PUBLIC KEY CRYPTOGRAPHY
ALGORITHM
Concept and Example
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RSA
1. It’s a block cipher algorithm.
2. Plain text and cipher text are integer
between 0 to n-1 for some n.
3. RSA algorithm involve the following
operations
1. Key Generation.
2. Encryption/Decryption
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RSA - Key Generation
• Each user generates a public/private key pair by:
1. selecting two large primes at random: p, q & p<> q
2. computing their system modulus n=p.q
3.
Calculate ø(n)=(p-1)(q-1)
‫ والقاسم‬n ‫ عدد األرقام الموجبة التي اقل من‬ø(n) is Euler Totient :
. 1 ‫المشترك بينها هو‬
4. selecting at random the encryption key e
– where 1<e<ø(n), gcd(e,ø(n))=1
5. solve following equation to find decryption key d
– e.d mod ø(n) = 1 and 0≤d≤n
6. publish their public encryption key: PU={e,n}
7. keep secret private decryption key: PR={d,n}
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–
RSA – Encryption/ Decryption
• to encrypt a message M the sender:
– obtains public key of recipient PU={e,n}
– computes: C = Me mod n, where 0≤M<n
• to decrypt the ciphertext C the owner:
– uses their private key PR={d,n}
– computes: M = Cd mod n
• Both Sender and Receiver know the value of
n, e.
• Only receiver knows d.
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Example
• Plain Text =88;
Steps
5
Values
1. Select two prime numbers
p=17 , q = 11
2. Calculate n = pq
N = 17 ×11 = 187
3. Calculate ø(n)=(p-1)(q-1)
ø(n)= 16 × 10 =
160
4. Select e such that e is relatively
prime to ø(n) = 160
e= 7
5. Determine d such that de mod
160 = 1
d = 23 , because 27 ×7 =
161 = ( 1×160) +1
Public key
{7, 187}
Private Key
{23, 187}
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Example
• Plain Text =88;
sample RSA encryption/decryption is:
given message M = 88 (nb. 88<187)
encryption:
C = 887 mod 187 = 11
decryption:
M = 1123 mod 187 = 88
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1. Diffie-Hellman Key Exchange
• The purpose is to share a secrete key securely
and use it in the encryption.
• Diffie- Hellman is using discrete logarithm
•
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1. Diffie-Hellman Key Exchange:
Discrete Logarithm
• Primitive root of a
prime number p is a
number whose power
generate all integers
from 1 to p-1 .
• Example : p = 7 , then a = 3
Power
– A mod p , a^2 mod p , a
^3 mod p .. Contain all
numbers from 1 to p-1
0
3^0 mod 7 = 1
1
3^ 1 mod 7 = 3
2
3^2 mod 7 = 2
3
3^3 mod 7 = 6
4
3^4 mod 7 = 4
5
3^5 mod 7 = 5
……
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Number
2. Diffie Hellman Setup
1. all users agree on global parameters:
– large prime integer q
– a being a primitive root mod q
2. each user generates their key
– User A
• chooses a secret key (number): xA < q
xA
• compute their public key: yA = a mod q
– User B
• chooses a secret key (number): xB < q
xB
• compute their public key: yB = a mod q
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2. Diffie Hellman Setup
3. Generation of Secret Key
xA
User A : K = (yB) mod q
xB
User B : K = (yA) mod q
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2. Diffie Hellman Key Exchange
• shared session key for users A & B is KAB:
xB
KAB = yA mod q (which B can compute)
xA
= yB mod q (which A can compute)
• KAB is used as session key in private-key encryption
scheme between Alice and Bob
• if Alice and Bob subsequently communicate, they will
have the same key as before, unless they choose
new public-keys
y
A
A
B
yB
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Man in the Middle Attack
1.
2.
3.
4.
5.
6.
7.

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Darth prepares by creating two private / public keys
Alice transmits her public key to Bob
Darth intercepts this and transmits his first public key to Bob.
Darth also calculates a shared key with Alice
Bob receives the public key and calculates the shared key (with
Darth instead of Alice)
Bob transmits his public key to Alice
Darth intercepts this and transmits his second public key to Alice.
Darth calculates a shared key with Bob
Alice receives the key and calculates the shared key (with Darth
instead of Bob)
Darth can then intercept, decrypt, re-encrypt, forward all
messages between Alice & Bob
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