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Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called "Montgomery's ladder" [2] addresses this concern.
This definition of exponentiation with negative exponents is the only one that allows extending the identity + = to negative exponents (consider the case =). The same definition applies to invertible elements in a multiplicative monoid , that is, an algebraic structure , with an associative multiplication and a multiplicative identity denoted 1 ...
The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because 10 4 = 10 × 10 × 10 × 10 = 10,000.
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The Cutler bar notation can be used to easily express other notation systems in exponent form. It also allows for a flexible summarization of multiple copies of the same exponents, where any number of stacked exponents can be shifted counterclockwise and shortened to a single variable.
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Multiplying can be viewed as adding exponents (i.e., logarithms). The text application leverages this by viewing concatenation as a variant of multiplication. Then the same techniques that are used for exponentiation (and thus that are available to multiplying exponents where "multiplying exponents" is just another name for exponentiation) are ...
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