[curves] A new curve

Michael Scott mike.scott at certivox.com
Thu Mar 19 10:03:55 PDT 2015

Its nice to find a new Elliptic curve that bucks the complexity curve.
Nothing nicer than more security for less cost.

So introducing the Edwards curve E-3363

x^2+y^2=1+11111.x^2.y^2 mod 2^336-3

The modulus works particularly well with the Granger-Scott approach to
modular multiplication. Observe that 336=56*6=28*12. The order is 8 times a
prime, the twist is 4 times a prime. 11111 is the smallest positive value
to yield a twist secure curve with cofactors less than or equal to 8. Not
only is it “rigid”, it even looks rigid!

This is merely billions of times more secure than the already secure
Curve25519. It fills a gap in terms of existing proposals, coming as it
does between WF-128 and WF-192. My implementation takes 333,000 cycles on a
64-bit Intel Haswell for a variable point multiplication, but it is also
32-bit-friendly. The modulus is 5 mod 8, but with Curve25519 we have gotten
over that already.

Note that with this curve we follow others in moving away from the
artificial constraint imposed by the desire to use a fully saturated
representation, whereby the modulus should be an exact multiple of the
word-length, and the associated idea of using a Solinas prime. In my view
this approach is (a) not necessarily optimal, (b) encourages non-portable
implementation, and (c) is harder to make side-channel secure.

At the very least Curve E-3363 provides a useful data-point on the
security-cost curve.
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