Book Review: BigNum Math
Reviewed by Cameron Laird
BigNum Math: Implementing Cryptographic Multiple Precision Arithmetic
By Tom St. Denis
Math has our children wary — they
know Daddy likes books and numbers and calculations, but isn't
doing arithmetic going backward? They suspect a trick.
The trick, of course, is that anything can be stretched,
expanded, abstracted, or reworked to become entirely more
challenging than its original form. Children's games become
million-dollar industries when played by sufficiently expert
athletes, and, treated with enough panache, peanut butter is
a gourmet item.
Or, as in the case at hand, when the additions and multiplications
contribute to "Implementing Cryptographic Multiple Precision Arithmetic"
(the book's subtitle), the numeric tables and rules we learned
in grade school appear in a whole new light.
Humans demand privacy and authentication with sufficient enthusiasm
to support a lively market in codes. For our purposes,
codes are cryptographic objects; their effective management requires
multiplication of numbers like 83567040975197715972768175 and
For your virtual private network (VPN) or DVD player to be fast
enough to satisfy you as a customer, then, depends crucially on
arithmetic that goes beyond the kind taught in elementary
school. At one level or another, you rely on "multiple-precision"
Many languages support arithmetic
within definite limits; you can
my_product = factor1 * factor2
but there are problems if
my_product gets too big or
too small or too unlucky. "Unlucky" here refers to the fact that,
while you might see nothing extraordinary in:
one_fifth = 1. / 5.
product = 5. * one_fifth
difference = product - 1.
most computing systems do not, for rather involved
historical and technical reasons, recognize that
difference should be 0.
This just won't do, especially not for codes and cyphers.
Keys have to be exactly right, and it's not enough to say
that the computer "lost a little precision" in unlocking
nuclear launch codes or directing space probes hundreds
of millions of kilometers distant.
Bignums, or multi-precision, arithmetics, give exact results
for operations beyond the ranges supported by conventional
hardware and languages. BigNum Math methodically
documents the algorithms and source code for one particular
"Methodically" here is a key word. Computing arithmetic has
been documented many times. No other work I know, though,
not even Knuth's Volume 2,
simultaneously takes such care to detail
mundane matters like signs and magnitude differences,
while also describing advanced and even deep results from
Karatsuba multiplication and modular exponentiation.
BigNum Math is unique.
Specialized, high in quality
It's quite specialized, both for good and bad. The languages
of expression are pseudocode and executable C. The
algorithms are almost exclusively for multi-precision
integer calculations, not rational, floating-point, or
Let's be clear on these distinctions. A modern CPU might
offer 32- or even 64-bit integer arithmetic in its
dedicated hardware. For the cryptographic purposes that
motivate him, BigNum Math author Tom St. Denis needs
a couple of hundreds of bits of full precision. To achieve
these, he defines C structures that represent multiple-precision
integers. For his purposes, though, the precision is fixed at
compile time. Given this context, he makes detailed and
subtle measurements that yield tiny but definite performance
With some effort, St. Denis's ideas and algorithms can be
translated to infinite-precision code. Infinite-precision
numbers, standard in such languages as Lisp and the latest
Python, are variable-length representations, ones
whose precisions are determined at the time of execution.
Number theorists typically need this sort of arithmetic
more than the multiple-precision forms of BigNum Math.
The text is clear and relentlessly dry. Typographical
errors are few and inconsequential: a single word
missing or out of place. Symbolic expressions have
been reviewed carefully; from everything I've noticed,
they're entirely correct.
Only the very few programmers who need to implement a
multiple-precision arithmetic library will want to read
BigNum Math through from cover to cover. There
are many more people, though, who might like to have
on their shelves a reference that supplies many unique
passages. Anyone tuning high- or variable-precision
arithmetic, anyone looking to implement specialized
primality tests or exponentiation speed-ups, and anyone
analyzing cryptography constraints will find in
BigNum Math descriptions and results published
is vice president of the
specializing in high-reliability and high-performance
applications managed by high-level languages.
He's implemented comprehensive multiple-precision
arithmetics for multiple companies and remains
passionate about details of numeric representations.
For UnixReview, he has reviewed more than 50 books.