
a, b r0=a r1=b
riri+1 =ri−1−qi.riqi
ri−1riu0= 1, u1= 0 v0= 0, v1= 1 (ui, vi)
ui+1 =ui−1−qi.uivi+1 =vi−1−qi.vi
ri
ui.a +vi.b =ri
i
N(ui)N≥i≥1
(vi)N≥i≥1(|ui|)N≥i≥1(|vi|)N≥i≥1
a∈IN b∈IN∗(q, r)a=bq +r
0≤r < b
a≥0b > 0
q←0r←a
r > b
r←r−b
q←q+ 1
q r
q
b a

a
t= log2(a)a
a≥0b > 0
a b
n←0
2nb≤a
n←n+ 1
α←2n−1β←2n
k1n−1
γ=α+β
2
γb ≤a
α←γ
β←γ
q=α r =a−bq
(G, ∗)
anO(log(n)) ∗
x∗y
x, y G 2G
g G n
G gn
(g, n)
u←1; v←g
n > 1
n
v←v∗v n ←n/2
u←u∗v;v←v∗v;n←(n−1)/2
u∗v
2
n
P`o˘u˚rffl n∈IN ˚t´e¨l `qfi˚u`e n > 1, `o“nffl ”n`o˘t´e In˜l„`e›n¯sfi`e›m˜b˝l´e `d`e˙s `é¨l´é›m`e›n˚t˙s ˚i‹n‹vfleˇr¯sfi˚i˜b˝l´e˙s `d`e ˜l„`a‹n‹n`e´a˚uffl
(Z/nZ,+,×).

...
5. P`o˘u˚rffl ˜l´e˙s `a˜l´g´o˘r˚i˚t‚h‹m`e˙s `d`e›m`a‹n`d`é˙s, `o“nffl ˚u˚tˇi˜lˇi¯sfi`eˇr`affl ˚u‹n˚i`qfi˚u`e›m`e›n˚t ˜l´e˙s `o¸p`éˇr`a˚tˇi`o“n¯s ×,+,ˆ
`eˇt ˜l´affl ˜f´o“n`cˇtˇi`o“nffl `d`e `d`eˇu‹x ”vˆa˚r˚i`a˜b˝l´e˙s `o˘ùffl `d`o“n‹n`e ˜l´e ˚r`e˙sfi˚t´e `d`e ˜l´affl `d˚i‹v˘i¯sfi˚i`o“nffl
`eˇu`c¨lˇi`d˚i`e›n‹n`e `d`e `affl ¯p`a˚rffl ˜b ¯p`o˘u˚rffl a∈IN `eˇt b∈IN∗. O”nffl ¯p`o˘u˚r˚r`affl `é´g´a˜l´e›m`e›n˚t ˚u˚tˇi˜lˇi¯sfi`eˇrffl `d`e˙s
˜bˆo˘u`c¨l´e˙s `d`e ˚t›y˙p`e
-
-
- `eˇt ˜l´affl `c´o“n¯sfi˚tˇr˚u`cˇtˇi`o“nffl
O”nffl ¯p˚r`é´cˇi¯sfi`eˇr`affl ˜l´e ˜l´oˆgˇi`cˇi`e¨l `d`e `c´a˜l´cˇu˜l ˜f´o˘r‹m`e¨l `o˘uffl ˜l´e ”m`oˆd`è¨l´e `d`e `c´a˜l´cˇu˜l´a˚tˇr˚i`c´e ˚u˚tˇi˜lˇi¯sfi`é.
5.1. É`cˇr˚i˚r`e ˚u‹n`e ¯p˚r`oˆc´é´d˚u˚r`e `a‹y´a‹n˚t `c´o“m‹m`e `a˚r`gˇu‹m`e›n˚t˙s `d`eˇu‹x `e›n˚tˇi`eˇr¯s ”n`a˚tˇu˚r`e¨l˙s
˛kffl `eˇt ”nffl `a‹vfle´c n > 1`a˜f¨fˇi`c‚h`a‹n˚t ‘‘1’’ ¯sfi˚iffl k∈In`eˇt ‘‘0’’ ¯sfi˚i‹n`o“nffl.
5.2. É`cˇr˚i˚r`e ˚u‹n`e ¯p˚r`oˆc´é´d˚u˚r`e `a‹y´a‹n˚t `c´o“m‹m`e `a˚r`gˇu‹m`e›n˚t ˚u‹nffl `e›n˚tˇi`eˇrffl n`a‹vfle´c n > 1
`a˜f¨fˇi`c‚h`a‹n˚t ˜l´e `c´a˚r`d˚i‹n`a˜l `d`e In.
5.3. É`cˇr˚i˚r`e ˚u‹n`e ¯p˚r`oˆc´é´d˚u˚r`e `a‹y´a‹n˚t `c´o“m‹m`e `a˚r`gˇu‹m`e›n˚t˙s `d`eˇu‹x `e›n˚tˇi`eˇr¯s ”n`a˚tˇu˚r`e¨l˙s
k`eˇt n`a‹vfle´c n > 1`a˜f¨fˇi`c‚h`a‹n˚t ˜l´affl ”vˆa˜l´eˇu˚rffl `d`e ω(k), ˜l„`o˘r`d˚r`e `d`e k`d`a‹n¯s (In,×), ¯sfi˚iffl k∈In`eˇt
"E˚r˚r`eˇu˚r" ¯sfi˚i‹n`o“nffl.