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# Topic: Exponentiation

###### In the News (Thu 25 Apr 19)

 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   (2356 words)

 Modular exponentiation - Wikipedia, the free encyclopedia Modular exponentiation is a type of exponentiation performed over a modulus. Modular exponentiation can be performed with a negative exponent e by finding the multiplicative inverse d of b modulo m using the extended Euclidean algorithm. It is a combination of the previous method and a more general principle called binary exponentiation (also known as exponentiation by squaring). en.wikipedia.org /wiki/Modular_exponentiation   (1510 words)

 Articles - Exponent   (Site not responding. Last check: 2007-10-22) One inverse of exponentiation is the logarithm; exponentiation is sometimes called the antilogarithm, i.e. Similarly, whereas addition or multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2Ã—3)Ã—4 = 24 = 2Ã—(3Ã—4)), exponentiation is not associative either: 2 to the 4th power is 8 or 4096, while 2 to the 3 power is 2 or 2417851639229258349412352. Exponentiation with base 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 Ã— 10 and then approximated as 2.998 Ã— 10 if this is useful. www.kimia-sains.com /articles/Exponent   (2644 words)

 Precedence and Prophylactic Parentheses The in-line operators ** and ^ are used for exponentiation in a way that obscures the visual appearance of the exponent as a part of the symbol. The unary minus and the exponentiation operator, **, are assigned to the highest priority group but, because that group is evaluated right to left, the ** in -2**2 will be evaluated before the change of sign. The exponentiation operator is not defined in a field. www.macnauchtan.com /pub/precedence.html   (6105 words)

 Euler's formula: This operation agrees with the original definition of exponentiation on their common domain, and also satisfies (1), (2) and (3). Because of this and other reasons, it is accepted as the modern definition of exponentiation. The key is to define an exponential function by (6) and work from there. db.uwaterloo.ca /~alopez-o/math-faq/node39.html   (620 words)

 Exponentiation over GF(2m) for Public Key Crypto System using Cellular Automata   (Site not responding. Last check: 2007-10-22) For more efficient exponentiation, it is necessary to develop more efficient AB The purpose of the current paper is to investigate and develop a simple and efficient architecture for the exponentiation over GF(2 And the proposed architecture for exponentiation is much more efficient in terms of space even though it is the same in terms of time. www.cs.utk.edu /~jerzy/para04/Abstracts/kyo_min_ku/kyo_min_ku.html   (255 words)

 RsaXli::Exp   (Site not responding. Last check: 2007-10-22) If the enumerated bit of the exponent is set, the current exponentiation result is multiplied by the current exponentiation factor and the product is placed in the RsaXli exponentiation result object. The exponentiation result object is re-sized according to its actual significance. The automatically-allocated RsaXli exponentiation result object is returned as the functional result of the Exp operation. www.grc.nasa.gov /WWW/price000/rsa/htm/rsaxli_exp.html   (224 words)

 Exponentiation - Simple English Wikipedia Exponentiation (power) is something which is done on numbers. There are some rules which help to calculate powers: It is possible to calculate exponentiation of matrices. simple.wikipedia.org /wiki/Exponentiation   (209 words)

 Exponentiation cryptosystems on the IBM PC Some of these are of the public key variety and offer notable advantages in cryptographic key management, both for secret communication and for message authentication. This paper describes a set of experimental programs that were developed to demonstrate that exponentiation cryptosystems can be efficiently implemented on the IBM Personal Computer (PC). The programs are organized into four layers, comprising procedures for: multiple precision integer arithmetic, modular exponentiation, prime number generation and testing, and cryptographic key generation. domino.research.ibm.com /tchjr/journalindex.nsf/e90fc5d047e64ebf85256bc80066919c/aba15a2ec768c0c185256bfa00685c4c?OpenDocument   (228 words)

 TIP #123 Version 1.5: Adding an Exponentiation Operator to the [expr] Command   (Site not responding. Last check: 2007-10-22) This TIP proposes to add a new operator to the operators recognised by the [expr] command: the exponentiation operator. Currently Tcl's [expr] command uses the exponentiation function pow() to calculate such expressions as "2 to the power 10". The precedence of the exponentiation operator is thus higher than the multiplication, division and remainder operations and lower than the unary operations, in accordance with common definitions. www.tcl.tk /cgi-bin/tct/tip/view/123?ver=1.5   (507 words)

 CARDIS '02 Technical Program - Abstract For example, for exponentiation-based cryptosystems (including RSA, DH or DSA), various exponentiation algorithms protected against sidechannel analysis are known. However, the exponentiation algorithm itself or the underlying cryptoalgorithm often involve division operations (for computing a quotient or a remainder). denormalization) process in fast exponentiation algorithms and the second case appears in the data processing before (resp. www.usenix.org /publications/library/proceedings/cardis02/joye.html   (253 words)

 RsaXli::ExpMod   (Site not responding. Last check: 2007-10-22) If the enumerated bit of the exponent is set, the current exponentiation result is multiplied by the current exponentiation factor and the product is placed in the RsaXli exponentiation result object, modulo rMod. The current exponentiation factor is squared by self-multiplication, modulo rMod. The automatically-allocated RsaXli exponentiation result object is returned as the functional result of the ExpMod operation. www.grc.nasa.gov /WWW/price000/rsa/htm/rsaxli_expmod.html   (248 words)

 Fast Exponentiation Using Data Compression We present the first exponentiation algorithm that uses the entropy of the source of the exponent to improve on existing exponentiation algorithms when the entropy is smaller than (1+w(S)/l(S)) For entropy 1 it is comparable to the best-known general purpose exponentiation algorithms. Yacobi, Yacov, "Fast Exponentiation Using Data Compression," SIAM Journal on Computing, Vol. research.microsoft.com /research/pubs/view.aspx?pubid=596   (80 words)

 The Prime Glossary: binary exponentiation In general, binary exponentiation will always take less than 2log(n)/log(2) multiplications. Another method, suggested by al-Kashi in 1427 (and similar to a method used by the Egyptians to multiply around 2000BC), is now called right-to-left binary exponentiation. Binary exponentiation does not always gives the fewest possible multiplications. primes.utm.edu /glossary/page.php?sort=BinaryExponentiation   (366 words)

 [No title] JUSTIFICATION: Reference 2 and reference 3 say: "Page 772, C.16, Standard arithmetic, second paragraph, item 3 talks about accuracy of exponentiation and item 8 talks about portable results; neither matches the specification on page 125 for exponentiation, where results are defined for positive exponents 1, 2, 3, and 4 and for any negative exponent. There are likely to be cases where exponentiation to a negative exponent does not exactly tie to the corresponding positive exponent. Item 3 should be structured like item 2, with implementor-defined results for negative exponents other than -1, -2, -3, and -4. www.ncits.org /tc_home/j4htm/m233/01-0551.doc   (401 words)

 [No title] These rules also apply to exponentiation; additional rules for exponentiation are specified in 8.8.1.4.2, Exponentiation in standard-binary arithmetic. Page 133, 8.8.1.5, Standard-decimal arithmetic: Add a new item 4 as follows: The rules for addition, subtraction, multiplication, division, unary-plus and unary-minus are specified in IEEE P754/D0.8.1-2004 November 29 2005, DRAFT Standard for Floating-Point Arithmetic, as applied to items in basic 128-bit decimal floating-point format, also described in that document. These rules also apply to exponentiation; additional rules for exponentiation are specified in 8.8.1.4.2, Exponentiation in standard-decimal arithmetic. www.cobolportal.com /j4/files/05-0096.doc   (594 words)

 Basic Mathematics The topic list for this project is: set theory, systems of equations, relations and functions, algebraic functions, and exponential and logarithmic functions. To recognize linear, quadratic, cubic, quartic, polynomial, rational, exponential, and logarithmic functions and their inverses. To become familiar with the principle theorems of exponential functions. www.madscitech.org /courses/bmath1/bmath1.htm   (1259 words)

 Exponentiation precedence - dBforums adopted opposite standards for the interpretation of exponentiation. Note that C and Java don't even have an exponentiation operator. Language designers are free to use the rule appropriate for that language. www.dbforums.com /t377245.html   (2415 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)

 CypherCalc - Cryptographer's Calculator. Big number arithmetic, for modular math, inverses, Montgomery exponentiation, ... It is intended for use in the design, testing, and analysis of cryptographic algorithms involving key exchanges, modular exponentiation, modular inverses, and Montgomery Math. Also included is a simple scripting program language that allows you to automate repetitive calculations and algorithms. The example at right shows a 320-bit modular exponentiation. www.cyphercalc.com   (812 words)

 TIP #123 Version 1.2: Adding an Exponentiation Operator to the [expr] Command   (Site not responding. Last check: 2007-10-22) TIP #123 Version 1.2: Adding an Exponentiation Operator to the [expr] Command This is not necessarily the current version of this TIP. This document is placed in the public domain. www.tcl.tk /cgi-bin/tct/tip/view/123?ver=1.2   (507 words)

 : Class Exponentiation A particular instance of this class is parameterized by a specific exponent to be used by that instance in exponentiation. The exponent is provided to this instance via an XML attribute called Extract the exponent and remember it for use in later computations. www.eclipse.org /articles/Article-Plug-in-architecture/doc/com/bolour/sample/eclipse/service/exponentiation/Exponentiation.html   (119 words)

 DiffEq Matrix Exponentiation Explorer   (Site not responding. Last check: 2007-10-22) This is a web software tool to explore the accuracy of numerically solving linear homogeneous differential equations using matrix exponentiation. Possibly usefull for hands on demonstration of the cool tool that matrix exponentiation is. If you use this for anything useful, I'd love to hear about it. www.davideous.com /matrixexp   (65 words)

 Looking for an x87 exponentiation function - GameDev.Net Discussion Forums   (Site not responding. Last check: 2007-10-22) Home » Community » Forums » Math and Physics » Looking for an x87 exponentiation function LucasArts is hiring in a city near you. Posted - 12/25/2004 3:28:22 PM Does anyone have or know where any source code is www.gamedev.net /community/forums/topic.asp?topic_id=290566&whichpage=1&   (241 words)

 SmallBASIC :: View topic - Exponentiation on PalmOS? Posted: Mon Nov 28, 2005 1:22 am Post subject: Exponentiation on PalmOS? I can't seem to find the character ^ on my Tungsten T5 (PalmOS 5) with SB 0.8.2b so I can do exponentiation. I looked in the ABC, abc and 123 sections but can't seem to find it. smallbasic.sourceforge.net /phpBB2/viewtopic.php?t=458   (167 words)

 Works Citing the On-Line Encyclopedia of Integer Sequences Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics, Electronic Journal of Combinatorics, Volume 3(2), 1996, article #R12. Williams, On Eliminating Square Paths in a Square Lattice, Master's Thesis, Rice University, 2000. Winkel, An exponential formula for polynomial vector fields II. www.research.att.com /~njas/sequences/cite.html   (9150 words)

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