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u=-(2^62 + 2^55 +1)

p=36*u^4+36*u^3+24*u^2+6*u+1
r=36*u^4+36*u^3+18*u^2+6*u+1
t=6*u^2+1
%gp

#-------------- Fp--------------
Fp=GF(p)
#-------------------------------
b=2
#--------------Fp^2--------------
K2.<i>=PolynomialRing(Fp)
Fp2.<i>=FiniteField(p^2,modulus=i^2+1)
#------------------------------------
#-------------Fp^6---------------------
K6.<v>=PolynomialRing(Fp2)
Fp6.<v>=Fp2.extension(v^3-i-1)
#-------------------------------------

#---------------Fp^12-------------------

K12.<z>=PolynomialRing(Fp6)
Fp12.<z>=Fp6.extension(z^2-v)
Fp12.is_field=lambda:True
#---------------------------------------
*** syntax error, unexpected --: #--------------Fp----- *** ^-------------------- *** at top-level: Fp=GF(p) *** ^----- *** not a function in function call *** syntax error, unexpected --: #--------------------- *** ^-------------------- 2 *** syntax error, unexpected --: #--------------Fp^2--- *** ^-------------------- *** syntax error, unexpected real number, expecting $end: K2 *** .<i>=PolynomialRing( *** ^-------------------- *** syntax error, unexpected real number, expecting $end: Fp2 *** .<i>=FiniteField(p^2 *** ^-------------------- *** syntax error, unexpected --: #--------------------- *** ^-------------------- *** syntax error, unexpected --: #-------------Fp^6---- *** ^-------------------- *** syntax error, unexpected real number, expecting $end: K6 *** .<v>=PolynomialRing( *** ^-------------------- *** syntax error, unexpected real number, expecting $end: Fp6 *** .<v>=Fp2.extension(v *** ^-------------------- *** syntax error, unexpected --: #--------------------- *** ^-------------------- *** syntax error, unexpected --: #---------------Fp^12- *** ^-------------------- *** syntax error, unexpected real number, expecting $end: K12 *** .<z>=PolynomialRing( *** ^-------------------- *** syntax error, unexpected real number, expecting $end: Fp12 *** .<z>=Fp6.extension(z *** ^-------------------- (Fp12)->lambda:True *** syntax error, unexpected --: #--------------------- *** ^--------------------
f1=Fp12.random_element()
f=(f1^(p^6-1))^(p^2+1)

#algorithm 1: New development of Fuentes Method.
def algorithm_1 (f,u):
    t0=f^(-u)
    t1=t0^2
    t0=t1^2
    t0=t1*t0
    t2=t0^(-u)
    
    t3=t2^(u+1)
    t3=t3^2
    t0=t0^(-1)
    t3=t3*t0
    t2=t2^(-1)
    t2=t3*t2
    t3=t3*f
    
    t1=t1*t2
    t0=f^(-1)
    t0=t1*t0
    
    t0=t0^(p^3)
    t0=t0*t3
    t1=t1^(p)
    t0=t0*t1
    t2=t2^(p^2)
    t0=t0*t2
    
    return t0


algorithm_1(f,u)==(f^((p^4-p^2+1)/r))^(2*u*(6*u^2+3*u+1))
True