The PGP implementation of DH is based on Galois Fields, aren't they broken?

No. There are two general types of Galois Fields with cryptographic significance, GF(p) with p prime, and GF(2n). When first introduced, GF(2n) was the preferred implementation, basically because it is easier to implement in hardware Sch96a, Odl83.

However, this was shown to be relatively insecure. The field GF(p) where p is around 2750 and is prime is thought to offer roughly the same security as GF(2n) where n is around 2000. Clearly, the Galois Field GF(p) offers better security for the same parameter size.

It is unfortunate that these two systems, though related, are both often discussed in the same breath - theory in one field isn't necessarily applicable in the other field. Anyway, PGP implements Diffie-Hellman over GF(p) which, as we'll see later, is still secure. If you are still interested in the relation between GF(p) and GF(2n) then I most highly recommend Odl83.

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