Let $N$ be an even integer. Consider the affine cipher on the space of plaintext messages $\mathcal M=\mathbb Z/N\mathbb Z$ with encryption function $e(m)=am$ where $a\;\epsilon\; \mathbb Z/N\mathbb Z$ and $a\neq1\pmod{N}$. Assume $a$ is invertible modulo $N$.

I know that $0$ is going to be a fixed message as $e(0)\equiv a(0)\equiv 0 \pmod{N}$. Also I know that the other fixed message is going to be of the form $$ e(m_1)\equiv am_1\equiv m_1\pmod{N}$$ $$ m_1\equiv a^{-1}m_1 \pmod{N}$$ but I am not sure how to find this without assuming it exists first. I tried to solve the same equation by: $$ am_1-m_1\equiv m_1(a-1)\equiv 0 \pmod{N}$$ but this is also using the same assumption, right?

  • $\begingroup$ You should use the hypothesis that $N$ is even. In that case, is $a$ even or odd? How about $a-1$? $\endgroup$ – FredH Feb 25 at 18:46
  • $\begingroup$ So then $a$ has to be odd because it is invertible.. so $a-1$ is even $\endgroup$ – joseph Feb 25 at 18:47
  • $\begingroup$ I did think of going this route but I couldn't figure out how that would eventually help me get my same input without assuming it already exists $\endgroup$ – joseph Feb 25 at 18:48

Write $N=2M$. Since $a$ is invertible mod $N$, you have $\gcd(a,N)=1$. Hence, $a$ cannot be even since $N$ is even. Thus $a$ is odd and so $a=1+2k$ for some $k$. Then $$ e(M)=aM=(1+2k)M=M + 2kM = M + kN = M \bmod N. $$ But clearly $M$ isn't the class of $0$ so this is your other fixed point.

  • $\begingroup$ why did you start by writing $N=2M$? I know N must be even but why does it have to be two times $m$? Or I guess how did you know this ahead of time? $\endgroup$ – joseph Feb 25 at 19:00
  • $\begingroup$ That's what an even number looks like. I used this to find out that $M$ was going to work. $\endgroup$ – Randall Feb 25 at 19:02
  • $\begingroup$ IOW, I showed that $N/2$ will always be fixed. So is $0$, so QED $\endgroup$ – Randall Feb 25 at 19:03
  • $\begingroup$ Yeah I know that an even number is given by $N=ex$ for some $x:\ \epsilon \mathbb Z$ I'm just wondering how you knew to write it as $a=M$? Did you just find that by playing with actual numbers ahead of time and then going back and using that from the beginning or was there some deeper intuition used? $\endgroup$ – joseph Feb 25 at 19:05
  • $\begingroup$ How does it matter? Write $N=2\beta$ and then prove that $\beta$ is a fixed message. In short, yes, I just played with it. I didn't call it "$M$" for "message" because I knew that was going to work ahead of time. $\endgroup$ – Randall Feb 25 at 19:05

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