Diffie Hellman Key Exchange | Asymmetric Encryption

Asymmetric Encryption-

 

Before you go through this article, make sure that you have gone through the previous article on Asymmetric Key Cryptography.

 

In asymmetric encryption,

• Sender and receiver use different keys to encrypt and decrypt the message.
• The famous asymmetric encryption algorithms are-

 

 

In this article, we will discuss about Diffie Hellman Key Exchange Algorithm.

 

Symmetric Key Cryptography-

 

In symmetric key cryptography,

• Both sender and receiver use a common secret key to encrypt and decrypt the message.
• The major issue is exchanging the secret key between the sender and the receiver.
• Attackers might intrude and know the secret key while exchanging it.

 

Read More- Symmetric Key Cryptography

 

Diffie Hellman Key Exchange-

 

As the name suggests,

• This algorithm is used to exchange the secret key between the sender and the receiver.
• This algorithm facilitates the exchange of secret key without actually transmitting it.

 

Diffie Hellman Key Exchange Algorithm-

 

Let-

• Private key of the sender = X_s
• Public key of the sender = Y_s
• Private key of the receiver = X_r
• Public key of the receiver = Y_r

 

Using Diffie Hellman Algorithm, the key is exchanged in the following steps-

 

Step-01:

 

• One of the parties choose two numbers ‘a’ and ‘n’ and exchange with the other party.
• ‘a’ is the primitive root of prime number ‘n’.
• After this exchange, both the parties know the value of ‘a’ and ‘n’.

 

Step-02:

 

• Both the parties already know their own private key.
• Both the parties calculate the value of their public key and exchange with each other.

 

Sender calculate its public key as- Ys = aXs mod n Receiver calculate its public key as- Yr = aXr mod n

 

Step-03:

 

• Both the parties receive public key of each other.
• Now, both the parties calculate the value of secret key.

 

Sender calculates secret key as- Secret key = (Yr)Xs mod n Receiver calculates secret key as- Secret key = (Ys)Xr mod n

 

Finally, both the parties obtain the same value of secret key.

 

PRACTICE PROBLEMS BASED ON DIFFIE HELLMAN KEY EXCHANGE-

 

Problem-01:

 

Suppose that two parties A and B wish to set up a common secret key (D-H key) between themselves using the Diffie Hellman key exchange technique. They agree on 7 as the modulus and 3 as the primitive root. Party A chooses 2 and party B chooses 5 as their respective secrets. Their D-H key is-

1. 3
2. 4
3. 5
4. 6

 

Solution-

 

Given-

• n = 7
• a = 3
• Private key of A = 2
• Private key of B = 5

 

Step-01:

 

Both the parties calculate the value of their public key and exchange with each other.

 

Public key of A

= 3^{private key of A} mod 7

= 3² mod 7

= 2

 

Public key of B

= 3^{private key of B} mod 7

= 3⁵ mod 7

= 5

 

Step-02:

 

Both the parties calculate the value of secret key at their respective side.

 

Secret key obtained by A

= 5^{private key of A} mod 7

= 5² mod 7

= 4

 

Secret key obtained by B

= 2^{private key of B} mod 7

= 2⁵ mod 7

= 4

 

Finally, both the parties obtain the same value of secret key.

The value of common secret key = 4.

Thus, Option (B) is correct.

 

Problem-02:

 

In a Diffie-Hellman Key Exchange, Alice and Bob have chosen prime value q = 17 and primitive root = 5. If Alice’s secret key is 4 and Bob’s secret key is 6, what is the secret key they exchanged?

1. 16
2. 17
3. 18
4. 19

 

Solution-

 

Given-

• n = 17
• a = 5
• Private key of Alice = 4
• Private key of Bob = 6

 

Step-01:

 

Both Alice and Bob calculate the value of their public key and exchange with each other.

 

Public key of Alice

= 5^{private key of Alice} mod 17

= 5⁴ mod 17

= 13

 

Public key of Bob

= 5^{private key of Bob} mod 17

= 5⁶ mod 17

= 2

 

Step-02:

 

Both the parties calculate the value of secret key at their respective side.

 

Secret key obtained by Alice

= 2^{private key of Alice} mod 7

= 2⁴ mod 17

= 16

 

Secret key obtained by Bob

= 13^{private key of Bob} mod 7

= 13⁶ mod 17

= 16

 

Finally, both the parties obtain the same value of secret key.

The value of common secret key = 16.

Thus, Option (A) is correct.

 

To gain better understanding about Diffie Hellman Key Exchange Algorithm,

Watch this Video Lecture

 

Next Article- Digital Signatures

 

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