# [curves] Curve with group order 2^255-19

Andrew Poelstra apoelstra at wpsoftware.net
Wed Mar 21 05:30:50 PDT 2018

```Hi all,

A spooky seeming-fact about j-invariant 0 prime-order curves over prime fields
is that you can "just swap the field and group order" to obtain a new prime
order curve of j-invariant [0].

This is very convenient, because many popular ZK systems work, or can be made
to work, over arithmetic circuits over a given field [1,2,3,4]. For EC-based
ZKPs this field typically has as many elements as the order of the curve
you're producing the ZKPs on.

This means that, e.g., you can prove in zero knowledge operations on secp256k1

y^2 = x^3 + 7   mod 2^256 - 2^32 - 977

by producing a ZKP on the curve "secq256k1" whose equation [5] is

y^2 = x^3 + 7   mod (group order of secp256k1)

which is a pretty nifty trick.

Doing ZKPs of EC operations on a target group is a generally very useful tool
because it lets you do ZKPs on deployed cryptosystems, which lets you "bolt on"
compression, audit trails, avoidance of semi-honest assumptions, etc., and
potentially layer new applications onto seemingly limited protocols [6].

Unfortunately, my trick of swapping the field and curve orders seems to only
work on j-invariant 0 prime-order fields, and ed25519 is neither. So my question
is: is there a standard (or at least well-known) (or at least easily findable)
DL-hard curve whose group of rational points has order 2^255 - 19?

Cheers
Andrew

[0] A j-invariant 0 curve has equation y^2 = x^3 + b and the various values
of b give you at most six different isomorphism classes. Not all have
prime order, you may have to try a few. But this seems to work very
reliably. See
https://mathoverflow.net/questions/249982/elliptic-curve-related-equivalence-between-fields-of-different-characteristic

[1] https://eprint.iacr.org/2013/507
[2] http://engineering.nyu.edu/events/2017/10/27/ligero-lightweight-sublinear-zero-knowledge-arguments
[3] https://eprint.iacr.org/2017/1066
[4] https://eprint.iacr.org/2018/046

[5] The fact that both equations have exactly the same coefficients is a
coincidence. In particular the two 7s, being in different ground fields,
are actually completely unrelated objects even though we use the same
symbol for them.

[7] https://www.nasdaq.com/article/scriptless-scripts-how-bitcoin-can-support-smart-contracts-without-smart-contracts-cm882818

--
Andrew Poelstra
Mathematics Department, Blockstream
Email: apoelstra at wpsoftware.net
Web:   https://www.wpsoftware.net/andrew

"A goose alone, I suppose, can know the loneliness of geese
who can never find their peace,
whether north or south or west or east"
--Joanna Newsom

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