Skip to content


Repository files navigation

License: MIT Docker Hub

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 \
    --anfread myfile.anf \
    --cnfwrite myfile.cnf \
    --solve --allsol
v x(0) x(1)+1 x(2) x(3)
v x(0) x(1)+1 1+x(2) 1+x(3)
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

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)
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
cd BRiAl
libtoolize --copy
automake --copy --add-missing
make -j4
sudo make install

Install CryptoMiniSat

git clone --depth 1
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
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/

./ solution_map cnf_solution
c solution below, with variables starting at 0, as per ANF convention.
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.


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

git clone --depth 1
cd bosphorus
mkdir build
cd build
ln -s ../utils/* .
./ /usr/bin/cryptominisat5

Where the argument to must be the location of the cryptominisat5 binary.


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.