[Cfrg] One question about MODP: the structure of DLP prime in a finite field

[Wang Guilin <[e-mail]>]

Dear everyone, 

Highly appreciate if anyone can help on the following question. 

RFC 3526 (https://tools.ietf.org/html/rfc3526) offers a number of DLP parameters in a finite field. An example is group ID 14, detailed specification copied below. 

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This group is assigned id 14.

   This prime is: 2^2048 - 2^1984 - 1 + 2^64 * { [2^1918 pi] + 124476 }

   Its hexadecimal value is: 
      FFFFFFFF FFFFFFFF C90FDAA2 2168C234 C4C6628B 80DC1CD1
      29024E08 8A67CC74 020BBEA6 3B139B22 514A0879 8E3404DD
      EF9519B3 CD3A431B 302B0A6D F25F1437 4FE1356D 6D51C245
      E485B576 625E7EC6 F44C42E9 A637ED6B 0BFF5CB6 F406B7ED
      EE386BFB 5A899FA5 AE9F2411 7C4B1FE6 49286651 ECE45B3D
      C2007CB8 A163BF05 98DA4836 1C55D39A 69163FA8 FD24CF5F
      83655D23 DCA3AD96 1C62F356 208552BB 9ED52907 7096966D
      670C354E 4ABC9804 F1746C08 CA18217C 32905E46 2E36CE3B
      E39E772C 180E8603 9B2783A2 EC07A28F B5C55DF0 6F4C52C9
      DE2BCBF6 95581718 3995497C EA956AE5 15D22618 98FA0510
      15728E5A 8AACAA68 FFFFFFFF FFFFFFFF

   The generator is: 2. 
=========================

The question is: What is the structure or factors of prime p-1, where the value of p is given above? Also, if we do not know the factors of p-1, it is risky to just use g=2 as a generator as the order of 2 could be quite small. In FRC 3526, the suggested exponent size for group ID 14 is 220 bits or more. 

My real reason to ask this question is: We want to test SPEKE (a PAKE protocol) by using group ID 14. However, to run SPEKE, we need to know a prime factor q of p-1, i.e. (p-1)=qk, where k is an integer. Ideally, the bit length of q is between 220-256. Once we know such a prime factor q for p-1, then both client and server in SPEKE can calculate a generator something like g=(H(pw, salt))^k. Then, they can run DH key exchange normally by using g. 

So, the difficulty here is: Without knowing the factors of p-1 in group ID 14, it seems not possible to generate such a generator g in SPEKE. 

Thanks in advance, 

Guilin