Tool to calculate/simulate key exchanges according to the Diffie-Hellman protocol based on mathematics and modular arithmetic.

Diffie-Hellman Key Exchange - dCode

Tag(s) : Modern Cryptography, Arithmetics

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The Diffie-Hellman key exchange is a mathematical/cryptographic protocol allowing 2 people (who may have never met) to agree on a secret number (a shared secret key), without disclosing it during their exchanges (i.e. say that a person who could monitor the exchanges could not deduce the secret number). This number can then be used as an encryption key to encrypt and decrypt messages/information between the two interlocutors

Two people Alice and Bob come into contact and choose 2 numbers, a prime number $ P $ and a number $ G $ (with $ P > G $). This choice can be communicated in plain text and made public.

Alice chooses a number $ a $ at random, called the private key (kept secret), and performs the calculation $ A = G^a \mod P $ whose value $ A $ is called Alice's public key, qu he sends to Bob publicly.

Similarly, Bob chooses a random number $ b $, called the private key (also kept secret), and performs the calculation $ B = G^b \mod P $ whose value $ B $ is called the public key from Bob, which he sends to Alice publicly.

Alice received the value $ B $ and can then calculate the value $ S = B^a \mod P $

Similarly, Bob who received the value $ A $ can calculate the value $ S = A^b \mod P $

Thanks to math (and modular arithmetic), the $ S $ value is the same for Alice and Bob, it's their shared secret key. They can then communicate by encrypting their messages with this key.

The publicly exchanged values ($ P $, $ G $, $ A $ and $ B $) do not allow to calculate $ S $ as long as the 2 private keys $ a $ and $ b $ remain hidden and protected by their owners.

P = | 101 |

G = | 12 |

a = | 123 |

b = | 345 |

A = | G^a%P = 35 |

B = | G^b%P = 60 |

S = | B^a%P = A^b%P = 62 |

The main advantage of DH is to allow secure key exchange over an insecure channel.

The second advantage is the simplicity of the implementation of the algorithm.

The DHKE (Diffie-Hellman Key Exchange) protocol is vulnerable to several types of attacks:

— Man-in-the-middle attacks: an attacker intercepts the communication of the 2 parties and pretends to be the other party.

— Attack by reflection: an attacker sends a fake message asking to perform a new key exchange with himself, authentication of the parties is therefore preferable.

— Attack by precalculation/factorization: private keys are generally less than 1024 bits, precalculation of combinations with low values is possible but very costly in time and resources.

When P is a prime number, mathematical calculations are more secure. The group of integers modulo P has better properties if P is prime.

However, it is possible to use a non-prime P, but in this case, a person knowing the factorization will be able to break Diffie-Hellman.

As their name suggests, the keys are private, they are never shared publicly.

Knowing the public key does not allow calculating the private key, this is a famous mathematical problem (known as the discrete logarithm problem).

Numbers/keys can be created with a random number generator.

It is preferable to change the private key with each new communication.

Whitfield Diffie and Martin Hellman presented their method in 1976

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Cite as source (bibliography):

*Diffie-Hellman Key Exchange* on dCode.fr [online website], retrieved on 2024-09-13,

- Secret Shared Key Calculator
- What is Diffie-Hellman Key Exchange? (Definition)
- How does Diffie-Hellman Key Exchange work?
- What are the Diffie-Hellman forces? (Advantages)
- What are the weaknesses of Diffie-Hellman? (Disadvantages)
- Why the number P must be prime?
- How to find out the private keys?
- When was Diffie-Hellman Key Exchange invented?

diffie,hellman,key,exchange,secret,public,dhke

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