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I want to make a following proof but I got some difficulties with it. Would be super if you people have any tips / advises.

Introduction:

Let (N,e) be our public key and (N,s) our private key with $N=pq$ and $ggT(\varphi(n),e)=1$, as well as $es\equiv 1\mod N$.

I want to show that since Lemma of Bézout shows, that there are $a,b\in \mathbb{N}$ so that $ap-bq=1$, the message of $x=ap\mod N$ is equal to the encrypted message $y=x^e \mod N$.

My Plan:

I started by using the Chinese remainder theorem. To show $x\overset{!}{=}y=x^e\mod N$ we need to show, that $x\overset{!}{=}x^e\mod p$ and $x\overset{!}{=}x^e\mod q$. I could also use the Chinese remainder theorem on $x=ap\mod N$.

So I need to show, that $ap \mod p=(ap)^e\mod p$. Since ap is a multiple of p, it should be $ap\equiv 0\mod p$ and so $(ap)^e=0\mod p$. It follows that $ap\mod p=(ap)^e\mod = p= 0 \mod p$.

I now need to show, that $ap\mod q=(ap)^e\mod q$. This is the part I am struggling with. I know that $ap-bq=1$ but I have no glue how this is going to help me. I was thinking about something like $ap=(bq+1)\mod q$ but i am not sure if this works / what to do next.

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  • $\begingroup$ Your last thought is entirely correct. It implies that $ap \equiv 1 \pmod q \implies (ap)^e \equiv 1 \pmod q$. $\endgroup$
    – user114966
    Jan 22 at 12:19
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    $\begingroup$ Oh my god, how could i oversee that ... Of course $q|bq$ and so we have $ap\equiv1\mod q$ $\endgroup$ Jan 22 at 14:25
  • $\begingroup$ Can you be more specific about what your question is? We are a question-and-answer site, so we require you to articulate a specific question -- asking whether we have any reactions to some work-in-progress isn't within the scope of this site. $\endgroup$
    – D.W.
    Jan 23 at 3:36
  • $\begingroup$ I am sorry about this. I wanted to know how to show, that $ap\equiv_p (ap)^e$. Dmitry already gave me the input needed and now i answered the question, so that other people can look it up. $\endgroup$ Jan 23 at 10:20
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You can show that $ap\mod q =(ap)^e\mod q$ by using the fact, that $ap-bq=1\Longleftrightarrow ap=bq+1$.

You know that $q|bq$ and therefor $ap\mod q = bq+1\mod q \equiv 1\mod q$.

This implicates that $(ap)^e\mod q = (bq+1)^e\mod q \equiv 1\mod q$. Therefor $ap\mod q \equiv (ap)^e\mod q.$

Now with the chinsese remainder theorem we know that $ap\equiv_N(ap)^e$.

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