adding_points.py

    # [input trace:]
    # p.x
    # p.y
    # p.infinity
    # q.x
    # q.y
    # q.infinity
    # [appended trace:]
    # coef -> coef * (trace[ap - 3] - trace[ap - 6]) - (trace[ap - 2] - trace[ap - 5]) = 0
    # x -> x - ( (coef**2 - trace[ap - 7] - trace[ap - 4]) * (1-p.infinity) * (1-q.infinity) + p.infinity*q.x + q.infinity*p.x ) = 0
    # y -> similar idea
    def trace_add(self, trace):
        ap = len(trace)
        trace.append((trace[ap - 2] - trace[ap - 5]) / (trace[ap - 3] - trace[ap - 6]))
        ap += 1
        coef = trace[ap - 1]
        # if p = O, q.x, if q = O, return p.x
        trace.append(
            (coef**2 - trace[ap - 7] - trace[ap - 4])
            * (1 - trace[ap - 5])
            * (1 - trace[ap - 2])
            + (trace[ap - 5] * trace[ap - 4])
            + (trace[ap - 2] * trace[ap - 7])
        )
        ap += 1
        x = trace[ap - 1]
        # if p = O, q.x, if q = O, return p.x
        trace.append(
            (coef * (trace[ap - 8] - x) - trace[ap - 7])
            * (1 - trace[ap - 6])
            * (1 - trace[ap - 3])
            + (trace[ap - 6] * trace[ap - 4])
            + (trace[ap - 3] * trace[ap - 7])
        )
        # infinity of p + q = inf(p) and inf(q)
        trace.append((trace[ap - 6]) * (trace[ap - 3]))