Elliptic_Diffie_Hellman

Elliptic Diffie-Hellman Key Exchange

Elliptic Diffie-Hellman Key Exchange is a Diffie-Hellman key exchange based on elliptic curve over a finite field.

Note

The security is based on the Elliptic Curve Discrete Logarithm Problem (ECDLP) and Elliptic Curve Diffie-Hellman Problem

Prerequisite

• Basic knowledge of elliptic curves
  • An Introduction to Mathematical Cryptography (Second edition)
  • Cryptography Standford Education

Algorithm

Public parameter creation

Trusted party chooses and publishes:

• Large prime $p$;
• Elliptic curve $E$ over $\mathbb{F}_p$;
• Point $P$ in $E(\mathbb{F}_p)$.

Private computations

Alice	Bob
Chooses a secret integer $n_A$. Computes $Q_A \equiv n_AP$.	Chooses a secret integer $n_B$. Computes $Q_B \equiv n_BP$.

Alice's private key is $n_A$ and her public key is $Q_A$.

Bob's private key is $n_B$ and his public key is $Q_B$.

Public exchange of values

Alice sends $Q_A$ to Bob.

Bob sends $Q_B$ to Alice.

Note

The public exchange can be with only the $x_a$ and $x_b$ value of the points $Q_A$ and $Q_B$.

Shared secret computations

Alice	Bob
Computes the number $n_AQ_B$.	Computes the number $n_BQ_A$.

The shared secret value is $n_AQ_B \equiv n_A(n_BP) \equiv n_B(n_AP) \equiv n_BQ_A$.

Resources

• An Introduction to Mathematical Cryptography (Second edition)