20200814, 04:08  #1  
"Sam"
Nov 2016
2^{3}·41 Posts 
Eliptic curve Jvariants
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+1u 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(F_{N}) = N+u1 (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 Nu+1 = 143. From wikipedia Quote:
I am completely lost at this point. For the Jinvariant (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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