Given some ElGamal private key, and an encrypted message, decrypt it.
Public key (F_q, g, b)
Private key a such that b=g^a
Message m encrypted so that r=g^k, t=mb^k
Decrypt: tr^-a = m
The Attempt at a Solution
My problem is finding r^-a. I don't quite get polynomial finite fields, and I can't figure out how to get x such that r^x = r^-a in F_q.
I have an example from the textbook:
Public key (F_23, 7, 4), private key a=6, encrypted message r=21, t=11
The textbook then does 11*21^-6 = 11*21^16 = 11*2^16 = 11*9 = 7
I understand why 11*2^16 = 7 in F_23, but I don't get how 21^-6 = 21^16 and how 21^16 = 2^16.
Any help is appreciated.