DiSSECT: Distinguisher of Standard & Simulated Elliptic Curves via Traits

DiSSECT is, to the best of our knowledge, the largest publicly available database of standardized elliptic curves (taken from our sister project) and offers generation of simulated curves according to the mentioned standards. The tool contains over 20 tests (which we call traits), each computing curve properties, ranging from classical algebraic ones to unconventional ones and those connected to implementations. After obtaining their empirical distributions, the traits allow us to compare the simulated curves to the standard ones. Finally, DiSSECT provides an easy-to-use interface for implementations of custom traits and their interactive visualization via Jupyter notebook.

DiSSECT is written in Python 3 and imports the SageMath library. The database of the standardized elliptic curves as well as the simulated ones with the results of the traits, including the visualization, can be found at https://dissect.crocs.fi.muni.cz/. DiSSECT is open-source and we welcome any collaborators who have an idea for a new trait, new simulation method, or just want to contribute in another way.

Authors

Vladimír Sedláček
Vojtěch Suchánek
Antonín Dufka

Thanks to Ján Jančár for help with the curve database and CRoCS members for fruitful discussions. Computational resources were supplied by the project "e-Infrastruktura CZ" (e-INFRA LM2018140) provided within the program Projects of Large Research, Development and Innovations Infrastructures.

Setup

Using virtual environment:

Clone with git clone --recurse-submodules https://github.com/crocs-muni/DiSSECT.git
Create virtual environment for python in sage: sage --python3 -m venv --system-site-packages environment
Activate the environment: source environment/bin/activate
Run pip install --editable . in DiSSECT folder

Alternatively without virtual environment (not recommended):
From the root directory, run sage --python3 setup.py develop --user to initialize the project.

Running the curve traits

To feed the trait results directly to a local MongoDB database, run ./run.py -n <trait_name> -c <curve_type> [-b <max_bit_length>] [-v] [-a <allowed cofactors>].

Alternatively, to get results as JSON files, run ./run_traits.py in directory traits. Use the -h flag to get the help menu. To merge the results of a trait ( a05 in this case) into single file, run ./merge_trait_results.py -n a05.

Example usage

To run trait a05 on all standard curves of bitsizes up to 192 with cofactor 1 or 2 in verbose mode using 3 cores and 100 jobs, run ./run_traits.py -n a05 -c std -v -b 192 -a 1 2 -t 3 -j 100.

Supported curve sets

std: all standard curves
sim: all simulated curves
sample: curves secp112r1, secp192r1, secp256r1
all: all curves in the database

Overview of available traits

name	description	implemented	computed*	time req.**	memory req.***
a01	group stucture of the curve in field extensions	✅	❌	high	low
a02	factorization of the CM discriminant	✅	✅	high	medium
a03	factorization of the quadratic twist cardinality	✅	✅	high	medium
a04	factorizations of $kn\pm 1$	✅	✅	high	high
a05	field extensions containing nontrivial/full $l$-torsion	✅	✅	medium	low
a06	factorizations of ratios of CM discriminants in extension fields and base fields	✅	✅	high	medium
a07	embedding degree	✅	✅	medium	low
a08	class number of the maximal order of the endomorphism ring	✅	✅	high	low
a12	multiplicative orders of small primes modulo curve order	✅	✅	medium	medium
a22	factorizations of small division polynomials	✅	✅	high	high
a23	volcano depth and crater degree in the $l$-isogeny graph	✅	✅	low	low
a24	field extensions containing nontrivial/full number of $l$-isogenies	✅	✅	medium	low
a25	trace in field extensions and its factorization	✅	✅	low	low
a28	Number of j-invariants adjacent to the curve by l-isogeny	✅	✅	medium	low
a29	Torsion order of the lift of E to curve over Q	✅	✅	low	low
i04	number of points with low Hamming weight	✅	✅	medium	low
i06	square parts of $4q-1$ and $4n-1$	✅	✅	low	low
i07	distance of $n$ from the nearest power of two and multiple of 32/64	✅	✅	low	low
i08	bit length of small inverted generator multiples	✅	✅	low	low
i13	$ a^3/b^2$, i.e. value used in x962, fips,secg	✅	✅	low	low
i14	overlap in curve coefficients	✅	✅	low	low
i15	curve coefficients in Weierstrass form	✅	✅	low	low

Notation: $n$ is the curve order, $q$ is the order of the base field
* on sim and std curves with at most 256 bits and cofactor 1
** this is very rough and subjective
*** on the above dataset: low is <100 MB, medium is 100-500 MB, high is >500 MB (measuring JSONs)

Overview of planned traits

name	description	fully specified
a09	existence of pairing-friendly cycles	❌
a10	existence of factorization bases	❌
a11	minimal codewords in elliptic codes	❌
a13	images of points under pairings	❌
a14	conductor and modularity	❌
a15	the lattice associated to the curve over $C$	❌
a16	the Neron model	❌
a17	the L-series	❌
a18	the invariant differential	❌
a19	local heights	❌
a20	$S$-integral points	❌
a21	Galois groups of various polynomials	❌
a26	lifts of curves to other fields	❌
a27	distribution and sizes of isogeny classes	❌
i01	curves under parameter bitflips	❌
i02	curves with the same $j$-invariant/group order, but different $q$	❌
i03	the number of modular reductions in various computations	❌
i05	vulnerability against $\rho$ and kangaroo	❌
s01	statistical properties of scalar multiplication	❌
s02	distribution of point coordinates in various intervals	❌
s03	properties of other curve models	❌
s04	modular polynomials in given $j$-invariant	❌
s05	images of points under isogenies	❌
s06	summation polynomials	❌
s07	distributions of curves with similar properties	❌
s08	properties of the function shifting a point by the generator	❌

Unit tests

Run sage --python3 -m unittest discover in directory traits/unit_tests/. Only unit tests starting with test will be run; those starting with local have to be run manually (as they require resources not available on the server).

Parameters and structure

From directory traits, parameter files can be (re)generated by sage --python3 params.py and structure files can be ( re)generated by sage --python3 traits/traits_structures.py -t all (both of these are already done during the setup).

Importing curves or results to a local database

After setting up a local database with MongoDB, you can run python3 utils/database_handler.py curves [database_uri] <curve_files...> to import curves from individual JSON files, or python3 utils/database_handler.py curves [database_uri] all to import all curves from their presumed directories.

Similarly, you can run python3 utils/database_handler.py results [database_uri] <trait_name> <results_file> to import trait results from a JSON file, or python3 utils/database_handler.py results [database_uri] <trait_name> auto to auto-import from the presumed location, or even python3 utils/database_handler.py results [database_uri] all to do this for all traits.

adamivora / DiSSECT