brainpoolP160t1.php
brainpoolP160t1
This curve is a twisted version of brainpoolP160r1 with A = -3. With brainpool, the curves ending in r1 are the "regular" curves and the curves ending in "t1" are the twisted version of the r1 curves. Per https://tools.ietf.org/html/rfc5639#page-7 you can convert a point on an r1 curve to a point on a t1 curve thusly:
F(x,y) := (x*Z^2, y*Z^3)
The advantage of A = -3 is that some of the point doubling and point addition can be slightly optimized. See http://hyperelliptic.org/EFD/g1p/auto-shortw-projective-3.html vs http://hyperelliptic.org/EFD/g1p/auto-shortw-projective.html for example.
phpseclib does not currently take advantage of this optimization opportunity
PHP version 5 and 7
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- brainpoolP160t1
- Curves over y^2 = x^3 + a*x + b