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Muath Almusharraf avatar Crypto

Plaintext message equals the encrypted message

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Muath Almusharraf

P=11 Q=29 Euler=280 e=3 d=187

When i take a message “88” for example; Encrypt: 88^3(mod 319) = 88 Decrypt: 88^187(mod 319) = 88

Why does the encrypted message equals the decrypted message and why does the plaintext message equals the encrypted message?

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kelalaka avatar
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kelalaka
Does this answer your question? [How many points in RSA, such that $m^e = m \bmod n$](https://crypto.stackexchange.com/questions/89803/how-many-points-in-rsa-such-that-me-m-bmod-n) and [Fixed point in RSA encryption](https://crypto.stackexchange.com/q/81128/18298) and another duplicated if one searched [What is the number of fixed plaintexts of RSA](https://crypto.stackexchange.com/q/89830/18298)
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Daniel S avatar Crypto
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Daniel S

There are always at least 4 values that are unchanged by RSA encryption. These are the four solutions to the two simultaneous equations $x\mod p=0,1$ and $x\mod q=0,1$. Two of these solutions are $x=0$ and $x=1$ which apply in all cases. In your case $x\mod p=0$ and $x\mod p=1$ is the solution 88. The other solution $x\mod p=1$ and $x\mod q=0$ is 232.

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Mark avatar Crypto
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Mark

It is worth mentioning that an encryption scheme sometimes having $\mathsf{Enc}_k(m) = m$ (for suitable choices of randomness) is something newcomers often find very unsettling, and wish to purposefully exclude in constructions.

In general this is a very bad idea, because (roughly) then if an adversary sees $c = \mathsf{Enc}_k(m)$, they know the message $c$ was not encrypted. This is to say that stopping encryption sometimes being a "no-op" explicitly leaks information, that attackers can then leverage. I quote from the wikpedia article of the Enigma machine, in a section discussing weaknesses of the Enigma Machine.

A letter could never be encrypted to itself, a consequence of the reflector.[19] This property was of great help in using cribs—short sections of plaintext thought to be somewhere in the ciphertext—and could be used to eliminate a crib in a particular position. For a possible location, if any letter in the crib matched a letter in the ciphertext at the same position, the location could be ruled out.[20] It was this feature that the British mathematician and logician Alan Turing exploited in designing the British bombe.

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