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Old 2020-08-14, 04:08   #1
carpetpool's Avatar
Nov 2016

2×163 Posts
Post Eliptic curve J-variants

I am interesting in understanding the theoretical aspect of the ECPP test, and how everything works.

Looking at this ECPP example so far I understand:

4*N = u^2 + D*v^2, with Jacobi(-D,N)=1

and tested with different D's until N+1-u has some large probable prime factor q. Then the test is repeated with q and so on until q is small. Makes sense so far, but the concept basic arithmetic, no group theory yet.

I am not sure how the curve used in the test is constructed from the above representation of 4*N:

E: y^2 = x^3 + a*x + b

nor how the cardinality of |E(FN)| = N+u-1

(E over the finite field of N elements)

In the Wikipedia example:

N = 167;
4*N = 25^2 + 43*(1)^2;

so u=25 and the cardinality of the constructed E is N-u+1 = 143.

From wikipedia

In order to construct the curve, we make use of complex multiplication. In our case we compute the J-invariant:

j = -960^3 ...

I am completely lost at this point. For the J-invariant (wiki page) j(r) there are only special cases, and formulas involving the discriminant of the cubic polynomial involved in the elliptic curve. I find that also linked on the wikipedia page:

j(i) = 12^3
j( (i*sqrt(163)+1)/2 ) = -640320^3

both of which are functions of the roots of quadratic polynomials. So probably is the case with the ECPP example that

j( (i*sqrt(43)+1)/2 ) = -960^3 ?

Is so, how is this derived... is there are simple formula to compute j(r) for any quadratic integer r as it is used in the ECPP test? There must be some way to understand this without knowing too much CM theory. Can anyone explain this to me? Thanks.
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