# 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- ## 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.

## 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 = Xs
• Public key of the sender = Ys
• Private key of the receiver = Xr
• Public key of the receiver = Yr

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.

## 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

= 3private key of A mod 7

= 32 mod 7

= 2

Public key of B

= 3private 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

= 5private key of A mod 7

= 52 mod 7

= 4

Secret key obtained by B

= 2private 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

= 5private key of Alice mod 17

= 54 mod 17

= 13

Public key of Bob

= 5private 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

= 2private key of Alice mod 7

= 24 mod 17

= 16

Secret key obtained by Bob

= 13private 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

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Summary Article Name
Diffie Hellman Key Exchange | Asymmetric Encryption
Description
Diffie Hellman Key Exchange Algorithm enables the exchange of secret key between sender and receiver. Diffie Hellman Key Exchange Example. Diffie Hellman Key Exchange is an asymmetric encryption technique.
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Gate Vidyalay
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