title: Can one private key be associated with several public keys?
body: <p>Ok, so my crpto lecturer in Uni posed this question at the end of the RSA key generation lecture as a brain teaser.</p>
  
  <p>I have been thinking about this and I think I have come up with a way ( I am aware its not practical and very vulnerable - But I would love for you guys to punch holes through it ):</p>
  
  <ul>
  <li><p>Use RSA to generate public and private key pair.</p></li>
  <li><p>Make an additive hash function that taken in any input and adds the<br>
  ascii value of each character and gives that as an output. ( I am
  aware additive hash has horrible distribution for ex ABC , BAC , ACB
  all have the same hash output -- but theoretically this can be used
  as an advantage to solve our problem )</p></li>
  <li><p>Now take your public key and generate several possible keys by
  jumbling up the on and off bits.</p></li>
  <li><p>Give these jumbled up keys to potential customers.</p></li>
  <li><p>Put our hash function on the server.</p></li>
  <li><p>When customer puts in his jumbled up public key - Hash function<br>
  calculates the hash - which should be similar to original public key 
  generated by RSA. Customer gets access.</p></li>
  </ul>
  
  <p>Once again I am aware that this is very vulnerable and terrible. It cant be used in the real world. But does it answer the question my lecturer posed ?</p>
  
  <p><strong>EDIT:</strong></p>
  
  <p>Guys I am sorry for the confusion. This question has nothing to do with collisions and expoiting the maths ( Those are definitely valid ways to do that - but they weren't what my lecturer was referring too - they were discussed in the lecture itself. ). The way my lecturer asked this question was - " come up with ways to make several public keys related to a single private key - nevermind how vulnerable your solution is ".  </p>