1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
|
## ecgen [](https://travis-ci.org/J08nY/ecgen) [](https://github.com/J08nY/ecgen/releases/latest)
Tool for generating Elliptic curve domain parameters.
### Usage
ecgen --fp/--f2m BITS
#### Field specification
- `--f2m` Binary field.
- `--fp` Prime field.
#### Generation options
- `-c`, `--count=COUNT` Generate multiple curves.
- `-i`, `--invalid` Generate a set of invalid curves, for a given curve (using Invalid curve algorithm).
- `-k`, `--cofactor=BOUND` Generate a curve with cofactor up to `BOUND`
- `-K`, `--koblitz` Generate a Koblitz curve (a = 0).
- `-n`, `--order=ORDER` Generate a curve with given `ORDER` (using Complex Multiplication).
- `-p`, `--prime` Generate a curve with prime order.
- `--points=TYPE` Generate points of given `TYPE` (random/prime/none).
-`r`, `--random` Generate a random curve (using Random approach).
-`s`, `--seed[=SEED]` Generate a curve from `SEED` (ANSI X9.62 verifiable procedure).
-`u`, `--unique` Generate a curve with only one generator.
#### IO options
- `-a`, `--append` Append to output file (don't overwrite).
- `-f`, `--input=FILE` Input from `FILE`.
- `-o`, `--output=FILE` Output into `FILE`. Overwrites any existing file!
- `-t`, `--format=FORMAT` Format to output in. One of [csv,json], default is json.
- `-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).
#### Examples
Generate a prime field, uniquely generated 192-bit curve, don't ask for input try random values:
> ecgen --fp -r -p -u 192
Generate 5 random, binary field, 163-bit koblitz curves:
> ecgen --f2m -r -k -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 *(show with example output here)*:
> ecgen --fp -r -p -u -v 128
+++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-++++-+++++++
{
"field": {
"p": "0xa5393890f26881d94ad670171ce67b79"
},
"a": "0x9c083973bdca36ea71078bbaabab4947",
"b": "0x3d986a0206bfbe1ba62c858df54385e9",
"order": "0xa5393890f26881d9394aece3bc2d9b47",
"generators": [
{
"x": "0x5acc17d6a44e8f8d30e877f4fef8712f",
"y": "0x6864dd64e80609abd1797c8de1febb9f",
"order": "0xa5393890f26881d9394aece3bc2d9b47",
"cofactor": "0x1"
}
],
"points": [
{
"x": "0x9c7878930ddf5bfb705102f652754e7",
"y": "0x4b15a7bb808cb3579fd4c2ce42f628de",
"order": "0xa5393890f26881d9394aece3bc2d9b47"
}
]
}
### 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]](https://www.cdc.informatik.tu-darmstadt.de/reports/reports/harald_baier.diss.pdf)
[Generation Methods of Elliptic Curves - [Baier, Buchmann]](https://www.ipa.go.jp/security/enc/CRYPTREC/fy15/doc/1030_Buchmann.evaluation.pdf)
##### 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).
##### 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
- [Validation of Elliptic Curve Public Keys - [Antipa, Brown, Menezes, Struik, Vanstone]](https://www.iacr.org/archive/pkc2003/25670211/25670211.pdf)
- [Differential Fault Attacks on Elliptic Curve Cryptosystems - [Biehl, Mayer, Muller]](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.107.3920&rep=rep1&type=pdf)
- [Practical Invalid Curve Attacks on TLS-ECDH - [Jager, Schwenk, Somorovksy]](http://euklid.org/pdf/ECC_Invalid_Curve.pdf)
##### Complex multiplication
- Capable of generating a curve of a given (prime) order.
- Generates a subset of all Elliptic Curves over a given field.
- Used with the `-n / --order` option
- [Constructing elliptic curves of prime order - [Broker, Stevenhagen]](https://arxiv.org/abs/0712.2022)
- [Generating Elliptic Curves of Prime Order - [Savas, Schmidt, Koc]](http://people.oregonstate.edu/~schmidtt/ourPapers/SavasKoc/ches01curve.pdf)
### Build
git clone https://github.com/J08nY/ecgen.git
cd ecgen
make
### Requirements
- PARI/GP
ecgen uses the [PARI/GP](http://pari.math.u-bordeaux.fr/) library for elliptic
curve arithmetic and it's SEA point counting algorithm implementation. It also requires the
additional [seadata](http://pari.math.u-bordeaux.fr/packages.html) package (seadata and seadata-big recommended for large curves).
### 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 3 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, see <http://www.gnu.org/licenses/>.
*© Eastern Seaboard Phishing Authority*
|
