calculate modular algorithm with large exponents

Question

I don't know how to calculate the following modulo: $$321^{654} \mod 1013$$

Are there some easy way to do this?

With or without a calculator? – Carl Schildkraut Sep 25 '16 at 17:50 — Carl Schildkraut, Sep 25 '16 at 17:50
Related: https://math.stackexchange.com/questions/81228 – Watson Sep 25 '16 at 17:55 — Watson, Sep 25 '16 at 17:55

score 0 · Answer 1 · answered Sep 25 '16 at 23:25

Working in the prime field $\Bbb F_{1013}$ we can use equalities.

There are several ways to solve according the easier we can. For example

$$321^{654}=(321^{109})^6$$ so from $321^{10}=52$ we have $$321^{100}=52^{10}=781\Rightarrow321^{109}=781\cdot(321)^9=185\cdot1013+35$$ It follows $$321^{654}=35^6=\color{red}{863}$$

calculate modular algorithm with large exponents

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