Short Signatures from the Weil Pairing Form

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okay we're back uh let me continue uh then we have defined uh the vape pairing uh rather complicated definition in terms of some functions f and g and it gives a root of unity let me give you an example um for these functions f so remember so for example take m equals to 2 then we're looking for a function f remember that f was a function such that the divisor of f is m times t minus m times o where t is an m torsion point right so what i can actually do is remember that if i write my elliptic curve in this format then i know a torsion point the two torsion point for example is the point uh e1 comma zero that is a detersion point and we proved uh long ago perhaps um i don't know how long a few lectures ago that the divisor of x minus e1 is none other than two times t minus two times o right so that works as f so i can pick f to be x minus e one right and then um i would need to compute g but now that i know that g what g needs to satisfy uh i can actually do this so compose g minus e1