Bob has a secret and Alice has the same secret. Why can’t they create a shared encryption key based on their secrets? Well, they can do this with Password Authentication Key Exchange (PAKE). So, let’s look at a simple method using discrete logs, and then we will convert it to elliptic curve methods. While discrete logs have been used in the past for Diffie-Hellman key exchange methods, we are increasing moving towards elliptic curve implementations.

## SPEKE (Simple Password Exponential Key Exchange) — Discrete Logs

SPEKE (Simple Password Exponential Key Exchange) supports password-authenticated key agreement. Bob and Alice share a secret password (*π*) and a shared prime number (*p*). This password then hashed and used to determine a generator (*g*):

*g*=*H*(*π*)² (mod *p*)

The square function of the hash makes sure that *g *is a generator for the prime number *p*. After this, we can use a standard Diffie-Hellman type exchange. For this, Alice generates a random number *a* and Bob generates a random number *b*. Alice then sends:

*A*=*g^a *(mod *p*)

and Bob sends:

*B*=*g^b *(mod *p*)

Alice computes the shared key as:

*K*1=*B^a*(mod *p*)