The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 59

Also, c s = R s /S s = s by Proposition 2.28. By Lemma 2.26,

R rφ q + s /S rφ q + s = c rφ q + s = c rφ q + c s =0+ s = s.

Therefore, R rφ q + s =0ifandonlyif p s .

2.10 Singular Curves
We have been working with y 2 = x 3 + Ax + B under the assumption that
x 3 + Ax + B has distinct roots. However, it is interesting to see what happens
when there are multiple roots. It will turn out that elliptic curve addition
becomes either addition of elements in K or multiplication of elements in K ×
or in a quadratic extension of K . This means that an algorithm for a group
E ( K ) arising from elliptic curves, such as one to solve a discrete logarithm
problem (see Chapter 5), will probably also apply to these more familiar
situations. See also Chapter 7. Moreover, as we'll discuss briefly at the end of
this section, singular curves arise naturally when elliptic curves defined over
the integers are reduced modulo various primes.
We first consider the case where x 3 + Ax + B has a triple root at x =0,so
the curve has the equation

y 2 = x 3 .

The point (0 , 0) is the only singular point on the curve (see Figure 2.7). Since

Figure 2.7
y 2 = x 3

any line through this point intersects the curve in at most one other point,