exponentiation$500322$ - definição. O que é exponentiation$500322$. Significado, conceito
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O que (quem) é exponentiation$500322$ - definição

METHOD OF EXPONENTIATION BY POSITIVE INTEGERS REQUIRING A MINIMAL NUMBER OF MULTIPLICATIONS
Addition chain exponentiation

Addition-chain exponentiation         
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain, with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base.
Modular exponentiation         
TYPE OF EXPONENTIATION PERFORMED OVER A MODULUS
Modpow; Modular Exponentiation; Discrete exponential function
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
exponent         
  • n}} tends to the infinity.
  • The three third roots of 1
  • Power functions for <math>n=1,3,5</math>
  • Power functions for <math>n=2,4,6</math>
MATHEMATICAL OPERATION
Power (mathematics); Exponent; Raised to the power; Raised to the power of; Power function; Exponential functions; Multiplying exponents; Laws of Indices; First Law of Indices; Second Law of Indices; Third Law of Indices; Negative Exponents; Exponents; Indices (maths); Complex numbers exponential; Exponents (Math); Fraction power; Laws of exponentiation; Laws of exponents; Exponentiate; Exponentiating; 2^x; Exponentation; Exponention; Exponentiation over sets; Indices Laws; Exponent (mathematics); Power Functions; Exponental relationships; Zeroth power; Power (math); To The Power Of; Exponentiation ofer sets; Power Function; Math.Pow; Rules of exponents; Mathematical power; Pow function in c; Exponent (algebra); Power (algebra); A^b; Exponetation; Raising to a power; Imaginary exponent; Negative exponents; Zero exponent; Binary exponential; Base 2 antilogarithm; Exponent of 2; Base two antilogarithm; Exponent of two; Base 2 anti-logarithm; Base two anti-logarithm; Base-two anti-logarithm; Common exponential; Base 10 antilogarithm; Exponent of ten; Exponent of 10; Base 10 anti-logarithm; 10^x; Base ten anti-logarithm; Base ten antilogarithm; Base-ten anti-logarithm; Base-ten antilogarithm; Exponentiation operator; Hyper3; Hyper-3; 3-ation; ^ (math); ^ (maths); ** (math); ** (maths); ^ (mathematics); ** (mathematics); Commutative exponentiation; Ninth power; Pow function in C; Tower of powers; Exponent rules; Law of indices; Laws of indices; X^y; Set exponentiation; Draft:Powers in mathematics; To the power of zero; To the power of; Complex exponentiation
<programming> (Or "characteristic") The part of a floating-point number specifying the power of ten by which the mantissa should be multiplied. In the common notation, e.g. 3.1E8, the exponent is 8. (1995-02-27)

Wikipédia

Addition-chain exponentiation

In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain, with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base. (This corresponds to OEIS sequence A003313 (Length of shortest addition chain for n).) 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 a15, where the binary method needs six multiplications but the shortest addition chain requires only five:

a 15 = a × ( a × [ a × a 2 ] 2 ) 2 {\displaystyle a^{15}=a\times (a\times [a\times a^{2}]^{2})^{2}\!} (binary, 6 multiplications)
a 15 = ( [ a 2 ] 2 × a ) 3 {\displaystyle a^{15}=([a^{2}]^{2}\times a)^{3}\!} (shortest addition chain, 5 multiplications).
a 15 = a 3 × ( [ a 3 ] 2 ) 2 {\displaystyle a^{15}=a^{3}\times ([a^{3}]^{2})^{2}\!} (also shortest addition chain, 5 multiplications).

On the other hand, the determination of a shortest addition chain is hard: no efficient optimal methods are currently known for arbitrary exponents, and the related problem of finding a shortest addition chain for a given set of exponents has been proven NP-complete. Even given a shortest chain, addition-chain exponentiation requires more memory than the binary method, because it must potentially store many previous exponents from the chain. So in practice, shortest addition-chain exponentiation is primarily used for small fixed exponents for which a shortest chain can be pre-computed and is not too large.

There are also several methods to approximate a shortest addition chain, and which often require fewer multiplications than binary exponentiation; binary exponentiation itself is a suboptimal addition-chain algorithm. The optimal algorithm choice depends on the context (such as the relative cost of the multiplication and the number of times a given exponent is re-used).

The problem of finding the shortest addition chain cannot be solved by dynamic programming, because it does not satisfy the assumption of optimal substructure. That is, it is not sufficient to decompose the power into smaller powers, each of which is computed minimally, since the addition chains for the smaller powers may be related (to share computations). For example, in the shortest addition chain for a15 above, the subproblem for a6 must be computed as (a3)2 since a3 is re-used (as opposed to, say, a6 = a2(a2)2, which also requires three multiplies).