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Coppersmith small roots

Coppersmith small roots method (solving polynomial equation over composite modulus on small bounds)

Features

  • multiple coppersmith equation support
    • univariate polynomial
    • multivariate linear polynomial (Herrmann-May method)
    • multivariate polynomial (Jochemsz-May heuristic method)

Mainly, dealing with theoretically established Coppersmith method applicable equation. We recommend using univariate or linear (instead of heuristic) if you know the type of equation.

  • automated parameter tuning

Firstly choose suitable parameters based on bounds and $\beta$, and then increment parameters until finishing to find an integer equation.

  • selectable internal LLL/BKZ method

  • selectable solving integer equation method

    • jacobian Newton method (numerical)
    • Hensel lifting
    • triangulation (Groebner basis + dealing with non-zero dimensional ideal partially)

Installation

$ git clone https://github.com/kionactf/coppersmith --sync

If you use fplll or flatter, need to build these packages. if you already installed them, you can use this by setting these install paths to fplll_path or flatter_path on lll.py.

Or you can create a docker image. It is based on sagemath/sagemath:latest image and also downloads well-known lattice libraries. (If you know good other libraries, let us know for including these.)

$ docker build -t coppersmith .
$ docker run --rm -it coppersmith /bin/bash

Usage

Call coppersmith_onevariable.coppersmith_onevariable or coppersmith_linear.coppersmith_linear with Sagemath PolynomialRing over Zmod(N), bounds, beta.

See example.py.

Also, you can use only LLL by calling lll.do_lattice_reduction with Sagemath matrix over ZZ and some optimization parameters. And lll.babai for CVP solver.

For integer equations solver, use rootfind_ZZ.rootfind_ZZ with a list of Sagemath polynomial over ZZ and bounds.

Custom Configuration

(The feature was introduced on 2024/3/8.)

Although we set up built-in default settings based on our some experiments, some users say the setting does not fit on their environments.

So we prepares a configuration interface context for lll.py and rootfind_ZZ.py.

You can use it by importing context just as the following example:

from coppersmith_linear import * # including context
from lll import FLATTER
from rootfind_ZZ import JACOBIAN, TRIANGULATE

# use flatter
context.lllopt = {'algorithm':FLATTER, 'use_pari_kernel':True}
# use jacobian method and triangulate in order,
# and for jacobian method set the number of iteration(loop) as 32 (much less comparing the default value 1024)
context.rootfindZZopt = {'algorithm':(JACOBIAN, TRIANGULATE), 'maxiternum':32}

# debug output enable
logger.setLevel(DEBUG)

P = PolynomialRing(...)
f = ...
beta = ...
bounds = [...]
coppersmith_linear(f, bounds, beta)

You can check the list of options and their default values at register_options_lll and register_options_rootfind_ZZ at contextclass.py.

Note (use_pari_kernel)

For computing LLL, we use pari.matker for eliminating linearly dependent vectors for defining lattice. The process needs to use flatter. Though pari.matker is fast and outputs correct results in many cases, it sometimes outputs wrong results. (You can check this by running lll.test().) You may disable to use pari.matker by setting the option use_pari_kernel=False, where it forces using Sagemath kernel (which do internally run computing hermite normal form (HNF).) Note that HNF tends to produce large elements, so it causes LLL process makes super slow.

Background

See Why we could not solve chronophobia… Analysis for Coppersmith method more. (it might be an old article, though.)

How to choose parameters?

Coppersmith small root method is to find a root of the following type equation:

$f(r_1,\ldots,r_n)=0 \pmod{b}$, where $| r_i | < X_i\ (i=1,\ldots,n)$, for known polynomial $f(r)$ and known $N$ such that $b\ |\ N$.

The package function requires the following parameters.

  • basepoly: the polynomial $f(r)$ over Zmod($N$)
  • bounds: the list $[X_1,\ldots,X_n]$ whose positive integers $X_1,\ldots,X_n$
  • beta: a positive floating point number $\beta$ such that $b \ge N^\beta$

For determining $\beta$, we recommend the following guideline.

  • If $b$ is known, then $\beta=\log_{N}(b)$
  • If $b$ is unknown but bitsize of $b$ is known, then $\beta=(\text{bitsize}(b)-1)/(\text{bitsize}(N))$

For example, $N=pq$ and $\text{bitsize}(p)=\text{bitsize}(q)$, then $\beta\simeq 0.499$.

Completeness of the Package

Q: Can you solve many modulus multivariate polynomial systems by the package?

A: Maybe no. It seems to be hard to create general modulus multivariate polynomial solver. It depends on monomials (such as [ $f=ax^2 + by + c$ ] v.s. [ $g=ax^2 + bxy + cy + d$ ]), coefficients (such as [ $f=ax^2-by+c$ ] v.s. [ $g=ax^2-ay+c$ ]), and bounds (such as [ $X \simeq Y$ ] v.s. [ $X \simeq Y^2$ ]). Many papers handle each specific cases.

Especially, we do not recommend to use heuristic method without understanding the polynomial. Heuristic method does not estimate bound condition (unlike univariate case or linear case), so you would be confused that the solver did not output as what you expected.

Alternatively, you may use linearization strategy and/or apply rootfind_ZZ directly. For example, if you want to solve $f=ax^2+by \pmod{N}$, first define ${f_\mathbb{Z}}=ax'+by+kN \in \mathbb{Z}[x',y,k]$, then apply rootfind_ZZ (or other integer polynomial solver) with bound $[X^2, Y]$ (we search $k$ with bruteforce). If we can assume $|k|$ is small, the system will be solved. Note that Coppersmith methods does not necessarily find algebraically independent polynomial sets. So you might have to solve same polynomial system $f_\mathbb{Z}$ even if you forced to apply Coppersmith method. rootfind_ZZ.solve_root_triangulate tries to solve non-zero dimensional linear system.

Note that rootfind_ZZ does not necessarily find all roots, but only a few roots. Finding roots over integer is not easy, so you should not use the package for multiple roots included system. You can devise some methods for avoiding multiple roots. Some method might be narrowing bounds range by converting variables. Some rootfind_ZZ internal functions assume that a root exist near bounds.

Contribution

The package must not be perfect, we want you to be reviewed and improved this. Welcome to post any issues, pull requests and forks. And let us know test cases from any CTF or other resources. Failed test cases may make us to improve the package.

Reference

Some of our codes are based on the following libraries.

multivariate coppersmith method: coppersmith.sage

multivariate coppersmith method: lbc_toolkit/problems/small_roots.sage. Some solvers for integer polynomial roots are at lbc_toolkit/common/systems_solvers.sage.

various coppersmith methods (including Herrmann-May, Blomer-May, Boneh-Durfee, Jochemsz-May, etc.): shared/small_roots/*.py. Some solvers for integer polynomial roots are at shared/small_roots/__init__.py.

jvdsn/crypto-attacks is distributed under:

MIT license (https://github.com/jvdsn/crypto-attacks/blob/master/LICENSE)

(c) 2020 Joachim Vandersmissen.

Copyright

This library is distributed under Apache 2.0 License. See LICENSE.

(C) 2023 kiona

https://github.com/kionactf/coppersmith

For redistribution, just say that it has been changed and note the link for our files.