In the RSA cryptosystem it is possible that M C that is the

In the RSA cryptosystem, it is possible that M = C, that is, the plaintext and the ciphertext may be identical.

Is this a security concern in practice?

For modulus N = 3127 and encryption exponent e = 17, find at least one non trivial message M (i.e. M > 1) that encrypts to itself

Solution

Given:

e=17

N=3127

N=p*q=53*59

Therefore,

p = 53 and q = 59

(n) = (p-1)(q-1)

         = 52*58

         = 3016

d*e(mod( (n)) = 1

d*17(mod3016) = 1

d = 2129

M and C can be calculated as:

C=M­­^emodN

M=C^dmodN

Compare M and C for different values of M>1:

For the value 4.

C=M­­^emodN = 4¹⁷mod3127 = -763

M=C^dmodN = 4²¹²⁹mod3127 =-763

Verifying the result,

M=C^dmodN = -763²¹²⁹mod3127 =-763

C=M­­^emodN = -763¹⁷mod3127 = -763

Thus, for the value 4, M=C and M and C are identical.

D is the trivial message for which M = C.