Let F_p be a finite field over a prime p, and
E_j(F_p) be the elliptic curve defined by:
y^2 = x^3 + 36*x/(j0 - 1728) + -1*(j0 - 1728)
We consider the Solinas primes
P_224 = N^7 - N^3 + 1
P_256 = N^8 - N^7 + N^6 + N^3 - 1
P_384 = N^12 - N^4 - N^3 + N^1 - 1
where N = 2^32.
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For each of these primes, we calculate the trace, T_f,
of each non-singular K_j = E_j(F_p) curve
(j != 0, 1728) for j < 2^20 and #K_j
prime.
Noting that the number of points on the curve and its twist are given by
#K = p - T_f + 1
#K^t = p + T_f + 1
the number of prime and doubly-prime curves in the
interval 0 < j < 2^20 are given by:
| p | pri | pri' | p(p'|p)/100 |
|-------|--------|--------|-------------|
| P224 | 2790 | 31 | 1.11 |
| P256 | 1956 | 15 | 0.77 |
| P384 | 1131 | 20 | 1.77 |
N^ 0 1 2 3 4 5 6 7 8 9 10 11 12
P224 +1 -1 +1
P256 -1 +1 +1 -1 +1
P384 -1 +1 -1 -1 +1
sol384mod4 = [2^384 - 2^128 - 2^96 + 2^32 - 1,
2^384 - 2^224 - 2^160 + 2^32 - 1,
2^384 - 2^288 + 2^192 - 2^32 - 1,
2^384 - 2^288 + 2^64 + 2^32 - 1,
2^384 - 2^320 + 2^192 + 2^128 - 1,
2^384 - 2^320 + 2^288 - 2^160 - 1,
2^384 - 2^320 + 2^288 - 2^96 - 1,
2^384 - 2^352 + 2^224 + 2^64 - 1]