hcrypto/libtommath/tommath.tex

183 The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation
238 It is possible to implement algorithms which require large integers with fixed precision algorithms.  For example, elliptic
271 package.  As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used
368 provoking.  Wherever possible the problems are forward minded, stating problems that will be answered in subsequent
444 processors.  Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window
448 (\textit{API}) has been kept as simple as possible.  Often generic place holder routines will make use of specialized
453 Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project.  Ideally the library should
495 effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely
679 A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them
705 temporary mp\_ints, and return as soon as possible.  The goal is to leave the system in the same state it was when the
1048 $j$ digit produces a result of at most $i + j$ digits.  It is entirely possible that the result is $i + j - 1$
1226 work for GCC and not MSVC.  As such it is ideal to find a common ground for as many compilers as possible.  Pointer
1506 signs are known to agree in advance.
1589 The first two steps compare the signs of the two inputs.  If the signs do not agree then it can return right away with the appropriate
1590 comparison code.  When the signs are equal the digits of the inputs must be compared to determine the correct result.  In step
1601 The two if statements (lines 23 and 24) perform the initial sign comparison.  If the signs are not the equal then which ever
1602 has the positive sign is larger.   The inputs are compared (line 32) based on magnitudes.  If the signs were both
1603 negative then the unsigned comparison is performed in the opposite direction (line 34).  Otherwise, the signs are assumed to
1913 Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three
1970 \cite{HAC}.  Also this algorithm is restricted by algorithm s\_mp\_sub.  Chart \ref{fig:SubChart} lists the eight possible inputs and
2321 complete.  It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.
2507 used.  This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible.    One important
3225 For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
3611 is in modular exponentiation algorithms.  That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible.  This operation is used in the
5292 arises as a possible  solution.  Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$.
6463 As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as
6526 Note that there are two possible tables.  When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes
6536 it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$.  However, this test is not absolute as it is possible that the order