# Any way to prove possession of a RSA 2048 private key using 256 bit?

Alice has a RSA 2048 private key and Bob has corresponding public key. They have a secure connection, but Alice can transmit only 256bit to Bob (Bob can transmit any amount of data to Alice).

Is there a way how Alice can proof to Bob that she is in possession of the private key?

Bob selects a random 2048 bit value $$m < n$$ and computes $$m^e \bmod n$$ and sends that and $$\text{SHA3_256}(m )$$ to Alice.
Alice, with the private key, recovers $$m$$, and verifies that $$\text{SHA3_256}(m)$$ matches the value she received from Bob. If so, she computes and sends $$\text{SHA3_256}(m || m)$$; that's 256 bits.