EEA[g_Integer,p_Integer]:=Module[{L,u,v,r0,r1,t0,t1,q},If[g===0,Return[{{0,1}}]];
L={{u,r0,t0,v,r1,t1}={p,0,1,g,1,0}};
While[v=!=0,q=Quotient[u,v];
AppendTo[L,{u,r0,t0,v,r1,t1}={v,r1,t1,u-q v,r0-q r1,t0-q t1}];];
Return[L];];

rat=RandomInteger[10^30]/RandomInteger[10^30]

p=NextPrime[10^60];
m=PolynomialMod[rat, p];

trace=Take[#,2]& /@ Rest[EEA[m,p]];

Animate[TraditionalForm[TableForm[{{HoldForm[v m == Mod[u ,p]]}, {u==trace[[k,1]]},
{v==trace[[k,2]]}}]],{k,1,Length[trace],1},AnimationRepetitions->1,AnimationRunning->True]