Elliptic curve example from book

 Elliptic curve

epoints := proc(ec, x, ub, p)

local ecurve, z, pct, k, i;

pct := 0;

for k from 0 to p-1 while pct <= ub do

        z := subs(x=k, ec) mod p;

        if z = 0 then

pct := pct + 1;

           ecurve[pct] := [k, z];

        fi:

if z &^ ((p-1)/2) mod p = 1 then

z := z &^ ((p+1)/4) mod p; ecurve[pct+1] := [k, z]; ecurve[pct+2] := [k, -z mod p]; pct := pct + 2;

fi: od:

    if pct > ub then

       pct := ub:

fi:

seq(ecurve[i], i = 1..pct): end:

addec := proc(le, re, c, p)

    local i, cle, cre, lambda, res, x3, y3;

    cle := le mod p;

    cre := re mod p;

    if cle = 0 or cre = 0 then

res := cle + cre;

elif cle[1] = cre[1] and cle[2] = -cre[2] mod p then

res := 0; else

       if cle[1] = cre[1] mod p and cle[2] = cre[2] mod p

       then

lambda := ((3*cle[1]^2+c)/2/cle[2]) mod p; else

lambda := (cre[2]-cle[2])/(cre[1]-cle[1]) mod p; fi:

x3 := (lambda^2-cle[1]-cre[1]) mod p;

y3 := (lambda*(cle[1]-x3)-cle[2]) mod p; res := [x3, y3];

fi:

res; end:

elgamal := proc(alpha, e, c, p)

    local calpha, n, y;

    read(addec);

    calpha := alpha;

    n := e;

    y := 0;

    while n > 0 do

        if irem(n, 2, ’n’) = 1 then

           y := addec(calpha, y, c, p):

fi:

        calpha := addec(calpha, calpha, c, p):

    od:

y; end:

a := epoints(eqn, x, 1, p);

b := elgamal(a, n, c, p);

k := 431235145514:

y := elgamal(a, k, c, p);

kb := elgamal(b, k, c, p);

w := [1704130304252114, 2018001913141413]:

z := [ kb[1]*w[1] mod p, kb[2]*w[2] mod p ];

ny := elgamal(y, n, c, p);

The field of post-quantum cryptography is about designing public-key algorithms that cannot be broken by a quantum computer; that is, they would be quantum safe and able to replace RSA and elliptic curve–based algorithms in a future