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Verificatum Elliptic Curve Library (VEC)


This library implements elliptic curves based on the GMP library, for simultaneous or fixed base modular exponentiation. The formulas used are taken from

The purpose of this library is twofold. Firstly, we want a self-contained and relatively small library that we hope can converge to a static state. Secondly, our particular application allow using certain optimizations and we need low-level access to these.

Our code is not intended to be secure against side-channel attacks, since we do not need it for the applications we have in mind. It is the responsibility of the user to make sure that this is the case in their application.

For a detailed account of such algorithms, a good source is Handbook of Applied Cryptography, Menezes, Oorshot, and Vanstone, which is available for free. We use additional techniques only suitable for very large number of exponentiations.

We have copied optimized code from the OpenSSL project for a few standard curves. These are P-224 (written by Emilia Käsper), P-256 and P-521 (written by David Langley). This code is in turn heavily inspired by the implementation of Curve25519 by Dan Bernstein. The optimized code is roughly a factor of three faster than the code based cleanly on top of GMP, which explains the difference in running time between curves of the same size.

Torbjorn Granlund helped us with the benchmarking. Emilia Käsper took the time to answer our questions about her code and interpreting the benchmarks. Dan Bernstein gave advice on how to implement the default curves and pointed to Emilias code.

Some of the algorithms are implemented using macros. This may be thought of as poor man's C++ templates. The rationale behind this approach is that it is a way to include specialized add/double routines implemented by others in such a way that their datatypes need not be modified. Instead all that is needed is a macro file that maps actual datatypes to macros.

This is necessary, since any conversion to/from a canonical type, e.g., based on GMP, is too costly when the curve operations are aggressively optimized. We avoid C++ to not involve yet another language.

The following assumes that you are using a release. Developers should also read


Building has been tested with GMP 6.1.2. Then LIBRARY_PATH must point to and C_INCLUDE_PATH must point to gmp.h. This is usually the case automatically after installing GMP.

Then use


to build the library.



    make install

to install the library libvec.{la,a,so}.

You may need to run sudo /sbin/ldconfig on some platforms which have flawed implementations of the cache that stores locations of libraries.

If you prefer to use the Clang compiler in place of GCC for the native code, then you may use ./configure CC=clang instead of the above to enable it.

Caution: Please understand that although it seems that Clang works as well as GCC, switching compiler is a large change for mature software.


The software is supposed to be used by other applications, but you can use

    make check


    vec test

to test the arithmetic of all implemented curves, and you can use

    make bench


    vec speed

to get some benchmarks. Please consult the code to see exactly what is measured before drawing any conclusions. Both commands also accept names of curves as input to restrict the execution to these, e.g.,

    vec test P-224 P-256

tests the arithmetic of the curves P-224 and P-256 and nothing else.

API Documentation

You may use

    make api

to build also some documentation using Doxygen (this assumes you have installed doxygen). The API is not installed anywhere. You can copy it to any location.

Reporting Bugs

Minor bugs should be reported in the repository system as issues or bugs. Security critical bugs, vulnerabilities, etc should be reported directly to Verificatum Project. We will make best effort to disclose the information in a responsible way before the finder gets proper credit.


Self-contained elliptic curve library that provides optimized routines for fixed-basis, and simultaneous, exponentiation and similar routines.





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