n - Maths Langella

103
Congruences dans , anneau /n . Applications. Système RSA (Terracher) Monier
De Biasi
Sorosina
Terracher
I. Congruences.
[
]
a b n
≡
[
]
[
]
22 14 4 ; 13 8 4
≡ ≡
/
[
]
n
≡
[
]
n
≡
ˆ
x
{
}
ˆ/x x n
λ λ = + ∈
{
}
ˆ ˆ
/ 0,1,..., 1
n n
= −
[
]
[ ]
[
]
[ ]
a b n a c b d n
c d n ac bd n
≡ + ≡ +
⇒
≡ ≡
[
]
n
≡
[
]
[
]
(
)
k k
a b n a b n
≡⇒≡
0 1
, ,...,
n
x x x
0
n
x
≠
1
1 1 0
0, , 0, 1 ... /
n n
i n n
i n x b a x b x b x b x
−
−
∀ ∈ ∈ − = + + + +
•          
•          
•         
II. Anneau  /n .
, /
x y n
∈
/
x y n
+ ∈
. /
x y n
∈
+
.
(
)
/ , ,.
n
+
1
a b
∧ =
/ / /
ab a b
×
[
]
[ ]
( )
x a
S
x b
α
β
≡
≡
[
]
[ ]
7 8
( )
1 13
x
S
x
≡
≡
ˆ
x
x
∈
1
x n
∧ =
1, 2, 4, 5, 7, 8
(
)
n
ϕ
1
a b
∧ =
(
)
(
)
(
)
ab a b
ϕ ϕ ϕ =
( )
1
1
r r
p p
p
ϕ = −
1
i
N
i
i
r
n p
=
=
∏
( )
1
1
1
N
i
i
n n
p
ϕ
=
= −
∏
( )
[
]
Si 1, 1
n
a n a n
ϕ
∧ = ≡
n
|
n a
/
[
]
1
1
n
a n
−
≡

103
Congruences dans , anneau /n . Applications. Système RSA (Terracher) Mon, Bia, Sor, Ter.
III. Applications à la cryptographie.
{
}
{
}
[ ]
0,1,..., 25 0,1,..., 25
:
( ) 17 22 26
f
x f x x
→
≡ +
[
]
1
1; ( 1)( 1) et ( 1)( 1) e p q e p q
=
∈ − − ∧ − −
(
)
(
)
[
]
! 1; ( 1)( 1) et 1 1 1
d p q ed p q∃ ∈ − − ≡ − −
[
]
,ed
m m m n
∀ ∈ ≡
[
]
1
1; ( 1)( 1) tq ( 1)( 1) e p q e p q
=
∈ − − ∧ − −
1,
c n
∈
[
]
e
m c m
≡
[
]
d
c n
IV. Notes.
[
]
a b n
≡
(
)
b a n
λ
− =
1
1 1 0
10 10 ... 10
n n
n n
a x x x x
−
−
= + + + +
[
]
[
]
10 0 2 5
k
et
≡
[
]
[
]
0
2 5
a x et
≡
[
]
[
]
10 1 3 9
et
≡
[
]
[
]
*,10 1 3 9
k
k et
∀ ∈ ≡
[
]
[
]
1 0
... 3 9
n n
a x x x ou
−
≡ + + +
[
]
10 1 11
≡ −
( )
[ ]
*,10 1 11
k
k
k∀ ∈ ≡ −
( ) ( )
[ ]
1 0
1
... 11
1 1
n n
n n
a x x x
−
−
≡ + + + − −
[
]
' 9
p rr≡
(
)
n
cl x
( )
/
n
n
x cl x
→
(
)
n
cl x
( ) ( ) ( )( )
/ / /
:;
ab a b
ab a b
f
cl x cl x cl x
ξ
→ ×
=
f
x
ξ
(
)
{
}
/
ab
y cl x
z x z ab
ξ ∈ = = ∈ − ∈
(
)
{
}
/
a
cl x
z x z a
= ∈ − ∈
( )
(
)
(
)
, i.e.
a a a
y cl x
cl x cl y
∈ =
(
)
{
}
/
b
cl x
z x z b
= ∈ − ∈
(
)
(
)
b b
cl x cl y
=
f
f
(
)
(
)
(
)
(
)
(
)
1 1 ; 1
ab a b
f cl cl cl=
f
f
f
(
)
(
)
(
)
(
)
(
)
0 ; 0
ab a b
f cl x cl cl=
1
a b
∧ =
f
1
a b
∧ =
/ 8 / 2 / 4
×
/

103
Congruences dans , anneau /n . Applications. Système RSA (Terracher) Mon, Bia, Sor, Ter.
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( ) ( )( )
; ;
( ) ;
et surjective assure .
a b a b
a b
S cl x cl x cl cl
f x cl cl
f x
α β
α β
⇔ = ⇔
=
∃
/
n
ξ ∈
/
n
ζ ∈
1
ξζ
=
1
x y
=
1
xy kn
− =
1
xy kn
− =
1
x n
∧ =
1
x n
∧ =
1
xu nv
+ =
ξ
1 0
xu nv xu nv xu nv u u
ξ ξ
= + = + = + = + =
ξ
f
f
( ) ( ) ( )
2
, * , 1 ( ) ( ) ( )
a b a b f ab f a f b
∀ ∈ ∧ = ⇒ =
(
)
(
)
{
}
/ * / , 1
n
n cl x x x n
= ∈ ∧ =
(
)
(
)
/ *
n Card n ϕ =
(
)
( ). ( )
ab a b
ϕ ϕ ϕ =
(
)
{
}
{ } { }
1, 1, / 1
1, / 1
r r r
r r r
Card k p k p k p
p Card k p k p
Card
ϕ
∈ ∈ ∧ =
/
= ∈ ∧ =
= −
{
}
1,
r
r
Card k p
p
∈
=
1,
r
k p
∈
{
}
1
1 | / 1
r r
k p p k k qp q p
−
∧ = ⇔ ⇔ ∈ ≤ ≤
/
{
}
1
1
/ 1
r
r
qp q p
Card p
−
−
≤ ≤ =
(
)
1
r r r
p p p
ϕ
−
= −
1
i
N
i
i
r
n p
=
=
∏
( )
1
1
1
N
i
i
n n
p
ϕ
=
= −
∏
( )
[
]
Si 1, 1
n
a n a n
ϕ
∧ = ≡
1
a n
∧ =
( )
[ ]
1
n
a n
ϕ
≡
n
|
n a
/
[
]
1
1
n
a n
−
≡
(
)
1
n n ϕ
= −
0
0
ab ab n
= = =
0
a
≠
0
b
≠