Supersingular Isogeny-Based Cryptographic constructions

This repository includes a python-code library named sibc, which allows a user-friendly interface to deal with some isogeny-based cryptographic primitive.

The current version of the sibc library has integrated CSIDH and B-SIDH schemes using traditional and velusqrt formulae on Montgomery curve x-only projective coordinates; in particular, sibc allows working with prime and quadratic field classes that permit operating field elements as integers. Additionally, the cryptographic primitives are implemented in constant-time concerning the number of field operations, where a constant-time procedure refers to its running time does not depend on the input or it possibly does from randomness as CSIDH does.

It is worthing to mention, the library is constantly extended, and some signature schemes will be integrated into the sibc library.

The sibc library aims to allow isogeny-contributors for building new primitives with a constant-time nature.

Installation

Install the sibc module which provides the sibc program:

sudo python3 setup.py install

For development:

For this installation method, any further modification in sibc directory will be reflect when running sibc library.

# Installing required package
# Before running the following commands, ensure you have the lastest version of pip
pip3 install dh click numpy progress matplotlib networkx stdeb setuptools-scm setuptools

# only pip3 install cpuinfo is missing for macOS (to be fixed in coming versions)
pip3 install pytest pytest-xdist

# Installing the library
sudo pip3 install -e .

Debian package build

To build a package for Debian or Ubuntu, we suggest the use of stdeb:

sudo apt install -y dh-python python3-click python3-progress\
  python3-numpy python3-matplotlib python3-networkx \
  python3-stdeb python3-setuptools-scm python3-setuptools python3-cpuinfo
python3 setup.py bdist_deb
sudo dpkg -i deb_dist/python3-sibc_0.0.1-1_all.deb

Usage

The syntax compilation can be viewed by running one of the following three commands:

# Corresponding with the key-exchange protocol
sibc --help

# Corresponding with benchmarking (only for CSIDH, which has a variable
# running-time cost independent from the key)
sibc csidh-bench

# Corresponding with the costs of KPs (Kernel Point computation), xISOG
# (isogeny construction), and xEVAL (isogeny evaluation)
sibc csidh-test

Usage for the sibc tool:

Usage: sibc [OPTIONS] COMMAND [ARGS]...

    ,-~~-.___.          
   / |  '     \        
  (  )         0        
   \_/-, ,----'         
      ====           // 
     /  \-'~;    /~~~(O)
    /  __/~|   /       |
  =(  _____| (_________|

Options:
  -p, --prime [p434|p503|p610|p751|p253|p255|p247|p237|p257|p512|p1024|p1792|p2048|p4096|p5120|p6144|p8192|p9216]
                                  [default: p512]
  -f, --formula [tvelu|svelu|hvelu]
                                  [default: hvelu]
  -a, --algorithm [sidh|sike|csidh|bsidh|bsike]
                                  [default: csidh]
  -s, --style [wd1|wd2|df]        [default: df]
  -e, --exponent [1|2|3|4|5|6|7|8|9|10]
                                  [default: 10]
  -m, --multievaluation           [default: False]
  -c, --curvemodel [edwards|montgomery]
                                  [default: montgomery]
  -b, --benchmark INTEGER         [default: 128]
  -t, --tuned                     [default: False]
  -u, --uninitialized             [default: False]
  -v, --verbose                   Not the kind of verbosity you might expect
                                  [default: False]
  --version                       Show the version and exit.
  --help                          Show this message and exit.

Commands:
  bsidh-main                   Random instance example of a key-exchange
  bsidh-precompute-parameters  Precomputation of tuned velusqrt parameters
  bsidh-precompute-strategy    Precomputation of optimal strategies
  bsidh-test                   GF(p²)-operation cost of kps, xisog, and...
  csidh-bench                  Average GF(p)-operation cost of a GAE
  csidh-bounds                 Greedy-based search of optimal exponents
  csidh-dh                     Derive shared secret key from CSIDH sk,...
  csidh-genkey                 Generate random CSIDH secret key
  csidh-header                 Optimal strategies as C-code headers files
  csidh-ijk                    Velusqrt parameters as C-code headers files
  csidh-main                   Random instance example of a key-exchange
  csidh-precompute-parameters  Precomputation of tuned velusqrt parameters
  csidh-precompute-strategy    Precomputation of optimal strategies
  csidh-pubkey                 Derive CSIDH public key from CSIDH secret key
  csidh-sdacs                  SDACs as C-code headers files
  csidh-test                   GF(p)-operation cost of kps, xisog, and xeval
  decaps                       (B)SIKE decapsulation
  encaps                       (B)SIKE encapsulation
  keygen                       Generate random (B)SIKE secret and public...
  plot-strategy                draw strategy graphs as a subgraph...
  print-timing
  sidh-precompute-strategy     Precomputation of optimal strategies

SIDH cryptographic API

CSIDH, BSIDH, SIDH, SIKE, and BSIKE objects are available from the sibc package and module.

Automatically generated documentation is available with pydoc after sibc is installed:

pydoc3 sibc.csidh
pydoc3 sibc.bsidh
pydoc3 sibc.sidh

Command Line Interface: examples

# CSIDH
sk_a="$(sibc csidh-genkey)"
pk_a="$(echo "$sk_a"|sibc csidh-pubkey -)"
sk_b="$(sibc csidh-genkey)"
pk_b="$(echo "$sk_b"|sibc csidh-pubkey -)"
ss_a="$(echo "$sk_a"|sibc csidh-dh - "$pk_b")"
ss_b="$(echo "$sk_b"|sibc csidh-dh - "$pk_a")"
echo $ss_a
echo $ss_b

# SIKE
sk="$(sibc -a sidh -p p434 keygen)"
pk3=`echo "${sk}" | tail -n1`
ck="$(echo "$pk3"|sibc -a sidh -p p434 encaps -)"
c0=`echo "${ck}" | head -1`
c1=`echo "${ck}" | tail -2 | head -1`
K=`echo "${ck}" | tail -n1`
K_="$(echo "$sk"|sibc -a sidh -p p434 decaps - "$c0 $c1")"
echo $K
echo $K_

# BSIKE
sk="$(sibc -a bsidh -p p253 keygen)"
pk3=`echo "${sk}" | tail -n1`
ck="$(echo "$pk3"|sibc -a bsidh -p p253 encaps -)"
c0=`echo "${ck}" | head -1`
c1=`echo "${ck}" | tail -2 | head -1`
K=`echo "${ck}" | tail -n1`
K_="$(echo "$sk"|sibc -a bsidh -p p253 decaps - "$c0 $c1")"
echo $K
echo $K_

Basic shared secret generation example with CSIDH

from sibc.csidh import CSIDH, default_parameters
csidh = CSIDH(**default_parameters)

# alice generates a key
alice_secret_key = csidh.secret_key()
alice_public_key = csidh.public_key(alice_secret_key)

# bob generates a key
bob_secret_key = csidh.secret_key()
bob_public_key = csidh.public_key(bob_secret_key)

# if either alice or bob use their secret key with the other's respective
# public key, the resulting shared secrets are the same
shared_secret_alice = csidh.dh(alice_secret_key, bob_public_key)
shared_secret_bob = csidh.dh(bob_secret_key, alice_public_key)

# Alice and bob produce an identical shared secret
assert shared_secret_alice == shared_secret_bob

Basic shared secret generation example with BSIDH

from sibc.bsidh import BSIDH, default_parameters
bsidh = BSIDH(**default_parameters)
sk_a, pk_a = bsidh.keygen_a()
sk_b, pk_b = bsidh.keygen_b()
ss_a, ss_b = bsidh.derive_a(sk_a, pk_b), bsidh.derive_b(sk_b, pk_a)
ss_a == ss_b

Basic example with BSIKE (BSIDH + key encapsulation)

from sibc.bsidh import BSIKE, default_parameters
bsike = BSIKE(**default_parameters)
s, sk3, pk3 = bsike.KeyGen()
c, K = bsike.Encaps(pk3)
K_ = bsike.Decaps((s, sk3, pk3), c)
K == K_

bsike255 = BSIKE('montgomery', 'p255', 'hvelu', True, False, False, False)
s, sk3, pk3 = bsike255.KeyGen()
c, K = bsike255.Encaps(pk3)
K_ = bsike255.Decaps((s, sk3, pk3), c)
K == K_

Basic shared secret generation example with SIDH

from sibc.sidh import SIDH, default_parameters
sidh = SIDH(**default_parameters)
sk_a, pk_a = sidh.keygen_a()
sk_b, pk_b = sidh.keygen_b()
ss_a, ss_b = sidh.dh_a(sk_a, pk_b), sidh.dh_b(sk_b, pk_a)
ss_a == ss_b

Basic example with SIKE (SIDH + key encapsulation)

from sibc.sidh import SIKE, default_parameters
sike = SIKE(**default_parameters)
s, sk3, pk3 = sike.KeyGen()
c, K = sike.Encaps(pk3)
K_ = sike.Decaps((s, sk3, pk3), c)
K == K_

sike503 = SIKE('montgomery', 'p503', False, False)
s, sk3, pk3 = sike503.KeyGen()
c, K = sike503.Encaps(pk3)
K_ = sike503.Decaps((s, sk3, pk3), c)
K == K_

Adding new primes

The field characteristic p should be stored in directory data/sop/, and CSIDH and BSIDH have different structures (see below):

# CSIDH format (here p = cofactor * l_1 * .... l_n - 1)
cofactor l_1 l_2 ... l_n

# BSIDH format
Hexadecimal representation of the prime p
4 l_1 l_2 ... l_n
c e_1 e_2 ... e_n
l'_1 l'_2 ... l'_m
e'_1 e'_2 ... e'_m

# SIDH format: p = 2^{e_2} * 3^{e_3} - 1
e_2 e_3

For the case of BSIDH, M := (4^c * l_1^{e_1} * l_2^{e_2} * ... * l_n^{e_n}) must divide (p + 1), and N := (l'_1^{e'_1} * l'_2^{e'_2} * ... * l'_n^{e'_n}) must divide (p-1). Additionally, the order-M generators PA, QA and PQA := PA - QA should be stored in directory gen/ as projective x-coordinate points. Similarly, the order-N generators PB, QB and PQB := PB - QB also should be stored it the same directory. Both 3-tuples of points must be stored in a single file with the following syntax:

Re(x(PA)) Im(x(PA)) Re(x(QA)) Im(x(QA)) Re(x(PQA)) Im(x(PQA))
Re(x(PB)) Im(x(PB)) Re(x(QB)) Im(x(QB)) Re(x(PQB)) Im(x(PQB))

where Re(X) and Im(X) denote the real and imaginary parts of X with respect to F_p[i]/(i^2 + 1), respectively. Moreover, all the above twelve integers should be stored in hexadecimal.

For SIDH, generators have order either M=2^{e_2} or N=3^{e_3}.

Examples

We summarize some examples of runs of the sibc tool as follows:

# CSIDH
# A single random intances of a key exchange
sibc -p p512 -f hvelu -a csidh -s df -e 10 csidh-main
sibc -p p512 -f hvelu -a csidh -s df -e 10 -m csidh-main
sibc -p p512 -f hvelu -a csidh -s df -e 10 -t csidh-main
sibc -p p512 -f hvelu -a csidh -s df -e 10 -m -t csidh-main
# Average GF(p)-operation cost of 64 random instances
sibc -p p512 -f hvelu -a csidh -s df -m -e 10 -b 64 csidh-bench
sibc -p p512 -f hvelu -a csidh -s df -m -e 10 -b 64 -m csidh-bench
sibc -p p512 -f hvelu -a csidh -s df -m -e 10 -b 64 -t csidh-bench
sibc -p p512 -f hvelu -a csidh -s df -m -e 10 -b 64 -t -m csidh-bench
# GF(p)-operation cost of kps, xisog, and xeval blocks
sibc -p p512 -f hvelu -a csidh csidh-test
sibc -p p512 -f hvelu -a csidh -m csidh-test
sibc -p p512 -f hvelu -a csidh -t csidh-test
sibc -p p512 -f hvelu -a csidh -t -m csidh-test

# BSIDH
# A single random intances of a key exchange
sibc -p p253 -f hvelu -a bsidh bsidh-main
sibc -p p253 -f hvelu -a bsidh -m bsidh-main
sibc -p p253 -f hvelu -a bsidh -t bsidh-main
sibc -p p253 -f hvelu -a bsidh -t -m bsidh-main
# GF(p²)-operation cost of kps, xisog, and xeval blocks
sibc -p p253 -f tvelu -a bsidh bsidh-test
sibc -p p253 -f svelu -a bsidh bsidh-test
sibc -p p253 -f hvelu -a bsidh -t bsidh-test

Remark, our implementation allows us to plot each optimal strategy required (only tested in Linux machines):

# CSIDH
sibc -p p512 -f tvelu -a csidh -s df -e 10 plot-strategy
sibc -p p512 -f svelu -a csidh -s wd1 -e 10 plot-strategy
sibc -p p512 -f hvelu -a csidh -s wd2 -e 5 plot-strategy

# BSIDH
sibc -p p253 -f hvelu -a bsidh plot-strategy
sibc -p p253 -f hvelu -a bsidh -m plot-strategy
sibc -p p253 -f hvelu -a bsidh -t -m plot-strategy

# SIDH
sibc -p p434 -a sidh plot-strategy
sibc -p p503 -a sidh plot-strategy
sibc -p p610 -a sidh plot-strategy
sibc -p p751 -a sidh plot-strategy

Additionally, one can created files with extension .h that includes all the required variables in a the sdacs, strategies, and velusqrt (at least for CSIDH implementations).

# Suitable bounds search with e = 10.
sibc -a csidh -p p512 -s df -f hvelu -e 10 -u csidh-bounds # The greedy-based algorithm on a large searching space, it could take hours or even days!: option -u is required
# SDACs (options -s and -e do not affect the output)
sibc -p p512 -f hvelu -a csidh -s df -e 10 csidh-sdacs
# Optimal sizes of I, J, and K required in velusqrt (options -s and -e do not affect the output)
sibc -p p512 -f hvelu -a csidh -s df -e 10 -t csidh-ijk  # option -t is required
# Optimal strategies
sibc -p p512 -f hvelu -a csidh -s df -e 10 -t csidh-header

BSIDH primes

Currently only p253, p255, p247, p237, and p257 are implemented and tested in the current API. Extending this to other primes is straight-forward.

SIDH primes

Currently only p434, p503, p610, and p751 are implemented and tested in the current API. Extending this to other primes is straight-forward.

Precomputing data for a new prime instances

Generating new data can be easily done by adding and running to either misc/create-csidh-data.sh or misc/create-bsidh-data.sh. The new prime number description should b stored as previously mentioned.

bash misc/create-csidh-data.sh
bash misc/create-bsidh-data.sh

Also, you can do it manually by doing something as follows:

# CSIDH
sibc-precompute-sdacs -p p512 -a csidh # SDACs
sibc -p p512 -f hvelu -a csidh -m -t -u csidh-precompute-parameters # Tuned velusqrt parameters
sibc -p p512 -f hvelu -a csidh -s df -m -u csidh-precompute-strategy # Strategies
# BSIDH
sibc-precompute-sdacs -p p253 -a bsidh # SDACs
sudo sibc -p p253 -f svelu -a bsidh -u bsidh-precompute-parameters # Tuned velusqrt parameters: the option -u is required
sudo sibc -p p253 -f svelu -a bsidh -u bsidh-precompute-strategy # Strategies
# SIDH
sibc -p p434 -a sidh -u sidh-precompute-strategy # Strategies
sibc -p p503 -a sidh -u sidh-precompute-strategy # Strategies
sibc -p p610 -a sidh -u sidh-precompute-strategy # Strategies
sibc -p p751 -a sidh -u sidh-precompute-strategy # Strategies

Furthermore, you can create tests by running bash misc/create-tests.sh and bash misc/test-cli.sh (only CSIDH, BSIKE, and SIKE are handled by now).

If either a new prime instace or primitive is included, then you should add it to misc directory. New primitives require new bash scripts. SIDH instances are simple, only one configuration (for now), and thus you can omit adding them in misc directory.

Remarks

The primes labeled as p253, p255, p247, and p237 correspond with the examples 2, 3, 5, and 6 from B-SIDH paper, respectively. In particular, p257 denotes the prime number given in velusqrt paper. The field airthmetic is centered on primes p = 3 mod 4. Multiplying and squaring in GF(p²) = GF(p)[u]/(u^2 + 1) have a cost of 3 and 2 multiplications in GF(p), respectively.

The current implementation does not have implemented the B-SIDH key validation, it will be included in the next library version.

Adding new prime instances

When adding a new isogeny-based instances (the prime number, and public parameters) should be included in parameter list of sibc/__main__.py (click option -p, --prime) by modifying sibc/constants.py (updating the lists/dictionary [csidh/bsidh]_primes). If a new primitive is included, then you need to update sibc/__main__.py by extending the click options -p (--prime) and -a (--algorithm), and also to include its branch in sibc/montgomery/curve.py and sibc/montgomery/isogeny.py files.

Changes

Significant changes are listed in the CHANGELOG file. Future integrations/modifications are listed in the TODOLIST file.

Authors

Gora Adj gora.adj@udl.cat, gora.adj@gmail.com;
Jesús-Javier Chi-Domínguez jesus.dominguez@tii.ae, chidoys@gmail.com; and
Francisco Rodríguez-Henríquez francisco@cs.cinvestav.mx.

Main cryptographer contributors (collaborators):

Jorge Chávez-Saab jchavez@computacion.cs.cinvestav.mx, jorgechavezsaab@gmail.com

Logo creator contributor

Fabiola-Argentina Hernández-Torres farg.cls@outlook.com

User-friendly interface contributors

Jacob Appelbaum j.appelbaum@tue.nl; and
Leif Ryge leif@synthesize.us.

All contributors are listed in the CONTRIBUTORS file.

How to cite this library

@misc{sibc,
  author = {{Gora Adj} and {Jes\'us-Javier Chi-Dom\'inguez} and {Francisco Rodr\'iguez-Henr\'iquez}},
  title = {{SIBC} Python library},
  year = {2021},
  howpublished = {\url{https://github.com/JJChiDguez/sibc/}}
}

License

This project is licensed under the GNU general public license - see the LICENSE file for details.

Funding

This project has initially received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 804476), while the second author was doing a postdoctoral research stay in Tampere University.

The third author received partial funds from the Mexican Science council CONACyT project 313572.

ioerror/sibc