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 verifierThe 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:

- The user will abort if he receives B == 0 (mod N) or u == 0.
- The host will abort if it detects that A == 0 (mod N).
- 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.

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