Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with HTTPS or Subversion.

Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

140 lines (139 sloc) 9.589 kb
\BOOKMARK [0][-]{chapter.1}{Introduction}{}
\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}
\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}
\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}
\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}
\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3}
\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3}
\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}
\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1}
\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5}
\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5}
\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1}
\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6}
\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6}
\BOOKMARK [0][-]{chapter.2}{Getting Started}{}
\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2}
\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2}
\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2}
\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2}
\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2}
\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2}
\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5}
\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5}
\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2}
\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6}
\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6}
\BOOKMARK [0][-]{chapter.3}{Basic Operations}{}
\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3}
\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3}
\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2}
\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2}
\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3}
\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3}
\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4}
\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4}
\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3}
\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5}
\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5}
\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3}
\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6}
\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6}
\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{}
\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4}
\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4}
\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2}
\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2}
\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4}
\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3}
\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3}
\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4}
\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4}
\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4}
\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4}
\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5}
\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5}
\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5}
\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{}
\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5}
\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5}
\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2}
\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2}
\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5}
\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3}
\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3}
\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{}
\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6}
\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6}
\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2}
\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2}
\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6}
\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3}
\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3}
\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6}
\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4}
\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4}
\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6}
\BOOKMARK [0][-]{chapter.7}{Exponentiation}{}
\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7}
\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1}
\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7}
\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2}
\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2}
\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7}
\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3}
\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7}
\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{}
\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8}
\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1}
\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1}
\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1}
\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8}
\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2}
\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2}
\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8}
\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8}
\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4}
\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4}
\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{}
\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9}
\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1}
\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9}
\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9}
\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3}
\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9}
\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4}
\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9}
\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5}
\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5}
Jump to Line
Something went wrong with that request. Please try again.