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I am preparing for the final exam in my Data Security class so I am trying to read and understand the textbook's exercise questions. Any hint would be great.

Question (Exercise 9.14 of textbook): enter image description here

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Observation: Similar to $RSA$ decryption that $e.d = 1$, we to find some number $r$ such that $E*r = 1$ so that $(m^E)^r \ (mod \ N) = m$. But what should that $r$ be? Any hint would be great.

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I will assume that Alice uses $D' = E^{-1}(\varphi(N) - 1)$ with $E^{-1}$ the inverse of $E$ modulo $\lambda(N)$. The question uses a very specific definition of $D'$, which requires the assumption that $E$ divides $\varphi(N) - 1$. But since such an $E$ is necessarily coprime to $\varphi(N)$, it is a special case of the more general definition of $D'$ provided above.

So we must find some number $E'$ such that $$E' E^{-1}(\varphi(N) - 1) \equiv 1 \pmod {\lambda(N)}$$ Now note that $E' = -E$ is a solution because $\varphi(N) \equiv 0 \pmod{\lambda(N)}$, so we have

$$m^{-ED'} \equiv 1 \pmod N$$.

So given $c = m^{E}$, compute $c^{-D'}$ to obtain $m$ again. Either you first take the inverse, or you just use $\lambda(N) - D'$ as the exponent.

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