\\Initialise an elliptic curve E
E = ellinit([0,0,0,2,3])
\\Compute the trace of E over F_{17}
ellap(E,17)
\\Deduce the cardinality of E(F_{17}), i.e p+1-t
17+1--4
\\Define E modulo 17
Emod = ellinit([0,0,0,2,3]*Mod(1,17))
\\Find a point on Emod
ellordinate(Emod,3)
\\Define P on Emod
P=[3,6]
\\Check that P is on the curve
ellisoncurve(Emod,P)
\\Check that [22]P = [0]
ellpow(Emod,P,22)
\\Same computation modulo a larger prime
p=nextprime(2^30):ellap(E,p)
\\Same computation modulo a larger prime
p=nextprime(2^80):ellap(E,p)
\\needs to allocate more memory, Baby step giant step method needs memory
allocatemem(100000000)
p=nextprime(2^80):ellap(E,p)
\\Load Schoof package
read("c:/ed/schoof.gp")
\\Test it on a large prime 
p = 2^60+33
\\Redefine Emod
Emod = ellinit([0,0,0,2,3]*Mod(1,p))
\\P is on Emod
P = [84831642191856994, 732955120815734277]
\\Check that P is a 31-torsion point
ellpow(Emod,P,31)
\\Compute the L-division polynomials for Emod up to l = 31
L = Ldivision(Emod,31,p)
\\Check that there is no 5-torsion rational point
factor(L[5,1])
\\Use schoof algorithm to compute t mod 5
Find_trace_mod_l(E, 5, L, p)
\\Use schoof algorithm to compute t mod 7
Find_trace_mod_l(E, 7, L, p)
\\Use schoof algorithm to compute t mod 11
Find_trace_mod_l(E, 11, L, p)
\\Use schoof algorithm to compute t mod 17
Find_trace_mod_l(E, 17, L, p)
\\Load the SEA package
read("c:/ed/sea.gp")
\\Consider a prime too big for Shanks-Mestre or Schoof algorithms
p = nextprime(random(2^100))
\\Find the cardinality of E over Fp using SEA
ellsea(E,p)
\\Find the cardinality of E over Fp using SEA and display additional information
ellsea(E,p,1)
\\Find the cardinality of E over a larger field Fp using SEA
p=nextprime(random(2^140));ellsea(E,p,1)
\\Find the cardinality of E over a larger field Fp using SEA
p=nextprime(random(2^200));ellsea(E,p,1)
\\Find the cardinality of E over a larger field Fp using SEA
p=nextprime(random(2^256));ellsea(E,p,1)
\\Find a suitable curve of size 100 bits for crypto applications 
ellcrypto(100);
\\Load the package to perform EC arithmetic over a field of char 2
read("c:/ed/ecc_ed.gp")
\\Initialise F_{2^17}
InitFieldF2n(17)
\\Initialise the elliptic curve E defined over F_{2^17}
E = [1,0,0,0,(x^4+1)*OneF2n]
\\Find a point on E
P = FindPtAffF2n(E)
\\Change the representation of P
DispPtAffF2n(P,3)
\\Load the package to compute the number of points on an elliptic curve using the AGM
read("c:/ed/AGM.gp")
\\Computes #E using the AGM
AGM(E)
\\Check that the result is correct
MulPtAffF2n(P,131164,E)