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In the RSA public key cryptosystem, the private and public keys are $(e, n)$ and $(d, n)$ respectively, where $n=p \times q$ and $p$ and $q$ are large primes. Besides, $n$ is public and $p$ and $q$ are private. Let $M$ be an integer such that $0<M<n$ and $\phi(n) = (p-1)(q-1)$. Now consider the following equations. 

  1. $M’ = M^e \text{ mod } n$
    $M = (M’)^d \text{ mod } n$

  2. $ed  \equiv 1 \text{ mod } n$

  3. $ed  \equiv 1 \text{ mod } \phi(n)$

  4. $M’ = M^e \text{ mod } \phi(n)$
    $M = (M’)^d \text{ mod }\phi( n)$

Which of the above equations correctly represents RSA cryptosystem?

  1. I and II
  2. I and III
  3. II and IV
  4. III and IV
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3 Answers

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The basic principle behind RSA is the observation that it is practical to find three very large positive integers $e, d$ and $n$ such that with modular exponentiation for all $m:$

${\displaystyle (m^{e})^{d}\equiv m{\pmod {n}}}$ and that even knowing $e$ and $n$ or even m it can be extremely difficult to find $d.$

Additionally, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies: ${\displaystyle (m^{d})^{e}\equiv m{\pmod {n}}}$

The keys for the RSA algorithm are generated the following way:

  • Choose two distinct prime numbers $p$ and $q.$
  • Compute $n = pq.$
  • Compute $\varphi (n)  = \varphi (p)\varphi (q) = (p − 1)(q − 1)$
  • This is more clearly stated as: solve for $d$ given $d⋅e \equiv 1 (\mod \varphi (n))$

So, B is answer.

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They have reversed the sign.in general (e,n) is public key and (d,n) is private key for a sender so do not confuse with this.

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If equation 2 is true i.e, ed mod n =1, then any one can calculate private key of receiver.

Hence any one can decrypt the message of sender.


Therefore equation 2 is false.

If equation 4 is true,  i.e $m=m^{e}$ mod $\phi(n)$ no one except the receiver can encrypt the message because calculating $\phi(n)$ without the prime numbers p,q which made n(i.e, n=p*q) is computationally very difficult .

Which is clearly not we want.

Hence equation 2,4 are false.

Hence option b is answer

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