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Singular curves

Description

If a curve is a singular one (i.e. its discriminant \(\Delta = -16(4a^3 + 27b^2) \mod p = 0\)), it is isomorphic to multiplicative group, which enables to solve DLP faster.

Task

Given arbitary curve \(E\) with \(\Delta = 0\), and some point \(P = d*G\), find \(d\).

Solution

In case of \(\Delta = 0\) the curve has double or triple root \(x_{0}\) and the point \((x_{0}, 0)\) is a singular point.

With a change of variable, you can come to two cases:

  1. Cusp \(y^2 = x^3\)

    The curve is isomorphic to additive group of \(F_{p}\) (there is a mapping to that group), where discrete logarithm is trivial.

  2. Node \(y^2 = x^2*(x - 1)\)

    The curve is isomorphic to multiplicative group of \(\mathbf F_{p^2}^*\), where discrete logarithm is easier to compute.

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