Math 135: Lecture 16: RSA
Public Key Systems
•
In a
each pair of users must possess the same key.
•
In a
keys are divided into two parts.
•
A public encryption key is shared in a repository.
•
A private decryption key is held secretly by each participant.
•
For user
A
to send a private message to user
B
,
A
would look up
B
’s public key, encrypt
the message and send it to
B
. Since
B
is the only person who possesses the secret key
required for decryption, only
B
can read the message.
Key Distribution
•
How do you manage keys among 200 embassies?
•
In a private key system, users must exchange
•
In a public key system, users must
•
The key distribution problem is solved.
RSA
•
The possibility of public key cryptography was first published in 1976 in a paper by Di
ﬃ
e,
Hellman and Merkle.
•
The RSA scheme, named after its discoverers Rivest, Shamir and Adleman is an example
of a commercially implemented public key scheme.
Messages Are Integers
•
In RSA,
•
How does one get an integer from plaintext?
•
One possibility: Do what we did with a Vigen`
ere cipher, assign a number to each letter of
the alphabet and then concatenate the digits together.
Example 16.1.
With
A
↔
00
, B
↔
01
, . . .
MATH
↔
1
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Square and Multiply Algorithm
1.
To compute
M
e
(mod
n
)
for large
e
, write
e
in binary as
e
=
(
r
t
...r
2
r
1
r
0
)
2
where each
r
i
= 0
OR
1
.
2.
Compute
M, M
2
, M
4
, M
8
, ..., M
2
t
−
1
, M
2
t
(
mod
n
)
by squaring
the previous term in the sequence.
3.
Multiply the appropriate terms together, modulo n, to obtain
M
e
≡
Π
M
2
i
(
mod
n
)
.
Eg R1
Use the square and multiply algorithm to compute
2
29
(mod 187)
.
1.
29 =
16 + 4 + 2 + 1
= (
11101
)
2
.
2. Compute
2
,
2
2
,
2
4
,
2
8
,
2
16
(mod 187)
.
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 Fall '08
 ANDREWCHILDS
 Math, Cryptography, Prime number, Alice, decryption key

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