Reveal a plaintext with 3 encryptions of Textbook RSA Assume we using Textbook RSA encryption with the public key $(N,3)$.
How can I learn message $m$ if I have the encryptions of $m,m+1,m+2$ in an efficient way?
I'm studying for a test and it's a question from one of the tests of the past
We have
\begin{align}
m^3 &amp; &amp; &amp; \pmod n\\
(m+1)^3 &amp;= m^{3} + 3 \, m^{2} + 3 \, m + 1 &amp; &amp; \pmod n\\
(m+2)^3 &amp;= m^{3} + 6 \, m^{2} + 12 \, m + 8 &amp; &amp;\pmod n\\
\end{align}
Now multiply $(m+1)^3$ with $\color{red}{2}$
\begin{align}
m^3 &amp; &amp; &amp;\pmod n\\
\color{red}{2}(m+1)^3 &amp;= 2m^{3} + 6 \, m^{2} + 6 \, m + 2 &amp; &amp;\pmod n\\
(m+2)^3 &amp;= m^{3} + 6 \, m^{2} + 12 \, m + 8 &amp; &amp; \pmod n\\
\end{align}
Now substruct first and third equation from the second
\begin{align}
2(m+1)^3 - (m+2)^3 - m^3&amp;= - 6 \, m -6 \pmod n\\
2 \cdot c_2 - c_3 - c_1 &amp;= - 6 \, m -6 \pmod n
\end{align}