Multiplication of elements of tower fields

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The following is a structural solution, selfexplanatory, using either structural maps or "canonical" lifts.

p = 13

R.<X> = PolynomialRing( GF(p) )
K.<v> = GF( p** 4, modulus=X^ 4 - 2 )
L.<w> = GF( p**16, modulus=X^16 - 2 )

embd  = Hom( K, L )( w^4 )   # embd : K -> L, v -> w^4

f1 = K(1)
for _ in range(34):
    A  = K.random_element()
    f1 = f1^2 * A

g1 = embd(f1)
print "f1 is in K = %s" % f1
print "we map f1 in K to g1 in L via\n", embd 
print "g1 is in L = %s" % g1

G1 = R(g1)    # the polynomial lift, the coefficients can now be easily extracted
g1coeff = G1.coefficients( sparse=False )
print "g1 has the coefficients", g1coeff
print "g1 has the hex coefficients", [ hex(ZZ(c)) for c in g1coeff ]

hexcoeff = [ hex(ZZ(c)) for c in g1coeff ]
hexcoeff . reverse()
hexrep   = ''.join( hexcoeff )
print "hex representation:", hexrep

I considered that making all computations in K, as long as possible, then at the end pass to L, should be optimal. Alternatively, one can push immediately every A in the loop to embd( A ) from K to L, and compute everything in L, if the real life intention / application needs it there .

Results in this run:

f1 is in K = v^3 + 9*v^2 + 8*v + 10
we map f1 in K to g1 in L via
Ring morphism:
  From: Finite Field in v of size 13^4
  To:   Finite Field in w of size 13^16
  Defn: v |--> w^4
g1 is in L = w^12 + 9*w^8 + 8*w^4 + 10
g1 has the coefficients [10, 0, 0, 0, 8, 0, 0, 0, 9, 0, 0, 0, 1]
g1 has the hex coefficients ['a', '0', '0', '0', '8', '0', '0', '0', '9', '0', '0', '0', '1']
hex representation: 100090008000a

This should be all you need, enjoy!