# User:TomYu/PKINIT notes

## Diffie-Hellman

PKINIT uses the well-known IKE (Oakley) MODP group 2 (RFC 2409, RFC 2412), along with IKE MODP groups 14, and 16 (RFC 3526) when doing Diffie-Hellman key agreement. These groups are modulo safe primes, i.e., *p* = 2*q* + 1. They use 2 as a generator, and the primes are chosen so that 2 generates the subgroup of order *q*, to prevent the leaking of the least significant bit of the private exponent *x* via the Legendre symbol of the public key *g*^{x}.

### D-H number theory

A safe prime is of the form *p* = 2*q* + 1, where *q* is prime. To be cryptographically useful, *p* is a large (odd) prime, therefore *p* ≡ 1 (mod 2). Also, *p* ≡ 2 (mod 3), as is *q*, because either *p* or *q* being congruent to 1 (mod 3) implies that the other is divisible by 3. (This is only true if *q* ≠ 3, which is true for cryptographically useful primes.) By the Chinese remainder theorem, this means *p* ≡ 5 (mod 6). 2 generates the subgroup of size *q* if 2 is a quadratic residue mod *p*. According to the law of quadratic reciprocity, for 2 to be a quadratic residue mod *p*, *p* ≡ ±1 (mod 8). If *p* is a safe prime, it can't be 1 (mod 8) because that would mean that *q* is not prime.

### OpenSSL issues

The OpenSSL DH_check() tests could fail on the IKE MODP groups, because DH_check() applies the test *p* ≡ 11 (mod 24) for a generator of 2. The prime consequently has to also satisfy the congruences *p* ≡ 2 (mod 3) and *p* ≡ 3 (mod 8). The congruence *p* ≡ ±3 (mod 8) is true if 2 is not a quadratic residue mod *p*, which means that DH_check() is checking that 2 will generate the entire group modulo *p*. (*p* ≡ 5 (mod 8) implies *q* is an even number.) The code in DH_check() in newer versions of OpenSSL does additional checks if the *q* parameter is given, which include *g*^{q} ≡ 1 (mod *p*) (*g* generates the subgroup of order *q* if *q* is prime), *p* ≡ 1 (mod *q*) (*q* divides *p* - 1), and that *q* is prime. These checks on the *q* parameter supersede the check that the generator would generate the entire group mod *p*.

### Windows 7 interop

- Windows 7 clients omit the
*q*value in DomainParameters when sending PA-PK-AS-REQ [krbdev.mit.edu #7596] - Even after allowing the omission of the
*q*value, Windows 7 doesn't seem to deal with Diffie-Hellman group negotiation. (The KDC has to accept the 1024-bit modulus, because the counterproposal of the 2048-bit modulus fails on the client somehow.)