// 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 := 589881151426658740854227725580736348850640632297373414091790995505756623268837;
Fr := GF(r);

h1 := Z^2 + V^530855;
h1q := Evaluate(h1,Z^q);
Ix := h1q*Z - 1;

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

N := #Fn - 1;

// Generator of GF(3^{12*163})^*
g := X + V^2;

// Encoding pi
Re := RealField(2000);
pival :=Pi(Re);
hp := 0;
for i := 0 to 162 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 := 426395951498279193713291391953449000732592554251132525672039784356054526194343;

// 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;