Public Key
Cryptography
Bryan Pearsaul
Outline
•
What is Cryptology?
•
Symmetric Ciphers
•
Asymmetric Ciphers
•
Diffie

Hellman
•
RSA (Rivest/Shamir/Adleman)
•
Moral Issues
Outline
•
Summary
•
References
What is Cryptology?
•
The science of keeping data secure
•
Two transformation algorithms:
Enciphering and Deciphering
•
Symmetric ciphers
•
Asymmetric ciphers
Symmetric Ciphers
•
Also known as private key
•
Both parties must agree on the key
in advance
•
D_K(E_K(P)) = P
•
Not very computationally intensive
•
Key must be securely sent to both parties
Symmetric Cipher Example
•
k
= 4
Enciphering
E
E_K(X)
Deciphering
D
D_K(E_K(X)) = X
X
K
•
Turn plaintext SECRET into
ciphertext
•
S+4=W, E+4=I, C+4=G, R+4=V, E+4=I,
T+4=X
Symmetric Cipher Example
•
Much more elaborate transformations
are available
•
Some that are so complicated that
even if the transformation was
public a key would still be needed
•
Still require a distributed key
Asymmetric cipher
•
Also known as public key
Enciphering
E
E_K(X)
Deciphering
D
D_K’(E_K(X)) = X
X
K’
K
•
Two keys: public
k
, private
k’
•
Private key not required for both
parties
•
More computationally intensive
Diffie

Hellman
•
One of the first public key
cryptographic systems
•
Developed by Martin Hellman, Ralph
Merkle, and Whitfield Diffie at
Stanford University in 1976
Diffie

Hellman
•
Based on a special case of the
subset

sum, or knapsack, problem
Subset

sum Problem
5
8
4
11
6
20
Diffie

Hellman Example
•
Block cipher
•
Block size of 7 bits. Possible 2
7
combinations
•
Private key (
a’
1
,
a’
2
, … ,
a’
n
) of 7 integers: (1, 2, 5, 11, 32, 87, 141)
•
Chose two special integers,
w
and
m
,
such that
w
and
m
are relatively prime,
meaning gcd(
w
,
m
) = 1:
w
= 901,
m
= 1234
•
Public key (
a
1
,
a
2
, … ,
a
n
)
of 7 integers using the equation:
a
i
=
w
*
a’
i
mod
m
:
(901, 568, 803, 39, 450, 645, 1173)
•
Partition SECRET into 7 bit blocks each block consisting of x
n
bits (
x
1
,
x
2
, …,
x
n
)
S
1010011
E
1000101
C
1000011
R
1010010
E
1000101
T
1010100
•
B
x
=
∑
x
i
a
i
i=1
n
•
S = 1 X (901) + 0 X (568) + 1 X (803) + 0 X (39) + 0 X (450) + 1 X (645) + 1 X (1173)
•
S = 3522
Diffie

Hellman Example
•
Encrypted blocks B
x
received. Special version of subset

sum problem
•
Which subset of (
a’
1
,
a’
2
, … ,
a’
n
) sums to
B’
x
where
B’
x
=
B
x
*
w

1
mod
m
•
w

1
is the modular inverse of
w
for
m
,
w
*
w

1
mod
m
= 1
•
B’
x
= 3522 X (901)

1
mod 1234
•
B’
x
= 3522 X 1171 mod 1234
•
B’
x
= 234
1.
sum
←
0
2. for
i = n
step

1 until 1 do
if
a
i
+
sum
<=
B’
x
then
sum
←
sum
+
a
i
;
subset(i)
←
1
else subset(i)
←
0
3. if sum =
B’
x
then exit with subset
else exit with “failure”
•
Private key (1, 2, 5, 11, 32, 87, 141),
B’
x
= 234, find subset (1, 0, 1, 0, 0, 1, 1) = S
Diffie

Hellman
•
An algorithm that solves the
particular problem on which a
cryptographic system is based.
•
An algorithm which solves NP

complete problems quickly
•
Two possible points of vulnerability
RSA
•
Factorization so far is unsolvable in
polynomial

time
•
Based on the difficulty of factoring
large numbers
•
Developed by Ron Rivest, Adi
Shamir, and Leonard Adleman at
MIT in 1977.
RSA Example
•
Find two large prime integers,
p
and
q
, and form product
n
=
pq
•
Find a random integer,
e
, that is relatively prime to Ф(n) = (
p

1)(
q

1)
•
p
and
q
are kept private, (
n
,
e
) are the public key
•
Message is partitioned into blocks,
b
, such that
b
<
n
•
Each block is encrypted using the equation:
c
=
b
e
mod
n
•
For the private key, calculate integer
d
which is the modular inverse of
e
for Ф(n), or
e
*
d
mod Ф(n) = 1
•
Once
d
is calculated it becomes your private key and all records of
p
and
q
should be destroyed
•
Each encrypted block,
c
, is decrypted using the equation:
b
=
c
d
mod
n
•
p = 61, q = 53, n = 3233, Ф(n) = 3120, e = 17, d = 2753
•
encrypt(123) = 123
17
mod 3233 = 855
•
decrypt(855) = 855
2753
mod 3233 = 123
RSA
•
Factorization cannot be done in
polynomial

time
•
Factoring is required to break the
system
•
Security of RSA relies on two
assumptions
Moral Issues
•
Information Theft
•
Who does the data belong to?
•
Privacy
Summary
•
Diffie

Hellman and RSA
•
Symmetric and Asymmetric ciphers
–
Pros and Cons
•
Cryptology
•
Moral Issues
References
•
A. Shamir, “A Polynomial

Time Algorithm for Breaking the Basic Merkle

Hellman
Cryptosystem", A
dvances in Cryptology

CRYPTO '82 Proceedings
, pp. 279

288,
Plenum Press, 1983.
IEEE Transactions on Information Theory
, Vol. IT

30, pp. 699

704, 1984.
•
A.K. Dewdney,
The New Turning Omnibus
, pp. 250

257, Henry Holt and Company,
2001.
•
RSA Cryptosystem, http://primes.utm.edu/glossary/page.php?sort=RSA.
•
Cryptology FAQ, http://www.faqs.org/faqs/cryptography

faq/part06/.
•
The Extended Euclidian Algorithm,
http://www.grc.nasa.gov/WWW/price000/pfc/htc/zz_xeuclidalg.html.
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