Arbitrary-precision arithmetic: Difference between revisions

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==Applications==

A common application is [[public-key cryptography]], whose algorithms commonly employ arithmetic with integers having hundreds of digits.<ref>[http://arstechnica.com/news.ars/post/20070523-researchers-307-digit-key-crack-endangers-1024-bit-rsa.html Researchers: 307-digit key crack endangers 1024-bit RSA]</ref><ref>{{cite web|url=http://www.rsa.com/rsalabs/node.asp?id%3D2218 |title=Archived copy |accessdate=2012-03-31 |deadurl=yes |archiveurl=https://web.archive.org/web/20120401144624/http://www.rsa.com/rsalabs/node.asp?id=2218 |archivedate=2012-04-01 |df= }} recommends important RSA keys be 2048 bits (roughly 600 digits).</ref> Another is in situations where artificial limits and [[arithmetic overflow|overflows]] would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining the optimum value for coefficients needed in formulae, for example the √⅓ that appears in [[Gaussian integration]].

Arbitrary precision arithmetic is also used to compute fundamental [[mathematical constant]]s such as [[pi|π]] to millions or more digits and to analyze the properties of the digit strings<ref>{{cite journal |author=R. K. Pathria |authorlink=Raj Pathria |title=A Statistical Study of the Randomness Among the First 10,000 Digits of Pi |year=1962 |journal=Mathematics of Computation |volume=16 |issue=78 |pages=188–197 |url=http://www.ams.org/journals/mcom/1962-16-078/S0025-5718-1962-0144443-7/ |accessdate=2014-01-10 |doi=10.1090/s0025-5718-1962-0144443-7}} A quote example from this article: "Such an extreme pattern is dangerous even if diluted by one of its neighbouring blocks"; this was the occurrence of the sequence 77 twenty-eight times in one block of a thousand digits.</ref> or more generally to investigate the precise behaviour of functions such as the [[Riemann zeta function]] where certain questions are difficult to explore via analytical methods. Another example is in rendering [[fractal]] images with an extremely high magnification, such as those found in the [[Mandelbrot set]].