In exponentiation , the base is the number var b var in an expression of the form var b sup n var sup . The number var n var is called the exponent and the expression is known formally as exponentiation of var b var by var n var or the exponential of var n var with base var b var . It is more commonly expressed as the var n var th power of var b var , var b var to the var n var th power or var b var to the power var n var . When the var n var th power of var b var equals a number var a var , or var a b sup n sup var , then var b var is called an nth root var n var th root of var a var . The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent. The inverse function to exponentiation with base var b var when it is well defined is called the logarithm to base var b var , denoted log sub var b var sub . Thus math log b b n b log b n n. , math Category Exponentials ... more details
Modular exponentiation is a type of exponentiation performed over a modular arithmetic modulus . It is particularly ... exponentiation means calculating the remainder when dividing by a positive integer m called the modulus ... with the property 0 c < m . Modular exponentiation can be performed with a negative exponent e by finding ... m math Modular exponentiation problems similar to the one described above are considered easy to do ... exponentiation a good candidate for use in cryptographic algorithms. Straightforward method The most ... better security, the value b sup e sup becomes unwieldy. The time required to perform the exponentiation ... exponentiation requires more operations than the first method. Because the required memory is substantially ... required to perform modular exponentiation. It is a combination of the previous method and a more general principle called exponentiation by squaring also known as binary exponentiation . First, it is required ... speed benefit over both of the previous two algorithms. Reversible and quantum modular exponentiation In quantum computing , modular exponentiation appears as the bottleneck of Shor s algorithm .... Furthermore, in Shor s algorithm it is possible to know the base and the modulus of exponentiation ... Optimized Quantum Circuits for Modular Multiplication and Exponentiation , Quantum Information and Computation ... languages Because modular exponentiation is an important operation in computer science, and there are efficient ... to perform modular exponentiation Python programming language Python s built in tt pow tt exponentiation ... method to perform modular exponentiation Perl s tt Math BigInt tt module has a tt bmodpow tt method http perldoc.perl.org Math BigInt.html bmodpow 28 29 to perform modular exponentiation Go programming language Go s tt big.Int tt type contains an tt Exp tt exponentiation method http golang.org pkg ... a tt bcpowmod tt function http www.php.net manual en function.bcpowmod.php to perform modular exponentiation ... http gmplib.org manual Integer Exponentiation.html to perform modular exponentiation See also Montgomery ... more details
for exponentiation do not provide defence against side channel attack s. Namely, an attacker observing ... The Euclidean method was first introduced in Efficient exponentiation using precomputation and vector ... exponentiation exponents modulo a number. Especially in cryptography , it is useful to compute powers ... http home.mnet online.de wzwz.de temp ebs en.htm Calculation of products of powers Exponentiation ... and generalizations main Addition chain exponentiationExponentiation by squaring can be viewed as a suboptimal addition chain exponentiation algorithm it computes the exponent via an addition chain ... those powers of x , one can sometimes perform the exponentiation using fewer multiplications ..., have fewer multiplications than exponentiation by squaring at the cost of additional bookkeeping ... O notation &Theta log n , so these algorithms only improve asymptotically upon exponentiation by squaring by a constant factor at best. See also Modular exponentiation Vectorial addition chain Montgomery ... precision algorithms Category Computer arithmetic ca Exponenciaci bin ria de Bin re Exponentiation es Exponenciaci n binaria fr Exponentiation rapide pl Algorytm szybkiego pot gowania ru simple Exponentiation by squaring sv Bin r exponentiering vi Thu t to n ... more details
In mathematics and computer science , optimal addition chain exponentiation is a method of exponentiation by positive integer powers that requires a minimal number of multiplications. It works by creating a shortest addition chain that generates the desired exponent. Each exponentiation in the chain can be evaluated by multiplying two of the earlier exponentiation results. More generally, addition chain exponentiation may also refer to exponentiation by non minimal addition chains constructed by a variety of algorithms since a shortest addition chain is very difficult to find . The shortest addition chain algorithm requires no more multiplications than binary exponentiation and usually less. The first example of where it does better is for math a 15 math , where the binary method needs six multiplies but a shortest addition chain requires only five math a 15 a times a times a times a 2 2 2 math binary, 6 multiplications math a 15 a 3 times a 3 2 2 math shortest addition chain, 5 multiplications . class wikitable Table demonstrating how to do Exponentiation using Addition Chains Number of br Multiplications Actual br Exponentiation Specific implementation of br Addition Chains to do Exponentiation 0 a sup 1 sup a 1 a sup 2 sup a a 2 a sup 3 sup a a a 2 a sup 4 sup a a b b 3 a sup 5 ... ref Even given a shortest chain, addition chain exponentiation requires more memory than the binary .... In practice, therefore, shortest addition chain exponentiation is primarily used for small fixed ... than binary exponentiation. Indeed, binary exponentiation itself is a suboptimal addition chain ... of fast exponentiation methods. J. Algorithms 27, 1 Apr. 1998 , 129 146. doi http dx.doi.org 10.1006 ... math , which also requires three multiplies . Addition subtraction chain exponentiation If both multiplication ... of division compared to multiplication makes this technique unattractive in general. For exponentiation ... division math a 31 a a 2 2 2 2 2 math addition subtraction chain, 5 mults 1 div . For exponentiation ... more details
Rcon may refer to A computer Remote administration remote control administration tool In Rijndael key schedule is the exponentiation of 2 to a user specified value R Con, in a Gasket seal, a type of joint disamb ... more details
This is a list of notable square roots Square root of two Square root of three Square root of 5 Square root of minus one , a number equaling 1 when being Exponentiation squared Square root of a matrix Category Mathematics related lists ... more details
The continuum function is math kappa mapsto 2 kappa math , i.e. raising 2 to the power of &kappa using cardinal exponentiation . Given a cardinal number , it is the cardinality of the power set of a set of the given cardinality. See also Continuum hypothesis Cardinality of the continuum Beth number Gimel function settheory stub Category Cardinal numbers ... more details
Wiktionary slog A slog is a type of shot in the game cricket. Slog may also be A super logarithm , the inverse function of super exponentiation A creature of fictional Oddworld A blog run by Seattle alternative weekly newspaper The Stranger newspaper The Stranger Slog Transformers , a fictional character disambig ... more details
Dunamis Ancient Greek is the scholarly term for the philosophical concept of potentiality and actuality. Dunamis may also refer to Dynamis Bosporan queen , a Roman Client Queen of the Bosporan Kingdom Dynamis beetle Dynamis beetle , a weevil genus of the tribe Rhynchophorini Dynamis Ensemble , an instrumental group from Italy See also Exponentiation disambig ... more details
The symbol , an upward pointing arrow can refer to Yn , a Romanian Cyrillic letter For the symbol, represented in Unicode, see Arrow symbol For the mathematical notation of undefined, see Defined and undefined For the mathematical notation of iterated exponentiation, see Knuth s up arrow notation For the mathematical game theory position Up , see Combinatorial game theory Up Up game theory It also resembles the medieval Roman numerals Roman numeral for 900. Up arrow key See also Arrow disambiguation disambig no ... more details
Merge to Closed form expression discuss Talk Closed form expression Proposed merger from Analytical expression date May 2011 Unreferenced date December 2009 In mathematics , an analytical expression or expression in analytical form is a mathematical expression constructed using well known operations that lend themselves readily to calculation. As is true for closed form expression s, the set of well known functions allowed can vary according to context but always includes the Arithmetic Arithmetic operations basic arithmetic operations addition, subtraction, multiplication, and division , extraction of nth root n th roots , exponentiation, logarithms, and trigonometric functions. However, the class of expressions considered to be analytical expressions tends to be wider than that for closed form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are Series mathematics Infinite series infinite series and continued fraction s. On the other hand, Limit of a sequence limits in general, and integral s in particular, are typically excluded. If an analytic expression involves only the algebraic operations, which are addition, subtraction, multiplication, division and exponentiation with integral or fractional exponents hence including the extraction of n th roots , then it is more specifically referred to as an algebraic expression . math stub DEFAULTSORT Analytical Expression Category Special functions ja zh ... more details
refimproveBLP date January 2011 Stephen Pohlig is an electrical engineer currently working at MIT Lincoln Laboratory . As a graduate student of Martin Hellman s at Stanford University in the mid 1970 s, he helped develop the Pohlig Hellman exponentiation cipher and the Pohlig Hellman algorithm for computing discrete logarithm s. Bibliography S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over GF p and its cryptographic significance Corresp. , Information Theory, IEEE Transactions on 24, no. 1 1978 106 110. Martin E. Hellman and Stephen C. Pohlig, http patft.uspto.gov netacgi nph Parser?Sect2 PTO1&Sect2 HITOFF&p 1&u 2Fnetahtml 2FPTO 2Fsearch bool.html&r 1&f G&l 50&d PALL&RefSrch yes&Query PN 2F4424414 United States Patent 4424414 Exponentiation cryptographic apparatus and method , January 3, 1984. References http www.cbi.umn.edu oh display.phtml?id 353 Oral history interview with Martin Hellman , 2004, Palo Alto, California. Charles Babbage Institute , University of Minnesota, Minneapolis. Persondata Metadata see Wikipedia Persondata . NAME Pohlig, Stephen ALTERNATIVE NAMES SHORT DESCRIPTION DATE OF BIRTH PLACE OF BIRTH DATE OF DEATH PLACE OF DEATH DEFAULTSORT Pohlig, Stephen Category American electrical engineers Category Living people US engineer stub ... more details
In axiomatic set theory , the gimel function is the following function mapping cardinal number s to cardinal numbers math gimel colon kappa mapsto kappa mathrm cf kappa math where cf denotes the cofinality function the gimel function is used for studying the continuum function and the cardinal number Cardinal exponentiation cardinal exponentiation function. Values of the Gimel function The gimel function has the property math gimel kappa kappa math for all infinite cardinals &kappa by K nig s theorem set theory K nig s theorem . For regular cardinals math kappa math , math gimel kappa 2 kappa math , and Easton s theorem says we don t know much about the values of this function. For singular math kappa math , upper bounds for math gimel kappa math can be found from Saharon Shelah Shelah s PCF theory . Reducing the exponentiation function to the gimel function All cardinal exponentiation is determined recursively by the gimel function as follows. If &kappa is an infinite successor cardinal then math 2 kappa gimel kappa math If &kappa is a limit and the continuum function is eventually constant below &kappa then math 2 kappa 2 kappa times gimel kappa math If &kappa is a limit and the continuum function is not eventually constant below &kappa then math 2 kappa gimel 2 kappa math The remaining rules hold whenever &kappa and &lambda are both infinite If &alefsym sub 0 sub &le &kappa &le &lambda then &kappa sup &lambda sup 2 sup &lambda sup If &mu sup &lambda sup &ge &kappa for some &mu &kappa then &kappa sup &lambda sup &mu sup &lambda sup If &kappa &lambda and &mu sup &lambda sup &kappa for all &mu &kappa and cf &kappa &le &lambda then &kappa sup &lambda sup &kappa sup cf &kappa sup If &kappa &lambda and &mu sup &lambda sup &kappa for all &mu &kappa and cf &kappa &lambda then &kappa sup &lambda sup &kappa References Thomas Jech , Set Theory , 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3 540 44085 2. Category Cardinal numbers cs Funkce gi ... more details
for the Lego Technic motor system Lego Technic Power Functions In mathematics , a power function is a function of the form nowrap f x cx sup a sup , where c and a are constant real number s and x is a variable. Power functions are a special case of power law relationships, which appear throughout mathematics and statistics. See also Exponentiation Category Types of functions math stub ca Funci potencial cs Mocninn funkce de Potenzfunktion eu Funtzio potentzial fr Fonction puissance kk nl Machtsfunctie pl Funkcja pot gowa ru sk Mocninov funkcia uk ... more details
A unary function is a function mathematics function that takes one Parameter computer science argument . In computer science , a unary operator is a subset of unary function. Example a here a is an operand which is operated by a unary operator . it can be written in the form a a 1. Many of the elementary function s are unary functions, in particular the trigonometric functions , logarithm with a pre specified base, exponentiation to a pre specified power or from a pre specified base, and hyperbolic function are unary. See also Arity Binary function Binary operator List of mathematical functions Ternary operation Unary operation References http www.cs.ucl.ac.uk staff W.Langdon FOGP Foundations of Genetic Programming Category Functions and mappings Category Types of functions maths stub bs Unarna funkcija Example a means a is a operand which is operated by a unary operator . it can be writen in this form a a 1 ... more details
Unreferenced stub auto yes date December 2009 Decimalization process refers to the conversion of a system of coinage or measurement to a decimal system, i.e. one where the all ratios between the different units are Exponentiation power s of ten. The metric system metric system of measurement , an early version of which was first introduced in France after the French Revolution , is a decimal system of measurement. decimalization of time of day Decimalization of time periods within the day sees occasional use. Decimalization is generally advocated for reducing effort of calculations involving quantities of different scale. DEFAULTSORT Decimalization Process Category Measurement Measurement stub ... more details
Wiktionary exponent According to the Oxford English Dictionary , to expone is to set forth , and an exponent is a person or a statement that sets something forth. The word has assumed a plethora of other meanings Mathematics List of exponential topics Exponentiation Exponential function Matrix exponential Periodic group Exponent group theory Statistics Exponential distribution Exponential family Exponential function Exponential growth Exponential decay Linguistics Exponent linguistics Music The Exponents Publications Purdue Exponent Woman s Exponent Companies Exponent consulting firm See also Exponential disambiguation List of exponential topics Mathematics related exponents disambig cs Exponent de Exponent ... more details
more efficient than exponentiation. Using Euler s theorem As an alternative to the extended ..., but is sometimes used when an implementation for modular exponentiation is already available. Some ... of a prime. exponentiation. Though it can be implemented more efficiently using modular exponentiation ... in the first place. Without the Montgomery method, we re left with standard binary exponentiation ..., any kind of modular exponentiation is a taxing operation with computational complexity Big O notation ... more details
The Itoh Tsujii inversion algorithm is used to invert elements in a finite field . It was introduced in 1988 and first used over GF 2 sup m sup using the normal basis representation of elements, however the algorithm is generic and can be used for other bases, such as the polynomial basis . It can also be used in any finite field, GF p sup m sup . The algorithm is as follows Input A GF p sup m Output A sup &minus 1 sup r p sup m sup &minus 1 p &minus 1 compute A sup r &minus 1 sup in GF p sup m sup compute A sup r sup A sup r &minus 1 sup A compute A sup r sup sup &minus 1 sup in GF p compute A sup &minus 1 sup A sup r sup sup &minus 1 sup A sup r &minus 1 sup return A sup &minus 1 sup This algorithm is fast because steps 3 and 5 both involve operations in the subfield GF p . Similarly, if a small value of p is used a lookup table can be used for inversion in step 4. The majority of time spent in this algorithm is in step 2, the first exponentiation. This is one reason why this algorithm is well suited for the normal basis, since squaring and exponentiation are relatively easy in that basis. References T. Itoh and S. Tsujii. A Fast Algorithm for Computing Multiplicative Inverses in GF 2 sup m sup Using Normal Bases. Information and Computation , 78 171 177, 1988. External links http www.win.tue.nl henkvt ItohTsujiiEnciclopediaOfInfoSec Submitted.pdf A thorough discussion of the Itoh Tsujii algorithm by Guajardo Category Finite fields Category Computational number theory ... more details
The Schmidt&ndash Samoa cryptosystem is an asymmetric cryptographic technique, whose security, like Rabin cryptosystem Rabin depends on the difficulty of integer factorization . Unlike Rabin this algorithm does not produce an ambiguity in the decryption at a cost of encryption speed. Key generation Choose two large distinct primes p and q and compute math N p 2q math Compute math d N 1 mod text lcm p 1,q 1 math Now N is the public key and d is the private key. Encryption To encrypt a message m we compute the ciphertext as math c m N mod N. math Decryption To decrypt a ciphertext c we compute the plaintext as math m c d mod pq, math which like for Rabin and RSA algorithm RSA can be computed with the Chinese remainder theorem . Example math p 7, q 11, N p 2q 539, d N 1 mod text lcm p 1,q 1 29 math math m 32, c m N mod N 373 math Now to verify math m c d mod pq 373 29 mod pq 373 29 mod 77 32 math Security The algorithm, like Rabin, is based on the difficulty of factoring the modulus N , which is a distinct advantage over RSA. That is, it can be shown that if there exists an algorithm that can decrypt arbitrary messages, then this algorithm can be used to factor N . Efficiency The algorithm processes decryption as fast as Rabin and RSA, however it has much slower encryption since the sender must compute a full exponentiation. Since encryption uses a fixed known exponent an addition chain exponentiation addition chain may be used to optimize the encryption process. The cost of producing an optimal addition chain can be amortized over the life of the public key, that is, it need only be computed once and cached. References http eprint.iacr.org 2005 278.pdf A New Rabin type Trapdoor Permutation Equivalent to Factoring and Its Applications crypto navbox public key Category Asymmetric key cryptosystems ... more details
wiktionary exponential exponentially Exponential may refer to any of several mathematical topics related to exponentiation , including Exponential function , also Matrix exponential , the matrix analogue to the above Exponential decay , decrease at a rate proportional to value Exponential discounting , a specific form of the discount function, used in the analysis of choice over time Exponential growth , where the growth rate of a mathematical function is proportional to the function s current value Exponential map , in differential geometry Exponential notation , also known as scientific notation, or standard form Exponential object , in category theory Exponential time , in complexity theory in probability and statistics Exponential distribution , a family of continuous probability distributions Exponential family , sometimes used in place of exponential family Exponential smoothing , a technique that can be applied to time series data Function type Exponential type or function type, in type theory Topics listed at list of exponential topics Exponential may also refer to Exponential Technology , a vendor of PowerPC microprocessors disambiguation Category Mathematical disambiguation Category Exponentials ar eo Eksponento fr Exponentielle ... more details
about the mathematical concept the musical interval Pythagorean apotome Apotom lang el is an archaic mathematics mathematical term which, according to Webster s 1828 Dictionary, is the Subtraction difference of two quantities that are Commensurability mathematics commensurable only in Exponentiation power . According to Bailey s 1761 Dictionary it is an irrational remainder or residual when from a rational line a part is cut off which is only commensurate in power to the whole line . ref An Universal Etymological English Dictionary , N Bailey, London 1761 blockquote APO TOME in Mathematicks is an irrational Remainder or Residual, when from a Rational Line a Part is cut off, which is only Commensurable in Power to the whole Line. blockquote ref It appears not to be in modern usage. This concept of the Apotome appears in book X of Euclid s Elements . References references Webster s 1828 Dictionary 1728 Category Mathematical terminology Category Greek loanwords numtheory stub ar cs Apotom de Apotome es Apotom nl Apotome pl P ton chromatyczny ru ... more details
Unreferenced date September 2009 Image Recorde The Whetstone of Witte equals.jpg thumb 400 px right The Whetstone of Witte is the shortened title of Robert Recorde s mathematics book published in 1557. The full title being, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke containing the extraction of rootes the cossike practise, with the rule of equation and the workes of Surde Nombers , the book covers topics including whole numbers, the extraction of roots and irrational numbers. The work is notable for containing the first recorded use of the equals sign and also the first book in English to use the plus and minus signs . However, exponentiation was represented by a cumbersome form of mathematical notation in which Exponent indices and surds were represented by their prime factors see Prime factor exponent notation . See also Prime factor exponent notation External links http www.scribd.com doc 13687028 The Whetstone Of Witte The Whetstone of Witte at Scribd DEFAULTSORT Whetstone of Witte Category Mathematics books Category British non fiction literature Category 1557 books Mathpublication stub ... more details
Other uses Kilo disambiguation Wiktionary kilo Kilo symbol k, lowercase is a SI prefix unit prefix in the metric system denoting multiplication of the unit by one thousand. For example one kilogram is 1000 gram s one kilometre is 1000 metre s one kilojoule is 1000 joule s one kilobaud is a rate of transfer used primarily in telecommunications one kilobyte is formally equal to 1000 byte s. However, a second definition and usage has historically been in practice in many fields of computer science and information technology, which defines the prefix kilo when used with byte or bit units of information as 1024 2 sup 10 sup this is due to the Mathematical coincidence Concerning base 2 mathematical coincidence that math 2 10 approx 10 3. math Thus, in these fields 1 kilobyte is equal to 1 kibibyte , a new unit standardized as part of the binary prefix es to resolve the ambiguity. ref http physics.nist.gov cuu Units binary.html Definition of binary prefixes at NIST ref The kilo prefix is derived from the Greek language Greek word chilioi , meaning thousand . It was originally adopted by Antoine Lavoisier s group in 1795, and introduced into the metric system in France with its establishment in 1799. Exponentiation When units occur in exponentiation , such as in square and cubic forms, any multiplier prefix is considered part of the unit, and thus included in the exponentiation. 1 km sup 2 sup means one square kilometre or the area of a Square geometry square of 1000 m by 1000 m or 10 sup 6 sup m sup 2 sup . 1 km sup 3 sup means one cubic kilometre or the volume of a cube of 1000 m by 1000 m by 1000 m or 10 sup 9 sup m sup 3 sup . See also International System of Units References reflist SI prefixes Category SI prefixes ar bg br Kilo ca Kilo cs Kilo da Kilo et Kilo es Kilo prefijo eu Kilo fa fr Kilo gl Quilo ko hy hi id Kilo it Chilo prefisso he ka ... ... more details
Other uses Mega disambiguation Mega symbol M is an SI prefix prefix in the metric system denoting a factor of million 10 sup 6 sup or 1000000 number gaps 1 000 000 . Confirmed in 1960, it comes from the Greek language Greek , meaning great . ref OED http dictionary.oed.com cgi entry 00304790 mega ref Other common examples of usage megapixel 1 million pixel s in a digital camera one TNT equivalent megatonne of TNT a unit often used in measuring the explosive power of nuclear weapon s is the approximate energy released on igniting one million tonnes of TNT. megahertz frequency of electromagnetic radiation for radio and television broadcasting, GSM , etc. 1 MHz 1,000,000 Hertz Hz . Exponentiation When units occur in exponentiation , such as in square and cubic forms, any size prefix is considered part of the unit, and thus included in the exponentiation. 1 Mm sup 2 sup means one square megametre or the size of a Square geometry square of 1,000,000 m by 1,000,000 m or 10 sup 12 sup m sup 2 sup , and not 1,000,000 square metre s 10 sup 6 sup m sup 2 sup . 1 Mm sup 3 sup means one cubic megametre or the size of a cube of 1,000,000 m by 1,000,000 m by 1,000,000 m or 10 sup 18 sup m sup 3 sup , and not 1,000,000 cubic metre s 10 sup 6 sup m sup 3 sup Computing In computing , mega can sometimes denote 1,048,576 2 sup 20 sup of information units example a megabyte , a megaword , but can denote 1,000,000 10 sup 6 sup of other quantities, for example, transfer rates 1 megabit s 1,000,000 bit s . The prefix mebi has been suggested as a prefix for 2 sup 20 sup to avoid ambiguity. SI prefixes See also SI prefix Binary prefix Mebibyte Orders of magnitude References references External links wiktionary mega http www.bipm.org BIPM website Category SI prefixes bg br Mega ca Mega cs Mega da Mega et Mega es Mega eo Mega eu Mega fa fr M ga gl Mega ko hy hi id Mega it Mega he ... more details
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