
	Where results make sense

Topic: Exponentiation by squaring

Related Topics

RSA

Advanced Encryption Standard

In the News (Sat 19 Dec 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Exponentiation by squaring - Wikipedia, the free encyclopedia
		Exponentiating by squaring is an algorithm used for the fast computation of large integer powers of a number.
		Exponentiation by squaring may also be used to calculate the product of 2 or more powers.
		However, exponentiating by squaring is simpler to set up and typically requires less memory.
	en.wikipedia.org /wiki/Exponentiation_by_squaring (1260 words)

	Exponentiation - Wikipedia, the free encyclopedia
		In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition.
		Exponentiation with various bases: red is to base e, green is to base 10, and purple is to base 1.7.
		Exponentiation is a basic mathematical operation that is used pervasively in other fields as well, including physics, chemistry, biology, computer science and economics, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
	en.wikipedia.org /wiki/Exponentiation (2787 words)

	Exponentiation by squaring Definition / Exponentiation by squaring Research (Site not responding. Last check: 2007-11-07)
		Exponentiating by squaring is an algorithmAn algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will result in a corresponding recognisable end-state (contrast with heuristic).
		The most familiar numbers are the whole numbers {0, 1, 2,...} denoted by W and the natural numbers {1, 2, 3,...} used for counting and denoted by N. If the negative whole numbers are included, one obtains the integers Z. Ratios of integers are named rational numbers or frac...
		It implicitly uses the binaryThe binary or base-two numeral system is a system for representing numbers in which a radix of two is used; that is, each digit in a binary numeral may have either of two different values.
	www.elresearch.com /Exponentiation_by_squaring (421 words)

	:::► Dictionary of Meaning www.mauspfeil.net ◄:::
		In mathematics, '''exponentiation''' is a process generalized from repeated (or iteration iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition.
		Exponentiation with base 10 (number) 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458 × 10
		Exponentiation to an arbitrary real exponent can then be defined by continuous function continuity.
	www.mauspfeil.net /exponentiation.html (2221 words)

	Exponentiation : search word (Site not responding. Last check: 2007-11-07)
		Exponentiation with a fractional exponent is defined as :
		Exponentiation of real numbers, and even complex numbers, can be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define : x
		(This is where the term "power set" comes from.) Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process.
	www.searchword.org /ex/exponentiation.html (2036 words)

	Exponentiation - (Site not responding. Last check: 2007-11-07)
		Exponentiation involves two numbers, the base and the exponent.
		The inverse of exponentiation is the logarithm; exponentiation is sometimes called the antilogarithm.
		Exponentiation with a rational exponent $m/n$ can now be defined as
	psychcentral.com /psypsych/Exponentiation (2455 words)

	Exponentiation Unearth Paramount Absolute Good Judgment Surrounding Exponentiation (Site not responding. Last check: 2007-11-07)
		KiwiPedia Home - Alpha index - "EX" index - Exponentiation by squaring Exponentiation by squaring Exponentiating by squaring is an algorithm used for the fast computation of large integer powers of a...
		Exponentiation with a lower-case 'e' Language one -> Comp Lang Mumps Reply by: George Timson -- Jul 19, 2004 21:31:17 As a followup to this thread from a couple of weeks back: VA has gotten word...
		Square the value by raising it to the 2nd power, like this ^2.
	ellipticcurve.sakecurve.com /exponentiation (833 words)

	Exponentiating by squaring (Site not responding. Last check: 2007-11-07)
		Exponentiating by squaring is an algorithm used for the fast computation of large powers of a number x.
		Essentially, this algorithm is equivalent to decomposing the exponent (often means by a base conversion to binary) into a sequence of squares and products: for example
		This is an implementation of the above algorithm in the Ruby programming language.
	www.teachersparadise.com /ency/en/wikipedia/e/ex/exponentiating_by_squaring.html (262 words)

	Exponentiating by squaring Definition / Exponentiating by squaring Research (Site not responding. Last check: 2007-11-07)
		Exponentiating by squaring Definition / Exponentiating by squaring Research
		Exponentiating by squaring is an algorithm used for the fast computation of large integerThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,...) and the number zero.
		The set of all integers is usually denoted in mathematics by Z (or Z in flboard bold,), which stands for Zahlen (German for "numbers").
	www.elresearch.com /Exponentiating_by_squaring (152 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-07)
		You really should be precise about what you are asking in this case, since (-9)^2 means -9 times -9, but the expression -9^2 could also be taken to mean -(9^2), that is, the negative of the square of 9, which is -81.
		When we're working with variables, if we see -x^2, we interpret it in the second way, as -(x^2), because squaring (or any exponentiation) takes precedence over negation (or any multiplication; -x is treated as -1*x.
		When you have numbers only, as in -9^2, it's not at all clear that we should treat it differently from -x^2.
	mathforum.org /library/drmath/view/55713.html (199 words)

	Exponentiation Definition / Exponentiation Research (Site not responding. Last check: 2007-11-07)
		(The next operation after exponentiation is sometimes named tetrationTetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation.
		Exponentiation is the process of taking a quantity b (the base) to the power of another quantity e (the exponent).
		exponentiation is a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition.
	www.elresearch.com /Exponentiation (148 words)

	Upto11.net - Wikipedia Article for Modular exponentiation (Site not responding. Last check: 2007-11-07)
		First, it is required that the exponent e be converted to binary notation.
		This code, based on that on page 244 of Bruce Schneier's Applied Cryptography, 2e, ISBN 0471117099, uses a single while loop to perform all work necessary to compute the modular exponentiation.
		However, the repeated squaring in the third line of code ensures that at the completion of every loop, the variable base is equivalent to
	www.upto11.net /generic_wiki.php?q=modular_exponentiation (1270 words)

	[Python-checkins] python/dist/src/Misc ACKS, 1.279, 1.280 NEWS, 1.1118, 1.1119 (Site not responding. Last check: 2007-11-07)
		Update of /cvsroot/python/python/dist/src/Misc In directory sc8-pr-cvs1.sourceforge.net:/tmp/cvs-serv18253/Misc Modified Files: ACKS NEWS Log Message: SF patch 936813: fast modular exponentiation This checkin is adapted from part 1 (of 3) of Trevor Perrin's patch set.
		Gradeschool + squaring was sped by about a factor of 2, by exploiting that about half + the digit products are duplicates in a square.
		Because exponentiation + uses squaring often, this also speeds long power.
	mail.python.org /pipermail/python-checkins/2004-August/043013.html (193 words)

	Math Forum Discussions
		If you are doing more than one squaring...
		If you are doing 10000 squares per second...
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?threadID=1176851&messageID=3849970 (178 words)

	Intro to Algorithms: CHAPTER 33: NUMBER-THEORETIC ALGORITHMS
		Modular exponentiation is also an essential operation in many primality-testing routines and in the RSA public-key cryptosystem.
		Whenever a squaring step is performed on line 5, however, lines 6-7 check to see if a nontrivial square root of 1 has just been discovered.
		Corollary 33.35 states that only if n is composite can there be a nontrivial square root of 1 modulo n, so that a demonstration that x is a nontrivial square root of 1 modulo n is a proof that n is composite.
	www.personal.kent.edu /~mlu3/CSCourses/AdvAlgorithms/CLR-BOOK/books/book6/chap33.htm (10376 words)

	Exponentiation By Squaring Life history @ 209.68.55.254 () (Site not responding. Last check: 2007-11-07)
		Exponentiation By Squaring Life history @ 209.68.55.254 ()
		Find More Information about "Exponentiation by squaring" in 209.68.55.254's:
		"Exponentiation by squaring" results in these other popular encyclopedia sites:
	209.68.55.254 /encyclopedia/Exponentiation_by_squaring (75 words)

	BU CAS CS 113: Assignment 1 (Site not responding. Last check: 2007-11-07)
		mod m, where raising to a power is done with repetitive squaring.
		mod m, and to use repetitive squaring instead of iteration.
		We also need to keep y - the intermediate result of exponentiation (it actually corresponds to x raised to a suffix of p, namely x raised to the power of some number of the least significant digits of p).
	www.cs.bu.edu /faculty/itkis/cs113/assignments/p3.html (802 words)

	YourArt.com >> Encyclopedia >> et:RSA (Site not responding. Last check: 2007-11-07)
		Both of these computations can be done efficiently using the square-and-multiply algorithm for modular exponentiation.
		When used in practice, RSA must be combined with some form of padding scheme, so that no values of M result in insecure ciphertexts.
		The values m = 0 or m = 1 always produce ciphertexts equal to 0 or 1 respectively, due to the properties of exponentiation.
	www.yourart.com /research/encyclopedia.cgi?subject=/et:RSA (2827 words)

	Modular exponentiation as a one-way function
		--it's as if we're doing the exponentiation in ordinary arithmetic because the numbers don't get big enough for the modular nature of the arithmetic to have any effect.
		In discussing the second of two applications below, we see that modular exponentiation has at least one weakness that makes it not quite suitable; there is a kind of security defect.
		Use of modular exponentiation as a one-way function also has a usability defect: even though we have a reasonably fast algorithm for modular exponentiation, the repeated-squaring algorithm, for some applications even this is not fast enough.
	www.cs.brown.edu /courses/cs007/oneway/node3.html (470 words)

	Discrete logarithm - (Site not responding. Last check: 2007-11-07)
		The naive algorithm is to raise b to higher and higher powers k until the desired g is found; this is sometimes called trial multiplication.
		This algorithm requires running time linear in the size of the group G and thus exponential in the number of digits in the size of the group.
		Computing discrete logarithms is apparently difficult (no efficient algorithm is known), while the inverse problem of discrete exponentiation is not (it can be computed efficiently using exponentiation by squaring, for example).
	psychcentral.com /psypsych/Discrete_logarithm (580 words)

	Python News (Site not responding. Last check: 2007-11-07)
		Gradeschool squaring was sped by about a factor of 2, by exploiting that about half the digit products are duplicates in a square.
		The cutoff for Karatsuba multiplication was raised, since gradeschool multiplication got quicker, and the cutoff was aggressively small regardless.
		The exponentiation algorithm was switched from right-to-left to left-to-right, which is more efficient for small bases.
	python.prokmu.com /2.4/NEWS.html (6146 words)

	Number Theory - Numericana (Site not responding. Last check: 2007-11-07)
		We see by inspection that the pairwise products and the squares of these special indices do not correspond to a -1 value of M. From this, we deduce that the lowest index n above 13 corresponding to a -1 value cannot be composite.
		The sequence M itself happens to have the lowest [exponential] growth among such sequences (we're ruling out 6 trivial cases with subexponential growth).
		A multiplicative function which is zero for squares of primes, and higher powers of prime numbers, is thus the Dirichlet inverse of a totally multiplicative function.
	home.att.net /~numericana/answer/numbers.htm (7445 words)

	CSE 30 -- Lecture 7 -- Oct 20 (Site not responding. Last check: 2007-11-07)
		Homework 3: Write exponentiation subroutine in MIPS, test with spim.
		You must use the new repeated-squaring exponentiation algorithm.
		Your turnin must include test cases, and should be well commented.
	www.bennetyee.org /ucsd-pages/Courses/cse30.f97/lec7 (253 words)

	Industrial Newsroom
		Firmware functions include modular multiplication, squaring, modular exponentiation, and RSA and DSA signature and verification.
		Firmware functions include fast modular multiplication, squaring, modular exponentiation (with or without Chinese Remainder Theorem), RSA and DSA signature and verification, and other calculations on operands having software-selectable lengths of up to 2,176-bits.
		The ST19 platform offers a wide range of products to suit both low cost and high performance applications and is certified to ISO 15408 Common Criteria at level EAL4+.
	news.thomasnet.com /printready.html?prid=28256 (671 words)

	Lambda Calculus and Lambda Calculators
		Curiously, none of the arithmetic operations involve the Y combinator; even the division gets by without Y. Addition, multiplication, squaring, and exponentiation are easily expressed in lambda-calculus.
		We discuss two approaches to their "opposites" -- subtraction, division, finding of a square root and of a discrete logarithm.
		The property says that an application of a Church numeral to two other Church numerals is itself a Church numeral, which corresponds to exponentiated integers.
	okmij.org /ftp/Computation/lambda-calc.html (1723 words)

	RSA (Site not responding. Last check: 2007-11-07)
		He knows N and e, which Alice has announced.
		This can be done quickly using the method of exponentiation by squaring.
		Alice receives c from Bob, and knows her private key d.
	bopedia.com /en/wikipedia/r/rs/rsa.html (1270 words)

	[No title]
		* This is the workhorse of modular exponentiation.
		Further, if we start with a pattern like "100" * in the lookahead window, rather than placing n into the buffer * and then starting to square it, we have already computed n^2 * to compute the odd-powers table, so we can place that into * the buffer and save a squaring.
		Add that * to the one squaring we have to do to compute the table, * and you'll see that a k-bit window saves k-2 squarings * as well as reducing the multiplies.
	www.rpi.edu /AFS/home/41/bailem2/rt/campus/bsd/bind/8.2/8.2.3/rs_aix43/src/lib/cylink/lbn16.c (7483 words)

	Math B
		For example, you are often using composite functions when you are
		When entering this computation on the calculator, you will press the button to square a value and the button to raise a value to any power (exponentiation).
		The buttons represent the functions for squaring and exponentiation.
	regentsprep.org /Regents/mathb/f/compositionfunctions.htm (303 words)

Try your search on: Qwika (all wikis)