ecgen #

Tool for generating Elliptic curve domain parameters.

Usage#

ecgen --fp/--f2m BITS

Field specification#

• --f2m Binary field.
• --fp Prime field.

Generation methods#

• --anomalous Generate an anomalous curve (of trace one, with field order equal to curve order).
• -i / --invalid Generate a set of invalid curves, for a given curve (using Invalid curve algorithm).
• -n / --order=ORDER Generate a curve with given ORDER (using Complex Multiplication).
• -s / --ansi[=SEED] Generate a curve from SEED (ANSI X9.62 verifiable procedure).
• -b / --brainpool[=SEED]Generate a curve using the Brainpool verifiably pseudorandom algorithm from the original paper.
• --brainpool-rfc[=SEED] Generate a curve using the Brainpool verifiably pseudorandom algorithm as per RFC 5639.
• --twist Generate a twist of a given curve.

Generation options#

• -c / --count=COUNT Generate multiple curves.
• -k / --cofactor=VALUE Generate a curve with cofactor of VALUE.
• -K / --koblitz[=A] Generate a Koblitz curve (a in {0, 1}, b = 1).
• -p / --prime Generate a curve with prime order.
• --points=TYPE Generate points of given TYPE (random/prime/all/nonprime/none).
• -r / --random Generate a random curve (using Random approach).
• -u / --unique Generate a curve with only one generator.

IO options#

• -f / --input=FILE Input from FILE.
• -o / --output=FILE Output into FILE. Overwrites any existing file!
• -a / --append Append to output file (don’t overwrite).
• -v / --verbose[=FILE] Verbose logging (to stdout or FILE).

Other#

• -d / --data-dir=DIR Set PARI/GP data directory (containing seadata package).
• -m / --memory=SIZE Use PARI stack of SIZE (can have suffix k/m/g).
• --threads=NUM Use NUM threads.
• --thread-stack=SIZE Use PARI stack of SIZE (per thread, can have suffix k/m/g).
• --timeout=TIME Timeout computation of a curve parameter after TIME (can have suffix s/m/h/d).

Examples#

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

> ecgen --fp -u -p -r 192

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

> ecgen --f2m -r -c5 163

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

> ecgen --fp -i --output=invalid.json 192
p: <input prime>
a: <input a param>
b: <input b param>

Generate a prime field, uniquely generated, prime order curve, don’t ask for input try random values, verbose output (shown with example output here):

> ecgen --fp -u -p -r -v 128
+++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-+++++++
{
    "field": {
        "p": "0xa5393890f26881d94ad670171ce67b79"
    },
    "a": "0x9c083973bdca36ea71078bbaabab4947",
    "b": "0x3d986a0206bfbe1ba62c858df54385e9",
    "order": "0xa5393890f26881d9394aece3bc2d9b47",
    "subgroups": [
        {
            "x": "0x5acc17d6a44e8f8d30e877f4fef8712f",
            "y": "0x6864dd64e80609abd1797c8de1febb9f",
            "order": "0xa5393890f26881d9394aece3bc2d9b47",
            "cofactor": "0x1",
            "points": [
                {
                    "x": "0x9c7878930ddf5bfb705102f652754e7",
                    "y": "0x4b15a7bb808cb3579fd4c2ce42f628de",
                    "order": "0xa5393890f26881d9394aece3bc2d9b47"
                }
            ]
        }
    ]
}

Docs#

See docs. Also:

make docs

for Doxygen.

Generation methods#

Four 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 Koblitz curve.
  • -u / --unique generates a uniquely generated curve (with one generator/cyclic group).
  • --twist generates a curve and then produces it quadratic twist.
  • etc..

Invalid curve generation#

• Generates invalid curves for a given curve.
• These curves have the same field, and A parameter in the short Weierstrass equation.
• Multiplication using some(most?) scalar multiplication algorithm proceeds the same way multiplication on the input curve would.
• Used with the -i / --invalid option.
• These curves are NOT SECURE and are useful for implementation testing.
• Validation of Elliptic Curve Public Keys - [Antipa, Brown, Menezes, Struik, Vanstone]
• Differential Fault Attacks on Elliptic Curve Cryptosystems - [Biehl, Mayer, Muller]
• Practical Invalid Curve Attacks on TLS-ECDH - [Jager, Schwenk, Somorovksy]

Complex multiplication#

• Capable of generating a curve of a given (prime) order.
• Only works over a prime field.
• Used with the -n / --order option.
• Uses generation method from: Constructing elliptic curves of prime order - [Broker, Stevenhagen] (see Algorithm 2.2)
• Creates the class polynomials using the built in PARI/GP polclass() function or the implemented class polynomial algorithm from IEEE P1363
• Constructing elliptic curves of prescribed order - [Broker (thesis)]
• Generating Elliptic Curves of Prime Order - [Savas, Schmidt, Koc]

Anomalous curve generation#

• Generates curves of order equal to field order.
• Used with the --anomalous option.
• These curves are NOT SECURE and are useful for implementation testing.
• Elliptic curves over F_p suitable for cryptosystems - [Miyaji]

Build#

ecgen can be built using Make or CMake.

Make#

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

CMake#

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

Requirements#

• PARI/GP
• gcc / clang

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).

ecgen uses some extensions supported by the gcc and clang compilers, namely __thread-local storage.

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.

License#

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301,
USA.

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