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Vectorial Addition Chain
In mathematics, for positive integers ''k'' and ''s'', a vectorial addition chain is a sequence ''V'' of ''k''-dimensional vectors of nonnegative integers ''v''''i'' for −''k'' + 1 ≤ ''i'' ≤ ''s'' together with a sequence ''w'', such that : : ::: ⋮ ::: ⋮ : : ''v''''i'' =''v''''j''+''v''''r'' for all 1≤''i''≤''s'' with -''k''+1≤''j'', ''r''≤''i''-1 : ''v''''s'' = 'n''0,...,''n''''k''-1: ''w'' = (''w''1,...''w''''s''), ''w''''i''=(''j,r''). For example, a vectorial addition chain for 2,18,3is :''V''=( ,0,0 ,1,0 ,0,1 ,1,0 ,2,0 ,4,0 ,4,0 0,8,0 1,9,0 1,9,1 2,18,2 2,18,3 :''w''=((-2,-1),(1,1),(2,2),(-2,3),(4,4),(1,5),(0,6),(7,7),(0,8)) Vectorial addition chains are well suited to perform multi-exponentiation: :Input: Elements ''x''0,...,''x''''k''-1 of an abelian group ''G'' and a vectorial addition chain of dimension ''k'' computing 'n''''0'',...,''n''''k''-1 :Output:The element ''x''0''n''0...''x''''k''-1''n''''r' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\ ex& = \underbrace_ \times \underbrace_ \\ ex& = b^n \times b^m \end In other words, when multiplying a base raised to one e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition Chain
In mathematics, an addition chain for computing a positive integer can be given by a sequence of natural numbers starting with 1 and ending with , such that each number in the sequence is the sum of two previous numbers. The ''length'' of an addition chain is the number of sums needed to express all its numbers, which is one less than the cardinality of the sequence of numbers. Examples As an example: (1,2,3,6,12,24,30,31) is an addition chain for 31 of length 7, since :2 = 1 + 1 :3 = 2 + 1 :6 = 3 + 3 :12 = 6 + 6 :24 = 12 + 12 :30 = 24 + 6 :31 = 30 + 1 Addition chains can be used for addition-chain exponentiation. This method allows exponentiation with integer exponents to be performed using a number of multiplications equal to the length of an addition chain for the exponent. For instance, the addition chain for 31 leads to a method for computing the 31st power of any number using only seven multiplications, instead of the 30 multiplications that one would get from repeated mul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 .) 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 ''a''15, where the binary method needs six multiplications but the shortest addition chain requires only five: :a^ = a \times (a \times \tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponentiation By Squaring
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases: b^n = \underbrace_. The exponent is usually shown as a superscript to the right of the base. In that case, is called "''b'' raised to the ''n''th power", "''b'' (raised) to the power of ''n''", "the ''n''th power of ''b''", "''b'' to the ''n''th power", or most briefly as "''b'' to the ''n''th". Starting from the basic fact stated above that, for any positive integer n, b^n is n occurrences of b all multiplied by each other, several other properties of exponentiation directly follow. In particular: \begin b^ & = \underbrace_ \\ ex& = \underbrace_ \times \underbrace_ \\ ex& = b^n \times b^m \end In other words, when multiplying a base raised to one exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-adjacent Form
The non-adjacent form (NAF) of a number is a unique signed-digit representation, in which non-zero values cannot be adjacent. For example: :(0 1 1 1)2 = 4 + 2 + 1 = 7 :(1 0 −1 1)2 = 8 − 2 + 1 = 7 :(1 −1 1 1)2 = 8 − 4 + 2 + 1 = 7 :(1 0 0 −1)2 = 8 − 1 = 7 All are valid signed-digit representations of 7, but only the final representation, (1 0 0 −1), is in non-adjacent form. The non-adjacent form is also known as "canonical signed digit" representation. Properties NAF assures a unique representation of an integer, but the main benefit of it is that the Hamming weight of the value will be minimal. For regular binary representations of values, half of all bits will be non-zero, on average, but with NAF this drops to only one-third of all digits. This leads to efficient implementations of add/subtract networks (e.g. multiplication by a constant) in hardwired digital signal processing. Obviously, at most half of the digits are non-zero, which was the reason it was int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
