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Diffie-Hellman Key Exchange

Last modified: 2023-11-05

Diffie-Hellman key exchange is a mathematical method for exchaning cryptographic key securely. The goal is to securely communicate with each other while preventing man-in-the-middle attack. It is also used for cryptographic algorithms such as AES, DES.

Overview

DH calculates shared secrets with public keys and secret keys. Below is a Python script to calculate shared secrets for A and B. These (ss_A and ss_B) should be the same value with each other.

# Calculate public keys (A, B)
# p: public key (a prime number)
# g: public key (a primitive root modulo `p`)
# a, b: secret key
A = pow(g, a, p)
B = pow(g, b, p)

# Calculate shared secret
ss_A = pow(A, b, p)
ss_B = pow(B, a, p)

# Shared secrets should be the same value each other
print(ss_A == ss_B) # True
print(pow(pow(g, a, p), b, p) == pow(pow(g, b, p), a, p)) # True

Key Exchange Flow using OpenSSL

Reference: https://tryhackme.com/room/cauldron#

Assume that Alice and Bob want to communicate each other secretly. In such a situation, we can use the Diffie-Hellman key exchange.

1. Generate DH Params for Key Agreement

First of all, we need to generate Diffi-Hellman parameters which include two prime numbers p and g. These parameters are used for calculating shared secret for Alice and Bob.
We can use openssl with dhparam option for doing that.

# 2048: 2048-bit
openssl dhparam -out dhparams.pem 2048

For more robustness, we can use more longer bit length such as 4096 instead of 2048, but it requires more computational power and time.

2. Generate Private Keys

Next, generate private keys for both Alice and Bob. We can use genpkey option of openssl for doing that.

# Alice
openssl genpkey -paramfile dhparams.pem -out alice_private.pem

# Bob
openssl genpkey -paramfile dhparams.pem -out bob_private.pem

3. Generate Public Keys

Using the private keys, we also generate public keys.

# Alice
openssl pkey -in alice_private.pem -pubout -out alice_public.pem

# Bob
openssl pkey -in bob_private.pem -pubout -out bob_public.pem

4. Generate Shared Secret

This time, Alice and Bob generate shared secrets using each other’s public key. Each shared secret will be the same (alice_shared_secret == bob_shared_secret).

# Alice
openssl pkeyutl -derive -inkey alice_private.pem -peerkey bob_public.pem -out alice_shared_secret.bin

# Bob
openssl pkeyutl -derive -inkey bob_private.pem -peerkey alice_public.pem -out bob_shared_secret.bin

By doing this, it’ll be difficult for an eavesdropper to recreate shared secrets unless he knows Alice and Bob’s private keys because.

5. Use Shared Secret for Secure Communication

Now Alice and Bob can use this shared secret with cryptographic algorithm such as AES.
Assume Alice encrypts hello.txt which contains greeting message and sends the encrypted file to Bob. Then Bob decrypts the encrypted file with the same cryptographic algorithm and read the message.

# Encryption: Alice encrypts `hello.txt` with AES.
openssl enc -aes-256-cbc -pass file:alice_shared_secret.bin -in hello.txt -out encrypted.enc

# Decryption: Bob decrypts the encrypted `encrypted.enc` with the same algorithm.
openssl aes-256-cbc -d -in hello_encrypted.enc -pass file:bob_shared_secret.bin -out decrypted.txt

Unless the shared secret is published or stolen, their secret messages will never be seen by eavesdroppers.


Decrypt Secret Message using Private Key and Public Key

If we have a private key and a public key with some reason, we can decrypt a secret message by recreating a shared secret using these keys.

1. Generate Shared Secret

Using a private key and a public key, we can generate a shared secret which is used for decrypting an encrypted message.

openssl pkeyutl -derive -inkey private.key -peerkey public.key -out shared_secret

2. Decrypt a Secret Message

Now we can decrypt a secret message using the shared secret.

openssl aes-256-cbc -d -in encrypted.enc -pass file:shared_secret -out decrypted.txt

References