https://en.wikipedia.org/wiki/Modular_exponentiation

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蒙哥馬利(Montgomery)冪模運算是快速計算a^b%k的一種算法，是RSA加密算法的核心之一。

蒙哥馬利模乘的優(yōu)點在于減少了取模的次數(shù)（在大數(shù)的條件下）以及簡化了除法的復(fù)雜度（在2的k次冪的進制下除法僅需要進行左移操作）。模冪運算是RSA 的核心算法，最直接地決定了RSA 算法的性能。 針對快速模冪運算這一課題，西方現(xiàn)代數(shù)學(xué)家提出了大量的解決方案，通常都是先將冪模運算轉(zhuǎn)化為乘模運算。

Modular exponentiation?is a type of?exponentiation取冪，求冪；乘方?performed over a?modulus模數(shù)，系數(shù).

It is useful in?computer science, especially in the field of?public-key cryptography.

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The operation of modular exponentiation calculates the remainder when an integer?b?底數(shù)(the base) raised to the?eth power (the exponent指數(shù)),?b^e, is divided by a?positive integer?m?(the modulus).

In symbols, given base?b, exponent?e, and modulus?m, the modular exponentiation?c?is:?c?≡?b^e?(mod?m). ? ? ? ?//c=b的e次方 %m

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For example, given?b?= 5,?e?= 3?and?m?= 13, the solution?c?= 8?is the remainder of dividing?5³?= 125?by 13. ? ? ?//c=5^3%13=125%13 ? 因為125=13*9+8 ，所以125對13求余，結(jié)果是8

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Given integers?b?and?e, and a positive integer?m, a unique solution?c?exists with the property?0 ≤?c?<?m.

Modular exponentiation can be performed with a?negative?exponent?e?by finding the?modular multiplicative inverse?d?of?b?modulo?m?using the?extended Euclidean algorithm. That is:

c?≡?b^e?≡?d^?e?mod?m?where?e?< 0?and?b???d?≡ 1 mod?m.

Modular exponentiation similar to the one described above are considered easy to compute, even when the numbers involved are enormous巨大的.

On the other hand, computing the?discrete logarithm離散對數(shù)?– that is, the task of finding the exponente?when given?b,?c, and?m?– is believed to be difficult.

This?one-way function?behavior makes modular exponentiation a candidate for use in cryptographic algorithms.

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轉(zhuǎn)載于:https://www.cnblogs.com/chucklu/p/5309297.html

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