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^ ~LQ[SWX\]MO NM £R]Jv NvJ]W|RM

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(12)

(13)

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DESTINATION

SOURCE CANAL

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[v JSLM NM

âïëáüÜ

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SMÖRO [RS [R[WS LWSM LM ]MOOJQM z]WOT

JvO Rv O[RW N{MÃJWXz LJ Oz|RMvM NM OWQvJRY N[WX ýXSM LJ LRO

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âæð þÞê

QS M

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âæð þÞïüæáþãÜ

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(14)

[SO PvWO MX |RML|RMO zLz]MvXO NM S[KJKWLWXzONM X\z[SWM NMO QSJ\MO MX NM

]JvWRLJXW[v NMO \JvMO NM O]K[LMOT

VM SM]WMS \JWXSM NM M [RSO SzOMvXM LMO [RXWLO ]JX\z]JXW|RMO MO

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XMS]Wv[vO M \JWXSM JS RvM MO|RWOOM NR ]zJvWO]M RXWLWOz [RS JOOMS NR

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LJ X\z[SWM NM L{Wv[S]JXW[v MX NM LJ []SMOOW[v LJ SX[QSJ\WM MX JRY

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[]]RvWJXW[vOT

(15)

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300000

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10 ³⁹

(16)

Mv OM[vNMO

U nivers = 15 milliards ∗ 365.25 ∗ 24 ∗ 60 ∗ 60 ≈ 5.10 ¹⁷ secondes

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S[v

3.10 ¹²

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2.10 ³⁰

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0, 83.10 ⁻³⁰

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U nivers = (2.10 ³⁰ /0, 83.10 ⁻³⁰ ) ∗ 200.10 ⁹ ∗ 3.10 ¹² ≈ 10 ⁸⁴ electrons

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& '()*+ ,-+.*)+-/

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01210 3456789:;< =>?@A48=<

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pgcd

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a

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(a) 522

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^T

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(17)

Z

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pgcd

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x

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y

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1

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NVMUOMU DEEF GHHIIIJWOJNMWHWOUVLEHFMNO EVNHWORFJDERU

(18)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Temps en secondes

Nombre de chiffres

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(19)

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u _i ≡ u _j [p]

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O( √ p)

^êÞáþ^ÜÝ

q X − 1 F [ X ] n V [ X ] / ( P ) V deg( P ) Z

Z

V [X]/(P )

V deg(P )

X n − 1

F q [X]

V = { 0, 1 }

j − i + q−1 2

| V | = 2

m

DESTINATION

SOURCE CANAL

PERTURBATION

m

d log[10](m) e

d log[2](m) e

10 9

300000

= 3.10 8

10 39

U nivers = 15 milliards ∗ 365.25 ∗ 24 ∗ 60 ∗ 60 ≈ 5.10 17 secondes

3.10 12

2.10 30

0, 83.10 −30

U nivers = (2.10 30 /0, 83.10 −30 ) ∗ 200.10 9 ∗ 3.10 12 ≈ 10 84 electrons

Z

pgcd

b < a

d

a − b

a − 2b

d

a − nb = 0

n = a/b

a

b

a − nb = a

b

G

G :

· a b

¸ 7−→

· 0 1 1 − a

b

¸ · a b

¸

a

b

g = pgcd(a, b)

x

x.b ≡ g (mod a)

pgcd

(a) 522

(b) 453

(c) 522 − 453 = 69 (a − b) (d

453 − 6 ∗ 69 = 39 (b − 6c) (e) 69 − 39 = 30 (c − d) (f ) 39 − 30 = 9 (d − e) (g) 30 − 27 = 3 (e − 3f ) (h) 9 − 9 = 0 (f − 3g)

Z

pgcd

x

y

x × 522 + y × 453 ≡

G

· 3 0

¸

=

· 0 1 1 − 3

¸ · 0 1 1 − 3

¸ · 0 1 1 − 1

¸ · 0 1 1 − 1

¸ · 0 1 1 − 6

¸

· 0 1 1 − 1

¸ · 522 453

¸

=

· 46 − 53

− 151 174

¸ · 522 453

¸

− 151 × 522 + 174 × 453 = 3

V ^deg(P ⁾

X ⁿ − 1

F _q [X]

j − i + ^q−1 ₂

10 ⁹

= 3.10 ⁸

10 ³⁹

U nivers = 15 milliards ∗ 365.25 ∗ 24 ∗ 60 ∗ 60 ≈ 5.10 ¹⁷ secondes

3.10 ¹²

2.10 ³⁰

0, 83.10 ⁻³⁰

U nivers = (2.10 ³⁰ /0, 83.10 ⁻³⁰ ) ∗ 200.10 ⁹ ∗ 3.10 ¹² ≈ 10 ⁸⁴ electrons

O(log ₂ (a + b))

O(log ² ₂ p)

O(log ³ ₂ p)

10 ⁿ

Z ^∗ _n

Z ^∗ _n = { x ∈ N : 1 ≤ x < n

O(log ₂ p. log ² ₂ log ₂ p. log ₂ log ₂ log ₂ p)

Z ^∗ _n

Z /p ^k Z

{ p, 2p, 3p, . . . , p ^k−1 p }

ϕ(p ^k ) = p ^k − p ^k−1 = (p − 1)p ^k−1

p ₁ , . . . , p _n

∀ (k ₁ , . . . , k _n ) ∈ N, ϕ ³Y p ^k _i ⁱ ´

(p _i − 1)p ^k _i ⁱ ⁻¹

Z ^∗ _n

u = x ⁻¹

Z ^∗ _n