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離散數學
›
Ch1 集合與基礎數論
第 14 題／共 15 題
◀
LS 14/15
錯題回報
00:00
14. Bezout's Identity、Modular Inverse、Chinese Remainder Theorem、RSA
#LS-01-014
中
Bezout's Identity
Modular Inverse
Chinese Remainder Theorem
RSA
📝
☆
Which of the following statements about integers are incorrect?
A
If
gcd
(
a
,
b
)
=
1
\gcd(a, b) = 1
g
cd
(
a
,
b
)
=
1
, then for any non-zero integer
n
n
n
, there is a pair of integers
p
p
p
and
q
q
q
such that
p
a
+
q
b
=
n
pa + qb = n
p
a
+
q
b
=
n
.
B
The inverse of
p
p
p
module
q
q
q
exists only when
gcd
(
p
,
q
)
=
1
\gcd(p, q) = 1
g
cd
(
p
,
q
)
=
1
and
q
>
1
q > 1
q
>
1
.
C
The system
{
x
≡
a
1
(
m
o
d
m
1
)
x
≡
a
2
(
m
o
d
m
2
)
⋮
x
≡
a
n
(
m
o
d
m
n
)
\begin{cases} x \equiv a_1 \pmod{m_1} \\ x \equiv a_2 \pmod{m_2} \\ \quad\vdots \\ x \equiv a_n \pmod{m_n} \end{cases}
⎩
⎨
⎧
x
≡
a
1
(
mod
m
1
)
x
≡
a
2
(
mod
m
2
)
⋮
x
≡
a
n
(
mod
m
n
)
has a unique solution modulo
m
m
m
, where
m
i
m_i
m
i
are primes and
m
=
m
1
m
2
⋯
m
n
m = m_1 m_2 \cdots m_n
m
=
m
1
m
2
⋯
m
n
.
D
If
p
p
p
is prime, then for every integer
a
a
a
we have
a
p
−
1
≡
1
(
m
o
d
p
)
a^{p-1} \equiv 1 \pmod p
a
p
−
1
≡
1
(
mod
p
)
.
E
In RSA, if Alice wants to send a secret message to Bob, Alice uses Bob's private key to encrypt the message and then send the ciphertext message to Bob. After Bob receives the ciphertext, Bob can use his public key to decrypt the ciphertext.
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