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Nitya Nigam Internet privacy is an important and relevant issue in today’s world, especially given recent moves by world governments to restrict the influence of foreign technology companies in their countries. This is why Integral has partnered with Bytesized Privacy to give our readers insight into encryption, one of the most effective ways to ensure private data stays private. This article will cover the maths behind the Diffie-Hellman key exchange, which is a secure and widely used method of public key cryptography. Bytesized has a post about the applications as well as the pros and cons of end-to-end encryption that you can find here. We hope you find this collaboration useful and interesting. Let us know in the comments if you’d like to see more posts like this!
Public key cryptography is a type of encryption where each person has a pair of cryptographic keys: a public key and a private key. The private key is kept secret, but the public key can be used by anybody. The Diffie-Hellman key exchange uses public and private keys to give both sender and receiver a shared key that can be used in future communications, as the process of using both public and private keys for every message is computationally expensive. The maths behind this key exchange is based in modular arithmetic, where integers that have the same remainder when divided by a set integer are considered the same, or “congruent”. We can notate this mathematically by saying that, for example, 6 ≡ 1 mod 5, where the ≡ symbol means “congruent to”, mod means modulo (x mod y means x is the remainder when a number is divided by y) and 5 is the set integer we are using as the base of this system. The Diffie-Hellman key exchange involves 5 steps, outlined below. Let our two people who want to communicate with each other be called Alice and Bob.
Alice and Bob now have a shared key, s, that they can use for secure communication, that is unknown to all other users. This method works because B ≡ g^b mod p and A ≡ g^a mod p, so B^a ≡ g^ba mod p and A^b ≡ g^ab mod p, which are equivalent. This method is secure because for large p, a and b, reversing the modular exponentiation is near-impossible via a brute force attack, because there are many different possible values of base and exponent that lead to the same value after reducing the result mod p. g is usually quite small, as most primitive roots are small integers like 2 and 3. g being small does not compromise the security of the system.
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