一个私钥可以与几个公钥关联吗？

Question

Ok, so my crpto lecturer in Uni posed this question at the end of the RSA key generation lecture as a brain teaser.

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 ):

• Use RSA to generate public and private key pair.

• Make an additive hash function that taken in any input and adds the
  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 )

• Now take your public key and generate several possible keys by jumbling up the on and off bits.

• Give these jumbled up keys to potential customers.

• Put our hash function on the server.

• When customer puts in his jumbled up public key - Hash function
  calculates the hash - which should be similar to original public key generated by RSA. Customer gets access.

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 ?

EDIT:

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 ".

Answer 1

I think I know what your lecturer was getting at. I'm not going to give you the full answer but I'll lead you along a few steps. Obviously your lecturer wants you to study the math of RSA. In particular, look at the public and private exponents and how they are related. Now, suppose you have an RSA modulus N, and RSA public exponent e, and an RSA private exponent d. Is there another exponent, say e + x, that works exactly the same way as e in the RSA math? The answer is yes, there is. In fact, there is an entire family of such exponents. Now you have to discover what this x value is, and how it related to the math of RSA.

Answer 2

I don't want to spoil too much, but I think the answer your lecturer was waiting for is more related to the mathematical properties of finite groups. Maybe see the definition of textbook RSA and see if you can find another public key that matches a given private key. Fermat's Little Theorem might also be interesting.

Answer 3

在查找有限组时，您想阅读有关等价类的信息。