Let p be a odd prime. For a,b ∈Z_p, define f_((a,b)):Z_p-→Z_p by the rule

f_((a,b)) (x)=〖(x+a)〗^2+b mod p

Prove that (Z_p,Z_p,Z_p×Z_p,{ f_((a,b) ):a,b∈ Z_p}) is a strongly universal (p,p)-hash family.

I think we have to find the soultion a and b ,

a= (x-x')^-1 (y^1/2 -Y'^1/2 ) mod p

then we can find b,

but i don not know if (y^1/2 -y'^1/2 ) mod p exists because we know (x-X')^-1 mod p exists.

any help would be appreciated, thanks