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Taken from,

$\mathcal{E}(x_1) \cdot \mathcal{E}(x_2) = x_1^e x_2^e$
$\bmod\; m = (x_1x_2)^e \;$
$\bmod\; m = \mathcal{E}(x_1 \cdot x_2)$

What is the superscript $^e$ representing in the first expression, and how do we get from the second line, to the third?

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  • $\mathcal{E}$ represents encryption, superscript $e$ represents exponentiation. Textbook RSA encryption is defined as $\mathcal{E}(x) = x^e \mod m$ where $e$ is public exponent and $m$ is the modulus.

  • $x_1^e\cdot x_2^e = (x_1\cdot x_2)^e$ is a basic property of exponentiation.

  • To get from the third line to the second you use the definition of $\mathcal{E}$ on $x_1\cdot x_2$ yielding $(x_1\cdot x_2)^e$.
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Thanks very much! What do you mean by the definition of $E$, is that the procedure of encryption? – Masutatsu Sep 20 '13 at 7:55

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