8.1 Diffie Hellman key exchange

  Diffie-Hellman key exchange (DHKE) The scheme provides a solution to the key distribution problem in practice , That is, it allows both sides to communicate through insecure channels , Get a common key .DHKE The basic idea is ,$Z_p^{\ast }$ Exponential operation in (p Prime number ) It's a one-way function , And the exponential operation is commutative , namely :$$k=(a^x{)}^y=\equiv (a^y{)}^{x}modp$$
   value $k=(a^x{)}^y=\equiv (a^y{)}^{x}modp$ Is a joint key , It can be used as the session key of both communication parties .

Diffie-Hellman Key exchange protocol in $Z_p^{\ast }$ How it works
   This agreement has two parties ,Alice and Bob, They will set up a shared key . There may be a trusted third party , This party can properly select the public parameters required for key exchange . However ,Alice or Bob It is also possible to generate public parameters . Strictly speaking ,DHKE The agreement consists of two agreements : Handshake agreement and main agreement ; The main protocol is responsible for the real key exchange .

  Diffie-Hellman handshake protocol

1. Choose a large prime number $p$.
2. Choose an integer $\alpha \in \{2,3,4,....,p-2\}$.
3. Open $p$ and $\alpha$

  Diffie-Hellman key exchange




   Let's prove the correctness of the agreement , namely Alice and Bob In fact, the same session key is calculated $k_{AB}$

   prove : Alice Calculation $$B^{\alpha }\equiv ({\alpha }^b{)}^a\equiv {\alpha }^{ab}modp$$
   and Bob Calculation $$A^b\equiv ({\alpha }^a{)}^b\equiv {\alpha }^{ab}modp$$

   therefore Alice and Bob Share the session key $k_{AB}={\alpha }^{ab}modp$. This key can be used in Alice and Bob Establish a secure communication between , For example, will $k_{AB}$ Used as similar AES or 3DES Key of equal symmetry algorithm .

Example ：Diffie-Hellman The domain parameter of is $p=29,\alpha =2$. The process of this agreement is as follows :


   The value calculated by both parties is $k_{AB}=16$, This value can be used as a union key , For example, as the session key in symmetric encryption .

Reference material ：《 In simple terms, cryptography 》–Christof Paar,Jan Pelzl Writing