Overview of Cryptography

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Overview of Cryptography

Oct. 29, 2002

Su San Im

CS Dept. EWU

Contents


Cryptography



Encryption/Decryption Methods



Encryption/Decryption Protocols

Cryptography


Description: The art and science of keeping
messages secure by altering or transforming them

m: Plaintext

Encryption

c: Ciphertext

Decryption

Original

Plaintext

Key


Key

Criteria of Good Cryptography

Confidentiality


Can decrypt only with a secret key


Authentication


Identify the person at the other end of the line


Integrity


No change during transit (message authentication) &



detecting the loss of integrity

Nonrepudiation


Know who sent the message &




Documented proof of identity of sender





Encryption Methods




Symmetric Key:
Secret Key Encryption



(Same key for encryption and decryption)


e.g.: DES(Data Encryption Standard),



AES(Advanced Encryption Standard)







Asymmetric Key:
Public Key Encryption




(Different keys for encryption and decryption)


e.g.: RSA(Rivest Shamir Adleman)

RSA


Named after Ronald
R
ivest, Adi
S
hamir, Leonard
A
dleman


Public Key: n, e such that



1. n=p


q



2. e is relatively prime to (p
-
1)

(q
-
1)



3. p and q are prime numbers which remain secret


Private Key: n, d and d is kept secret









=>
1

= (e

d) mod


Encryption: c =


Decryption: m =

))
1
)(
1
((


q
p
))
1
)(
1
mod((
1




q
p
e
d
n
m
e
mod
n
c
d
mod
Example: RSA


n=3337 (p=47 and q=71, 47
∙71=3337
)


Choose e =79


Let m=688 be the message




d=1019 (


find x 1=(79


x) mod (46


70=3220) )


c=688 mod 3337 = 1570 => Encrypted message


m=1570 mod 3337 = 688 => Decrypted message

79
1019
Encryption/Decryption Protocols

M

M, K

CK

CK

CK

CM,

K

M

H

H

No|Yes

H

S

S

start

a

b

c

d

e

f

g

h

j

k

l

m

n

n

In this chart, boxes contain information, and paths denote activity working with or changing the information.
Initially, Alice has a message M that she wishes to send signed to Bob, via a security protocol.

a.
Alice generates a random key K for DES encryption.

b.
Alice hashes M to create H.

c.
Alice encrypts the key K with Bob’s public key to create CK

Encryption/Decryption Protocols

M

M, K

CK

CK

CK

CM,

K

M

H

H

No|Yes

H

S

S

start

a

b

c

d

e

f

g

h

j

k

l

m

n

n

d. Alice encrypts M using DES with key K to create CM.

e. Alice encrypts the hash H with her private key to create signature S.

f. Alice sends the encrypted form CK of the key K to Bob.

g. Alice sends the encrypted form CM of the message M to Bob.

h. Alice sends her “signature”, the encrypted form S of the hash H, to
Bob.

Encryption/Decryption Protocol

M

M, K

CK

CK

CK

CM,

K

M

H

H

No|Yes

H

S

S

start

a

b

c

d

e

f

g

h

j

k

l

m

n

n

j. Bob uses his private key to decrypt CK to recover the key K.

k. Bob uses K to decrypt CM to recover the message M.

l. Bob uses Alice’s public key to decrypt her signature S to recover the
hash H.

m. Bob hashes M to create his own version of the hash H.

n. Bob compares for equality his version of the hash H with the version
decrypted from Alice’s signature.

Public Key
Encryption/Decryption Protocols

Start with a letter


s

Convert to a number


19

Encrypt
(
public key

of 3)


39

Decrypt
(
private key

of 27)


19

Convert to a letter


s

Public Key
Encryption/Decryption Protocols


Encryption:


n = 55, e = 3, p = 5, q = 11


Let m = 19




Decryption:




3d = 1 mod 40


1= (3d) mod 40


d = 27


m =






= 584,064 mod 55



= 19

3
39
55
mod
6859
55
mod
19
3



c
))
1
11
(
)
1
5
mod((
1
3





d
55
mod
)
39
39
39
39
(
55
mod
39
2
8
16
27




55
mod
)
39
36
26
16
(




Digital Signature



Author authentication




Message authentication


-

Assures recipients that



the message was not altered in transit (integrity)




Backward of Public Key Encryption & Decryption Processes


Use Private Key to encrypt




Public Key to decrypt

Mathematical Background


Information Theory: How to convey info.






through number


Complexity Theory: How complex it is



Ex) O(n)


Number Theory: Find properties, patterns, and




relationships of numbers.



Ex) Prime Test


Probability, Statistics: How to make it secure

Number Theory(Why Prime?)


Prime Number: 1 and itself as factors


When prime numbers are large enough,
they're nearly impossible to factor the prime
numbers into p and q.


Number Theory(Theorems)



Fermat’s Little Theorem


if 0<m < p,


p: prime


Then





Euler’s Theorem


if n = p
∙ q

p,q : prime


and if 0<m<n<p


Then

1
mod
1


p
m
p
1
mod
)
1
)(
1
(



n
m
q
p
(so


m
n
m
q
p
k




mod
1
)
1
)(
1
(
)


m
m
m
m
m
m
m
k
k
q
p
q
p
k
ed
d
e












1
)
(
)
(
)
1
)(
1
(
1
)
1
)(
1
(
References


Bruce Schneier,
APPLIED CRYPTOGRAPHY:
Protocols, Algorithms, and Source Code in C (2
nd

Eds),

John Wiley & Sons, 1996. (ISBN 0
-
471
-
12845
-
7)


Bruce Schneier,
SECRETS AND LIES: Digital
Security in a networked world,

John Wiley &
Sons, 2000. (ISBN 0
-
471
-
25311
-
1)


H.M. Mel and Doris Baker,
CRYPTOGRAPHY
DECRYPTED,

Addison
-
Wesley, 2001. (ISBN 0
-
201
-
61647
-
5)

Thank you for your attention.