Difference between revisions of "User:TomYu/PKINIT notes"
Line 1:  Line 1:  
== DiffieHellman == 
== DiffieHellman == 

−  * Oakley MODP groups (used in PKINIT) have safe primes as moduli 

+  PKINIT uses the wellknown Oakley MODP groups ({{rfcref2412}}) when doing DiffieHellman 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 via the [[wp:Legendre symbolLegendre symbol]] of the public key. 

−  ** These primes don't satisfy the OpenSSL DH_check() tests, so there can be some confusion when debugging 

−  ** The generator generates the subgroup of order ''q'' instead of the whole group. (OpenSSL wants it to generate the whole group  the test is ''p'' = 11 mod 24, which includes the test ''p'' = 3 mod 8, which is false if 2 is a quadratic residue mod ''p''.) 

=== DH number theory === 
=== DH 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 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 Chinese Remainder Theorem, this means ''p'' ≡ 5 (mod 6). 2 generates the subgroup of size ''q'' if 2 is a quadratic residue mod ''p''. For 2 to be a quadratic residue mod ''p'', it must be ±1 (mod 8), and it can't be 1 (mod 8) because that would mean that ''q'' is not prime. 

+  
+  === OpenSSL issues === 

+  
+  The OpenSSL DH_check() tests cannot succeed on the Oakley 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''. 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. 

=== Windows 7 interop === 
=== Windows 7 interop === 
Revision as of 17:47, 11 April 2013
DiffieHellman
PKINIT uses the wellknown Oakley MODP groups (RFC 2412) when doing DiffieHellman key agreement. These groups are modulo safe primes, i.e., p = 2q + 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 via the Legendre symbol of the public key.
DH number theory
A safe prime is of the form p = 2q + 1, where q is prime. To be cryptographically useful, p is a large 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 Chinese Remainder Theorem, this means p ≡ 5 (mod 6). 2 generates the subgroup of size q if 2 is a quadratic residue mod p. For 2 to be a quadratic residue mod p, it must be ±1 (mod 8), and it can't be 1 (mod 8) because that would mean that q is not prime.
OpenSSL issues
The OpenSSL DH_check() tests cannot succeed on the Oakley 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. 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.
Windows 7 interop
 Windows 7 clients omit the q value in DomainParameters when sending PAPKASREQ [krbdev.mit.edu #7596]
 Even after allowing the omission of the q value, Windows 7 doesn't seem to deal with DiffieHellman group negotiation. (The KDC has to accept the 1024bit modulus, because the counterproposal of the 2048bit modulus fails on the client somehow.)