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== Diffie-Hellman ==

PKINIT uses the well-known IKE (Oakley) MODP group 2 ({{rfcref|2409}}, {{rfcref|2412}}), along with IKE MODP groups 14, and 16 ({{rfcref|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 [[wp:Legendre symbol|Legendre symbol]] of the public key ''g''<sup>''x''</sup>.

=== 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 [[wp:Chinese remainder theorem|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 [[wp:Quadratic reciprocity|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''<sup>''q''</sup> ≡ 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 {{bug|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.)