# Overview of Cryptography

AI and Robotics

Nov 21, 2013 (4 years and 5 months ago)

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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),

Asymmetric Key:
Public Key Encryption

(Different keys for encryption and decryption)

RSA

Named after Ronald
R
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

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,