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Bosphorus, ANF simplifier and solver, and ANF-to-CNF converter

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# meelgroup/bosphorus

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### requirements.test.txt

Bosphorus is an ANF simplification and solving tool. It takes as input an ANF over GF(2) and can simplify and solve it. It uses many different algorithms, including XL, SAT, Brickenstein's ANF-to-CNF conversion, Gauss-Jordan elimination, etc. to simplify and solve ANFs.

The main use of the system is to simplify and solve ANF problems. It should give you highly optimised ANFs and CNFs that it can solve. Its ANF simplifications should be useful is many areas, not just direct ANF-to-SAT solving. For example, it could be useful for helping to break post-quantum cryptograpy problems.

This work was done by Davin Choo and Kian Ming A. Chai from DSO National Laboratories Singapore, and Mate Soos and Kuldeep Meel from the National University of Singapore (NUS). If you use Bosphorus, please cite our paper (bibtex) published at DATE 2019. Some of the code was generously donated by Security Research Labs, Berlin.

## ANF simplification and solving

Suppose we have a system of two equations:

``````x1 ⊕ x2 ⊕ x3 = 0
x1 * x2 ⊕ x2 * x3 + 1 = 0
``````

Put this in the ANF file `test.anf`:

``````\$ cat test.anf
x1 + x2 + x3
x1*x2 + x2*x3 + 1
``````

or, you can use the more detailed description:

``````\$ cat test-detail.anf
x(1) + x(2) + x(3)
x(1)*x(2) + x(2)*x(3) + 1
``````

Let's simplify, output a simplified ANF, a simplified CNF, solve it and write out the solution:

``````\$ ./bosphorus --anfread test.anf --anfwrite out.anf --cnfwrite out.cnf --solvewrite solution
``````

The simplified ANF is in `out.anf`:

``````\$ cat out.anf
c -------------
c Fixed values
c -------------
x(2) + 1
c -------------
c Equivalences
c -------------
x(3) + x(1) + 1
c UNSAT : false
``````

The simplified CNF is in `out.cnf`:

``````3 0
2 4 0
-2 -4 0
``````

This CNF represents all the solutions to the ANF, i.e. it's equivalent to the ANF.

A solution to the problem is in `solution`:

``````\$ cat solution
v -0 1 2 -3
``````

This means x0 is `false`, x1 is `true`, x2 is `true` and x3 is `false`.

Explanation of simplifications performed:

• The first linear polynomial rearranged to `x1 = x2 + x3` to eliminate x1 from the other equations
• The second polynomial becomes `(x2 + x3) * x2 + x2 * x3 + 1 = 0`, which simplifies to `x2 + 1 = 0`
• Substituting `x2 + 1 = 0` yields `x1 + x3 + 1 = 0`

## List all solutions of an ANF

To find all solutions to `myfile.anf`:

``````./bosphorus \
--cnfwrite myfile.cnf \
--solve --allsol
[...]
s ANF-SATISFIABLE
v x(0) x(1)+1 x(2) x(3)
s ANF-SATISFIABLE
v x(0) x(1)+1 1+x(2) 1+x(3)
s ANF-UNSATISFIABLE
c Number of solutions found: 2
``````

Where `x(0)` means `x(0)` must be FALSE and `x(1)+1` means `x(1)` must be TRUE.

To convert `myfile.anf` to `myfile.cnf` with all the simplifications:

## Counting solutions of an ANF

Sometimes, there are too many solutions to an ANF to list them all (e.g. 2**40). You can count the number of solutions of an ANF in `test.anf` by using the standard translation and taking advantage of the projection written inside the CNF. This projection set is written as `c ind var1 var2 ... varn 0`. Many counters, such as ApproxMC are able to use this format to count the solutions in the CNF. Here is how to do it with ApproxMC:

``````./bosphorus --anfread test.anf --cnfwrite out.cnf
./approxmc out.cnf
[...]
c [appmc] Number of solutions is: 256*2**6
s mc 16384
``````

If the number of solutions is low (say, less than 1000) you can also use CryptoMiniSat to do the counting:

``````./bosphorus --anfread test.anf --cnfwrite out.cnf
./cryptominisat --maxsol 100000 out.cnf
[...]
c Number of solutions found until now:    16384
s UNSATISFIABLE
``````

## CNF simplification

This usage of the tool is EXPERIMENTAL. Do not, under any circumstances, rely on its correctness or veracity. In general `--cnfread` is not well-supported. If you are still interested, then Bosphorus can simplify and solve CNF problems. When simplifying or solving CNF problems, the CNF is (extremely) naively translated to ANF, then simplifications are applied, and a sophisticated system then translates the ANF back to CNF. This CNF can then be optinally solved.

Let's say you have the CNF:

``````\$ cat test.cnf
-2  3  4 0
2 -3 0
2  3 -4 0
-2 -3 -4 0
1  5 0
-1 -5 0
``````

Let's simplify and get the ANF:

``````\$ ./bosphorus --cnfread test.cnf --anfwrite out2.anf
\$ cat out2.anf
x(1)*x(2)*x(3) + x(1)*x(2) + x(1)*x(3) + x(1)
x(1)*x(2)*x(3) + x(1)*x(2) + x(2)*x(3) + x(2)
x(1)*x(2) + x(1) + x(2) + 1
x(1)*x(2)*x(3)
x(1) + x(2) + x(3)
c -------------
c Equivalences
c -------------
x(4) + x(0) + 1

``````

The system recovered XOR `x(1) + x(2) + x(3)` using ElimLin from the top 4 equations that encode the CNF's first 4 clauses. This resoution is in fact non-trivial, and can lead to interesting facts that can then be re-injected back into the CNF. Note that the first 4 clauses encode an XOR because the 2nd clause can be extended to the weaker clause `2 -3 4 0`, giving the trivial encoding of `x(1) + x(2) + x(3)` in CNF.

# Building, Testing, Installing

You must install M4RI, BriAl, and CryptoMiniSat to use compile Bosphorus. Below, we explain how to compile them all. First, the dependencies:

``````sudo apt-get install build-essential cmake zlib1g-dev \
libboost-program-options-dev libm4ri-dev libboost-test-dev
``````

### Install BRiAl

Try installing using `apt-get install brial-dev`. If that does not work, compile and install from source:

``````git clone --depth 1 https://github.com/BRiAl/BRiAl
cd BRiAl
aclocal
libtoolize --copy
automake
autoconf
./configure
make -j4
sudo make install
``````

### Install CryptoMiniSat

``````git clone --depth 1 https://github.com/msoos/cryptominisat.git
cd cryptominisat
mkdir build
cd build
cmake ..
make -j4
sudo make install
``````

Note (For MacOS): If you encounter `cryptominisat.h:30:10: fatal error: 'atomic' file not found` in `#include <atomic>` during compilation, you may need to use `CFLAGS='-stdlib=libc++' make` instead of just `make`.

### Build Bosphorus

``````git clone --depth 1 https://github.com/meelgroup/bosphorus
cd bosphorus
mkdir build
cd build
cmake ..
make -j4
./bosphorus -h
``````

## Mapping solutions from CNF to ANF

Let's take a simple ANF:

``````\$ cat test.anf
x(1) + x2 + x3
x1*x2 + x2*x3 + 1
``````

Let's simplify and it to CNF:

``````./bosphorus --anfread test.anf  --cnfwrite test.cnf --solmap solution_map
``````

Let's solve with any SAT solver:

``````lingeling test.cnf > cnf_solution
``````

Let's map the CNF solution back to ANF using the python script under `utils/map_solution.py`:

``````./map_solution.py solution_map cnf_solution
c solution below, with variables starting at 0, as per ANF convention.
s ANF-SATISFIABLE
v x(0) 1+x(1) 1+x(2) x(3)
``````

This means that `x(0)=FALSE`, `x(1)=TRUE`, `x(2)=TRUE`, and `x(3)=FALSE`.

If you want all solutions:

``````./cryptominisat x --maxsol 10000000 > cnf_solutions
``````

Then take the solutions from `cnf_solutions` individually, put them in a file, and call `map_solution` on it, as before.

## Fuzzing

The tool comes with a built-in ANF fuzzer. To use, install cryptominisat, then run:

``````git clone --depth 1 https://github.com/meelgroup/bosphorus
cd bosphorus
mkdir build
cd build
ln -s ../utils/* .
./build_normal.sh
./fuzz.sh /usr/bin/cryptominisat5
``````

Where the argument to `fuzz.sh` must be the location of the cryptominisat5 binary.

## Testing

The test suite uses LLVM lit and stp OutputCheck. It's convenient to use a Python virtual environment:

``````cd bosphorus
python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.test.txt
``````

Now you can run the tests via:

``````lit tests
``````

# Known issues

• PolyBoRi cannot handle ring of sizes over approx 1 million (1048574). Do not run `bosphorus` on instances with over a million variables.

Bosphorus, ANF simplifier and solver, and ANF-to-CNF converter

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## Releases 3

Bosphorus 3.1.0 Latest
Apr 30, 2023

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