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).
• --supersingular Generate a supersingular curve.
• -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), order can be a comma-separated(no spaces) list of factors.
• -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.
• --nums Generate a curve using the NUMS procedure (as per draft-black-numscurves-02).
• --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.
• -B / --smooth=BOUND Generate a smooth order curve with bit-lengths of factors bounded by BOUND.
• -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.
• --hex-check=HEX Generate a curve with HEX string in the hex expansion of its parameters.
• --metadata Compute the curve metadata (j-invariant, discriminant, trace of Frobenius, CM discriminant, embedding degree)

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"
                }
            ]
        }
    ]
}

Generate a prime field curve with given order (given as a list of factors) of size 128 bits and do not output any points on subgroups:

> ecgen --fp -n 7,43,83,503,653,823,863,887,1019,1279,2011,2311,2381,2957 --points=none 128
[{
    "field": {
        "p": "0xa5dad2cb65b65ab89ed2248be2971c0d"
    },
    "a": "0x2407ec02a74565af4952f552fee14588",
    "b": "0x5e80456d719dcb64df00923e8b7aa726",
    "order": "0xa5dad2cb65b65ab704d75427e1e4f81b",
    "subgroups": [
        {
            "x": "0x3219e70cb3b557dd8f8ddf9e6264a28d",
            "y": "0x0aa1274c0272f13411833fd7eb7e86d8",
            "order": "0xa5dad2cb65b65ab704d75427e1e4f81b",
            "cofactor": "0x1"
        }
    ]
}]

Generate a prime field, uniquely generated random curve, of size 192 bits, also display its metadata:

> ecgen --fp -r -u --metadata 192
[{
   "field": {
       "p": "0xf0886302f64dfdd33035064e0f0a346a74c0cf0c22e22a79"
   },
   "a": "0x54c517a28dd47c296842f20273a0ec1840e576e6c06bb4e8",
   "b": "0x3ee3025dc7d8c26104e6fb3e11f3bdaabc15c73b9df27c8f",
   "order": "0xf0886302f64dfdd33035064df1a3dc1e3c4577ad590bfaa6",
   "subgroups": [
       {
           "x": "0x9e886fbe1f155acdd790183bf1ceb5c92cfa112e693d34ef",
           "y": "0x9723f733f48229287d173ce4a5147cd947de0637def3f3ed",
           "order": "0xf0886302f64dfdd33035064df1a3dc1e3c4577ad590bfaa6",
           "cofactor": "0x1",
           "points": [
               {
                   "x": "0x1f7b95e47d5224164880e9809082c60194152901d1bc613e",
                   "y": "0x000000000000000000000000000000000000000000000000",
                   "order": "0x2"
               },
               {
                   "x": "0xce3e0c542060d6e2b244ba5850e7ccdda413854454b464ff",
                   "y": "0x38c193df5bb67817128fbf9f63d4c7af55535110553d247f",
                   "order": "0x3"
               },
               {
                   "x": "0x73d0035006dc8c9e63c65d76f96cbb5da7ab9735980ece74",
                   "y": "0x420a85069882811d44458106bba9b2aedd4820f25ca7761c",
                   "order": "0x9d"
               },
               {
                   "x": "0x68f998d413b99520fe2ba1501f8ea22db0357cf88e4ab80f",
                   "y": "0x6321a63274a60d2d0db49800e806d8d7923a5eb8bcd990a6",
                   "order": "0x1ca3"
               },
               {
                   "x": "0xd81d2b8eb95474be0e456dfd3e85d17d7832775a9e867ad4",
                   "y": "0x6dc2c4c566616b1b959935192e14d403d7cfcaac4a9f0b90",
                   "order": "0x67a21"
               },
               {
                   "x": "0x919460438753800b0332685fc9294d0ae44d61fd8d7d4403",
                   "y": "0xefa307fb3d2946939461fa3b6ac2a1c658821e69837d0122",
                   "order": "0x1053aa57a15ae784f7"
               },
               {
                   "x": "0xce1773b3b404b78923f0e8ba5493e806c088ef527c712dfb",
                   "y": "0x451aa2ca72326983381c77a1965517787333ffe8c036073d",
                   "order": "0x5869ef97eaf9885325d01"
               }
           ]
       }
   ],
   "meta": {
       "j": "1634327697577408300542081333151383474217426785000945527408",
       "discriminant": "4412636550389189038392233746786780978750224170644136456036",
       "embedding_degree": "529827387818798217347035546078799382319810259857040",
       "frobenius": "9098792692692684052427583444",
       "cm_discriminant": "-23508596506131498450876986604994502392929597678959240749652",
       "conductor": "1"
   }
}]

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]

Supersingular curves#

• CONSTRUCTING SUPERSINGULAR ELLIPTIC CURVES - [Broker]

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. ecgen uses git submodules for testing at:

• test/lib/assert.sh pointing at https://github.com/J08nY/assert.sh
• test/lib/JSON.sh pointing at https://github.com/jimklimov/JSON.sh
• test/lib/criterion pointing at https://github.com/Snaipe/Criterion

these need to be initialized for make test to work.

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
• posix-macos-addons for macOS

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.

© Eastern Seaboard Phishing Authority