Question

Someone sends you a message of "9". They also send you a copy of their message encrypted with their private key. The "signed", or encrypted copy is "4". Their public key is (exponent 23, clock 55), Show how to authenticate that the message is from them

Answer 1

This question can be answered using the basics of Public-Key Cryptography.

Someone sent you a mesage, "9". This can be treated as the plaintext.

Then, they sent you the "signed" or "encrypted" copy "4". This can be treated as the cipher text generated by encrypting with their private key.

Remember, you do not have access to their private key, but you have the access to their public key.

As given in the question, their public key is (exponent 23, clock 55).

We need to authenticate that the message is from them. For doing so, we need to check by converting the ciphertext (4) into plaintext (9).

Given plaintext = 9.

Using the standard public key encryption formulae, we know that

Ciphertext = Plaintext ^{public encrypt exponent} (mod n)

Here, Paintext = 9, public encrypt exponent = 23, and clock = n = 55, given.

Thus Ciphertext = 9²³ mod 55

which upon calculation, comes out to be 8862938119652501095929 mod 55.

Hence, you can clearly see the mod function upon input 55 and 9²³ will generate the output as 4, since 4 will be the common remainder when 9²³ is divided by 55.

Hence, till now, we have verified that the plaintext 9 when encrypted using the key details given in question, matches with the Ciphertext we have received. This is an indicator of authenticity.

Further, we can decrypt the signed copy that we have received, by applying the following formula:

Plaintext = Ciphertext ^exponent mod n

where ciphertext = 4, exponent = 23, n = clock = 55.

Hence, plaintext = 4²³ mod 55.

which upon calculation results into 70368744177664 mod 55.

Which clearly indicates that 4²³ upon division by 55 will leave a remainder in terms of 9.

Hence, the Plaintext evaluates to 9.

The commonly agreed public key perfectly interconverts the message and the signed copy.

By the above two comparisions, we can be sure about the authenticity of the two messages, one being the plaintext 9 and other being the signed text 4, and hence can be sure about the fact that 9 and 4 are from the same source.