Generating EC domain parameters - ecgen

ecgen is a tool for generating Elliptic curve domain parameters. While working on generating some interesting EC domain parameters for ECTester I found out there aren’t really any good tools for doing so. Cryptographic libraries don’t offer generating custom curves, they just offer preset ones or let you set custom ones. The only tools I found were ECB, LiDIA/GEC, MIRACL and cm. They all have their drawbacks however, ECB is limited in what parameters are modifiable and is closed source, LiDIA is unmaintained, and cm also doesn’t support the flexibility in parameters I needed. To add to that I lost the link to MIRACL and could not remember it’s name. So ecgen is what I started working on.

Since Elliptic Curve domain parameters are a rather complex object, generating them with various constraints and parameters is also complex and there are various algorithms for doing so. Generally two methods are used in practice. One is randomized and works as you might imagine from the name, by generating random domain parameters within some constraints, computing the rest of the parameters and hoping they satisfy the rest of the constraints. A variation of this is the ANSI X9.62 verifiably random algorithm. The other method is based on the theory of Complex Multiplication and is able to directly generate Elliptic Curve domain parameters with required constraints.

Generally the biggest problem when generating Elliptic Curve domain parameters is calculating the order of the curve specified by the curve equation. Since when this order is known the rest of the parameters (generators, group structure…) are found rather easily.

Computing this order given the equation is hard, there are several rather complex algorithms, some of which are fast for curves over $\mathbb{F}_{2^m}$ (AGM), some for curves over $\mathbb{F}_p$ (Schoof’s, SEA).

Usage#

ecgen --fp/--f2m BITS


Field specification#

• --f2mBinary field.
• --fpPrime field.

Generation options#

• -c / --count=COUNTGenerate multiple curves.
• -i / --invalidGenerate a set of invalid curves, for a given curve (using Invalid curve algorithm).
• -k / --cofactor=BOUNDGenerate a curve with cofactor up to BOUND TODO - NOT FINISHED
• --anomalous Generate an anomalous curve (of trace one, with field order equal to curve order).
• -K / --koblitzGenerate a Koblitz curve (a = 0).
• -n / --order=ORDERGenerate a curve with given ORDER (using Complex Multiplication). TODO - NOT IMPLEMENTED
• -p / --primeGenerate a curve with prime order.
• --points=TYPEGenerate points of given TYPE (random/prime/none).
• -r / --randomGenerate a random curve (using Random approach).
• -s / --seed[=SEED]Generate a curve from SEED (ANSI X9.62 verifiable procedure). TODO - NOT IMPLEMENTED
• -u / --uniqueGenerate a curve with only one generator.

IO options#

• -t / --format=FORMATFormat to output in. One of [csv,json], default is json.
• -f / --input=FILEInput from FILE.
• -o / --output=FILEOutput into FILE. Overwrites any existing file!
• -a / --appendAppend to output file (don’t overwrite).
• -v / --verbose[=FILE]Verbose logging (to stdout or FILE).

Other#

• -d / --data-dir=DIRSet PARI/GP data directory (containing seadata package).
• -m / --memory=SIZEUse PARI stack of SIZE (can have suffix k/m/g).
• --threads=NUMUse NUM threads.
• --thread-stack=SIZEUse PARI stack of SIZE (per thread, can have suffix k/m/g).

Examples#

Generate a prime field, prime order, uniquely generated 192-bit curve, don’t ask for input try random values:

&gt; ecgen --fp -r -p -u 192


Generate 5 random, binary field, 163-bit koblitz curves:

&gt; ecgen --f2m -r -k -c5 163


Generate invalid curves to a file, for a given prime field 192-bit curve:

&gt; ecgen --fp -i --output=invalid.json 192
p: &lt;input prime&gt;
a: &lt;input a param&gt;
b: &lt;input b param&gt;


Generation methods#

Three different EC curve parameters generation methods are implemented.

Efficient Algorithms for Generating Elliptic Curves over Finite Fields Suitable for Use in Cryptography - [Baier]

Generation Methods of Elliptic Curves - [Baier, Buchmann]

Random approach#

• Generates field and equation parameters:
• randomly
• using ANSI X9.62 verifiably random method(from seed), until a curve with requested properties appears.
• given input
• Can generate curves repeatedly until one satisfies requested properties:
• -p / --prime generates curves until a prime order curve is found.
• -K / --koblitz generates a curve with fixed A = 0 parameter.
• -u / --unique generates a uniquely generated curve (with one generator/cyclic group).
• etc..

Build#

git clone https://github.com/J08nY/ecgen.git
cd ecgen
git submodule update --init
make


Requirements#

• PARI/GP

ecgen uses the PARI/GP library for elliptic curve arithmetic and it’s SEA point counting algorithm implementation. It also requires the additional seadata package (seadata and seadata-big recommended for large curves).

Libraries#

• lib/parson ©MIT
• lib/sha1 ©MPL / GPLv2 or later

parson is used to input and output JSON and is included in the lib/ directory.

A SHA-1 implementation by Paul Kocher, based on the SHA 180-1 Reference Implementation (for ANSI X9.62 algorithm) is used and also included in the lib/ directory.