p = randomprime(2^100)
a = Mod(2,p);
type(a)
a^(p-1)
a.mod == p
lift(a)

T = x^2+1;
b = Mod(x+a, T);
type(b)
b^(p+1)
b.pol
b.mod == T

c = ffgen(3^8,'c) \\generator of F_3^8 as a field
type(c)
c.p
c.mod \\defining polynomial, lifted to Z
polisirreducible(c.mod*Mod(1,3))
c.f \\degree over F_3
d = c^9+1
d.pol
type(d.pol)

ffinit(3,5)

ffgen(x^2+Mod(1,3))

nf = nfinit(y^8 - 2*y^7 + 9*y^6 - 2*y^5 + 38*y^4 - 34*y^3 + 31*y^2 - 6*y + 1);
pr = idealprimedec(nf,2)[1]; [pr.e,pr.f]
g = nfmodpr(nf,y,pr)

modpr = nfmodprinit(nf,pr);
nfmodpr(nf,y^2+1,modpr)
nfmodprlift(nf,g+1,modpr) \\find a preimage

[c,c+1;2*c,1]^-1
d = random(c) \\random element in the field
issquare(d)
trace(d) \\over F_3
norm(d)
minpoly(d^82)
factor(x^5+x^3+c) \\variants factormodSQF and factormodDDF
polrootsmod(x^7+x+c)

fforder(c)
z = ffprimroot(c)
fforder(z)
n = fflog(c,z)
c == z^n

E = ellinit([c,1]);
E.cyc \\structure of the group of points
ellissupersingular(E)
hyperellcharpoly(x^7+c*x+2) \\y^2 = x^7+c*+2

d = ffgen([3,24],'d)
Mcd = ffembed(c,d); \\compute some embedding
ffembed(d,c)
break
c2 = ffmap(Mcd,c^5+c+1) \\apply the map
F = fffrobenius(d,8); \\8-th power of Frobenius
ffmap(F, d) == d
ffmap(F, c2) == c2

T = ffinit(3,5)
[e,Mde] = ffextend(d, T, 'e);
e.f
fforder(e)
ffmap(Mde, d)

Mce = ffcompomap(Mde,Mcd);
ffmap(Mce, c) == ffmap(Mde, ffmap(Mcd, c))
ffcompomap(F,Mcd) == Mcd
ffcompomap(F,F) == fffrobenius(d,16)

Mdc = ffinvmap(Mcd);
ffmap(Mdc, ffmap(Mcd, c^3+c+1))
ffmap(Mdc, d)
Mec = ffcompomap(Mdc, ffinvmap(Mde));
ffmap(Mec, ffmap(Mce, c))
ffinvmap(fffrobenius(c,3)) == fffrobenius(c,5)