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Constexpr C++20 Big-Integer / Finite-Field library


This is a header-only template library for fixed-width "small big-integer" computations, for use during run-time as well as compile-time. By "small big integers", we mean numbers with a few limbs (in other words, a few hundred bits), typically occurring in cryptographic applications.

Important note: not all functions in the library are constant-time (when used at run-time); only those for which this is explicitly indicated.

Currently, the library is a work in progress and supports the following operations

  • addition, formal verification: correctness using SAW and constant-timeness using ct-verif new
  • subtraction,
  • multiplication (naive O(n^2) "schoolbook" multiplication) constant-time-verified using ct-verif new
  • division: short division (single-limb divisor) and Donald Knuth's "algorithm D"
  • division: Granlund--Montgomery division by invariant integer (gives constant-time modulo reduction),
  • comparison constant-time-verified using ct-verif new
  • modular addition,
  • extended GCD and modular inverse,
  • Barrett reduction,
  • Montgomery reduction,
  • Montgomery multiplication,
  • Modular exponentiation (based on Montgomery multiplication)
  • Compile-time initialization from a base-10 literal
  • Serialization to ostream as base-10 string (binary serialization is trivial, by just copying the limbs)

Playground new

ctbignum is available in Matt Godbolt's Compiler Explorer! Play around with ctbignum's API, and see the assembly code it gets compiles down to, for the compiler of your choice..!


Because this is a header-only library, installation is as easy as downloading and copying the contents of the include directory into your system's include directory (e.g. /usr/local/include or /opt/local/include).

Required compiler

  • newer: C++20 compliant compiler
  • Until v0.3: Clang 5 or Apple LLVM/Clang 9.00 (C++17 compliant, older compilers may work but are untested)


  • C++ Standard library

Dependencies for compiling and running the benchmarks


// compile with: -std=c++20 (or a more recent standard)
#include <ctbignum/ctbignum.hpp>

// Initialization via (user-defined) literal 
// (with automatic deduction of number of limbs)

using namespace cbn::literals;
constexpr auto number = cbn::to_big_int(6513020836420374401749667047018991798096360820_Z);
constexpr big_int<3> expected_result = {1315566964, 326042948, 19140048};

static_assert(number == expected_result, "initialization failure");

(See unit tests for more examples.)


ctbignum - Multi-Precision Arithmetic for Cryptology in C++, at Run-Time and at Compile-Time

Citing our work

If you would like to mention our library in your academic publication, then please cite the following work: Multiprecision Arithmetic for Cryptology in C++ - Compile-Time Computations and Beating the Performance of Hand-Optimized Assembly at Run-Time, Niek J. Bouman, 2018

    author       = {Bouman, Niek J.},
    title        = {Multiprecision Arithmetic for Cryptology in C++ - Compile-Time Computations and Beating the Performance of Hand-Optimized Assembly at Run-Time},
    howpublished = {arXiv:1804.07236}, 
    year         = {2018},
    note         = {\url{}},


To run the tests in a docker image using .gitlab-ci.yml:

  • Get gitlab-runner
  • Run gitlab-runner exec docker build from the project root directory