Problem - Kevin Tafuro

7.4 Manipulating Big Numbers
Problem
You need to do integer-based arithmetic on numbers that are too large to represent
in 32 (or 64) bits. For example, you may need to implement a public key algorithm
that isn’t supported by the library you’re using.
Solution
Use a preexisting library for arbitrary-precision integer math, such as the BIGNUM
library that comes with OpenSSL(discussed here) or the GNU Multi-Precision (gmp)
library.
Discussion
Most of the world tends to use a small set of public key primitives, and the popular
libraries reflect that fact. There are a lot of interesting things you can do with public
key cryptography that are in the academic literature but not in real libraries, such as
a wide variety of different digital signature techniques.
If you need such a primitive and there aren’t good free libraries that implement it,
you may need to strike off on your own, which will generally require doing math
with very large numbers.
In general, arbitrary-precision libraries work by keeping an array of words that represents
the value of a number, then implementing operations on that representation in
software. Math on very large numbers tends to be slow, and software implementation
of such math tends to be even slower. While there are tricks that occasionally
come in handy (such as using a fast Fourier transform for multiplication instead of
longhand multiplication when the numbers are large enough to merit it), such libraries
still tend to be slow, even though the most speed-critical parts are often implemented
in hand-optimized assembly. For this reason, it’s a good idea to stick with a
preexisting library for arbitrary-precision arithmetic if you have general-purpose
needs.
In this recipe, we’ll cover the OpenSSLBIGNUM library, which supports arbitrary
precision math, albeit with a very quirky interface.
Initialization and cleanup
The BIGNUM library generally lives in libcrypto, which comes with OpenSSL. Its
API is defined in openssl/bn.h. This library exports the BIGNUM type. BIGNUM objects
always need to be initialized before use, even if they’re statically declared. For example,
here’s how to initialize a statically allocated BIGNUM object:
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