In the description below:

alpha

is a Diffie-Hellman base used system wide

p

is a Diffie-Hellman modulus used system wide

Rx

is a random number of the same order as p.

PKx

the public key of x

SKx

the private key of x

CERTx

the public key of x signed by the certifying authority

sign(x)

represents x signed with n's private key

In the following, the parties are labelled 0 and 1.

Each sends its public key and certificate to the other together with a computation alpha**r0 mod p (alpha**r1 mod p) based on the Diffie-Hellman parameters contained in the certificate:

0 -> 1  alpha**r0 mod p, CERTu0, PKu0
1 -> 0  alpha**r1 mod p, CERTu1, PKu1

Each can now use the CA's public key and the certificate received to check that each has the other's public key.

Finally, each user signs values known to both that each can then verify:

0 -> 1  sig0(alpha**r0 mod p, alpha**r1 mod p)
1 -> 0  sig1(alpha**r0 mod p, alpha**r1 mod p)

At this point 0 and 1 can calculate the shared secret alpha**(r0*r1), and can use it to encrypt later communications.