//Definition of the extension fields Fq := F3(U) and Fq3 := Fq(V)

q      := 3^4;
F3     := FiniteField(3);
P3<u>  := PolynomialRing(F3);
poly   := u^4 + u^2 + 2;
Fq<U>  := ext<F3|poly>;
Pq<v>  := PolynomialRing(Fq);
poly   := v^3 + v + U^2 + U;
Fq3<V> := ext<Fq|poly>;
Pq3<Z> := PolynomialRing(Fq3);
r      := 33098280119090191028775580055082175056428495623;
Fr     := GF(r);

h0     := V^326196*Z^2 + V^35305*Z + V^204091;
h0q    := Evaluate(h0,Z^q);
F      := Z - h0q;
Ix     := Factorization(F)[2][1];

Fn<X>  := ext<Fq3|Ix>;

N := #Fn - 1;

// Generator of GF(3^{12*137})^*
g := X + V^113713;

// Encoding pi
Re := RealField(2000);
pival :=Pi(Re);
hp := 0;
for i := 0 to 136 do
        hp := hp + V^(Floor(pival*(#Fq3)^(i+1)) mod #Fq3)*(X^i);
end for;

// This is the logarithm challenge
cofactor := N div r;
h := hp^cofactor;

// log_g(h) mod r is:
x := 27339619076975093920245515973214186963025656559;

// Define the exponent y to be used in the verification:
y  := IntegerRing()!(Fr!(x/cofactor));

// Check that h = (g^cofactor)^y
h eq (g^cofactor)^y;