SRP Protocol Design

SRP is the newest addition to a new class of strong authentication protocols that resist all the well-known passive and active attacks over the network. SRP borrows some elements from other key-exchange and identification protcols and adds some subtle modifications and refinements. The result is a protocol that preserves the strength and efficiency of the EKE family protocols while fixing some of their shortcomings.
  N    A large safe prime (N = 2q+1, where q is prime)
       All arithmetic is done modulo N.
  g    A generator modulo N
  s    User's salt
  U    Username
  p    Cleartext Password
  H()  One-way hash function
  ^    (Modular) Exponentiation
  t    Security parameter
  u    Random scrambling parameter
  a,b  Secret ephemeral values
  A,B  Public ephemeral values
  x    Private key (derived from P and s)
  v    Password verifier
The host stores passwords using the following formula:
  x = H(s, p)               (s is chosen randomly)
  v = g^x                   (computes password verifier)
The host then keeps {u, s, v} in its password database. The authentication protocol itself goes as follows:
User -> Host:  U, A = g^a                  (identifies self, a = random number)
Host -> User:  s, B = v + g^b, u           (sends salt, b = random number,
                                            u = t-bit random number)

        User:  x = H(s, p)                 (user enters password)
        User:  S = (B - g^x) ^ (a + ux)    (computes session key)
        User:  K = H(S)

        Host:  S = (Av^u) ^ b              (computes session key)
        Host:  K = H(S)
Now the two parties have a shared, strong session key K. To complete authentication, they need to prove to each other that their keys match. One possible way:
User -> Host:  M = H(H(N) xor H(g), H(U), s, A, B, K)
Host -> User:  H(A, M, K)
The two parties also employ the following safeguards:
  1. The user will abort if he receives B == 0 (mod N) or u == 0.
  2. The host will abort if it detects that A == 0 (mod N).
  3. The user must show his proof of K first. If the server detects that the user's proof is incorrect, it must abort without showing its own proof of K.
A paper describing this protocol is also available.