Department of Mathematics
Boise State University
Mathematics 124
Fall 2004

E-Poker and the Massey-Omura protocol

The Massey-Omura protocol is as follows: Two parties, Bob and Alice, want to communicate securely. They choose a common large prime number P. Then each privately and independently choose encryption keys eBob and eAlice where

  1. gcd(eBob,P-1) = 1;
  2. gcd(eAlice,P-1)=1.

They do not make these keys known to anyone. Each independently compute their decryption keys as follows:

  1. dBob   = 1/eBob mod (P-1 );
  2. dAlice = 1/eAlice mod (P-1).

The protocol for communicating a message, say M (<P), is as follows: Assume Bob wants to communicate it to Alice.

  1. Bob encrypts M to get E1:=MeBob mod P and sends this to Alice;
  2. Alice then encrypts E1 to get E2:=E1eAlice mod P, and returns this to Bob;
  3. Bob decrypts E2 to get E3:=E2dBob mod P, and returns this to Alice;
  4. Alice decrypts E3 to get E4:=E3dAlice mod P.

Note that E3 does not carry any encryption by Bob anymore, but still carries Alice's encryption. E4 is equal to M, the original message. One could envision this as follows: Bob places the message in a tamperproof container, and locks it with a lock for which only Bob has a key. This container with lock is then sent to Alice. Alice then also puts a lock for which only she has a key on the container, and returns the container (now carrying two locks) to Bob. Bob then removes the lock he originally put on the container, and returns the container - now carrying only Alice's lock, to Alice. Finally, Alice removes her lock and opens the container to retrieve the message.

This protocol can be used as follows by two remote parties to play e-poker. Say Alice is the dealer, and Bob the shuffler. First, Alice encrypts all cards in the deck, and send these to Bob. Bob now shuffles the deck, and also encrypts the cards, and returns the twice-encrypted, shuffled deck to Alice. Alice now removes ten cards from the deck, decrypts five of them, and sends these ten cards back to Bob. Bob then decrypts all ten. Five will now be visible to Bob, and this is Bob's hand. The remaining five are returned to Alice, who decrypts them, and this will be Alice's hand.