Singular Manual: generateG

| #### D.12.2.33 generateG Procedure from library `crypto.lib` (see crypto_lib). Usage: generateG(a,b,m); Return: m-th division polynomial Note: generate the so-called division polynomials, i.e., the recursively defined polynomials p_m=generateG(a,b,m) in Z[x, y] such that, for a point (x:y:1) on the elliptic curve defined by y^2=x^3+a*x+b over Z/N the point m*P=(x-(p_(m-1)*p_(m+1))/p_m^2 :(p_(m+2)*p_(m-1)^2-p_(m-2)*p_(m+1)^2)/4y*p_m^3 :1) and m*P=0 if and only if p_m(P)=0 Example:
    LIB "crypto.lib";
    ring R = 0,(x,y),dp;
    generateG(7,15,4);
    ==> 4xy6+140xy4+1200xy3-980xy2-1680xy-8572x

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