FactHacks 
RSA factorization in the real world 

SageThe FactHacks code snippets use Sage, a mathematical frontend to Python. The big advantage of Sage over Python is that it has many builtin math libraries.The algorithms we discuss are built from simple arithmetic operations such as addition, subtraction, multiplication, division, etc. Our goal in each code snippet is for the reader to understand the sequence of arithmetic operations in the algorithm. There are many ways to speed up the code without changing the sequence of operations: Python and Sage often use unnecessarily slow methods to perform these arithmetic operations, and often impose severe overheads beyond the arithmetic operations. The simplest way to save time, without changing programming language, is to use Sage's Integer type, which generally offers much faster arithmetic operations than Python's long type: sage: # laptop performance sage: x = randrange(2**1000000) sage: y = randrange(2**500000) sage: time r = x % y Time: CPU 9.15 s, Wall: 9.21 s sage: time s = x * y Time: CPU 1.06 s, Wall: 1.06 s sage: x2 = Integer(x) sage: y2 = Integer(y) sage: time r2 = x2 % y2 Time: CPU 0.05 s, Wall: 0.05 s sage: time s2 = x2 * y2 Time: CPU 0.02 s, Wall: 0.02 s sage: r2 == r True sage: s2 == s True Changes in syntax between Sage and PythonIf you use Python then you won't have any serious trouble reading or writing Sage code, but there are a few important syntactic differences to keep in mind:
Using SageThere are three different ways that you can try Sage:
$ sage   Sage Version 5.5, Release Date: 20121222   Type "notebook()" for the browserbased notebook interface.   Type "help()" for help.   sage: 2+2 4 sage: Version: This is version 2012.12.27 of the sage.html web page. 