# Decoding Message-Signature Pairs

I've been given a set of message-signature pairs to decrypt using a public RSA key I set up. I need to sort through the decoded messages, and find which of those have a valid signature. My public key is

e=13


and

m=23988477718194197355263570318118557072073635564327068738051383543307247873821867


My code so far comprises of

decryptionRSA[x_?IntegerQ, d_?IntegerQ, m_?IntegerQ] := PowerMod[x, d, m];
SetAttributes[decryptionRSA, Listable]
messages2 = Table[i -> decryptionRSA[messages, d, m], {i, messages}]


It then outputs messages2, which is too big to copy onto here. Note that: d=PowerMod[e,-1,m]. A message signature pair will come in the form (a,b), with b being the signature. To tell if the pair is valid and authentic, using the public key, you compute z=PowerMod[b,e,m] and if this equal to a then the pair is valid. However, I don't know how I would verify this with a large quantity of data, would I use the Table function?

This answer will try to address a number of shortcomings/inconsistencies in the body of the question and finally suggest a possible solution.

## Issues

1. In the definition of the function decryptionRSA[] the second parameter, namely parameter d, is not an actual parameter. According to the body of text, d=PowerMod[e,-1,m], where e and m are given.
2. Assigning the attribute Listable to symbol decryptionRSA will allow it to thread over lists. Applying a listable function over a list of pairs eg f[{{a,1},{b,2},...}] does not produce {f[a,1],f[b,2],...} which is the desirable behavior, but rather {{f[a],f[1]},{f[b],f[2]},...}.

## A solution

Assuming that messages is a list containing pairs of message and signature, then the following code will produce a list of booleans, where True would indicate the position of a valid pair in the original list.

d = PowerMod[e,-1,m]

The valid pairs can be retrieved using eg pos = Position[mask,True] and then valids = Extract[messages,pos].