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English: PDF version of the Think Python Wikibook. This file was created with MediaWiki to LaTeX . The LaTeX source code is attached to the PDF file (see imprint).
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
Karatsuba multiplication is an O(n log 2 3) ≈ O(n 1.585) divide and conquer algorithm, that uses recursion to merge together sub calculations. By rewriting the formula, one makes it possible to do sub calculations / recursion. By doing recursion, one can solve this in a fast manner.
A Byte of Python: Author: Swaroop C H: Software used: DocBook XSL Stylesheets with Apache FOP: Conversion program: Apache FOP Version 1.1: Encrypted: no: Page size: 595.275 x 841.889 pts (A4) Version of PDF format: 1.4
Classical modular multiplication reduces the double-width product ab using division by N and keeping only the remainder. This division requires quotient digit estimation and correction. The Montgomery form, in contrast, depends on a constant R > N which is coprime to N, and the only division necessary in Montgomery multiplication is division by R.
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. [2] [3] Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9.
CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and field-programmable gate arrays or FPGAs), as the only operations they require are additions, subtractions, bitshift and lookup tables.
However, division can be expensive and, in cryptographic settings, might not be a constant-time instruction on some CPUs, subjecting the operation to a timing attack. Thus Barrett reduction approximates 1 / n {\displaystyle 1/n} with a value m / 2 k {\displaystyle m/2^{k}} because division by 2 k {\displaystyle 2^{k}} is just a right-shift, and ...