No-Key Cryptography

1. No-Key Cryptography Nathan Marks Based on Massey-Omura US Patent # 4,567,600

2. Overview • Introduction and Overview • Analogy of No-Key Cryptography • Basic over-view of No-Key Cryptography • Original Massey-Omura Algorithm • Variations of Massey-Omura • Conclusion • Questions

3. Briefcase Analogy of No-Key • Alice wishes to pass Bob information. • Alice locks one lock on case. • Bob receives case, and locks other half. • Alice then unlocks her half. • Bob unlocks his half. • Bob is able to easily open case.

4. No-Key Cryptography Overview • Alice wishes to send Bob a message. • Alice encrypts message with secret key. • Bob encrypts message with secret key. • Alice decrypts message with her key. • Bob decrypts message with his key. • Bob can easily retrieve original message.

5. Differences of Case and Crypto • Transferring a physical case much different than digital data. • Brute force on case easy (break the case open). • Brute force on cipher hard because of encryption with completely secret key. • Locks on case do not interfere with each other. • Encrypting twice and then decrypting in same order may cause interference.

6. Problems with No-Key Crypto • There are 3 public transfers the size of the message – not one. • Not all encryption/decryption algorithms are associative with each other – meaning that encrypting and then decrypting in the incorrect order causes interference. • Need of separate authentication for all transmissions.

7. Original Massey-Omura • An algorithm that satisfies necessary mathematical requirements to make No-Key work. • Uses finite fields. • Relies on Discrete Log Problem for security.

8. Original Massey-Omura (cont.) • Operates in finite field world of GF(2m) • As shown by Diffie and Hellman in “New Directions in Cryptography” exponentiation in GF(2m) is easy (>m, but <2m operations). • Taking the logarithm in GF(2m) is hard DLP (approx. 2m/2 operations)

9. Original Massey-Omura (cont.) • The message M is encoded as an element of GF(2m) and represented as m binary digits in the manner: M=[bm-1, bm-2,…,b1,b0] such that bm-1 is the first bit of the message. • Both Alive and Bob generate a random # E such that 0<E<2m-1 • Both calculate D for their respective E’s such that: E*D=1 mod 2m-1

10. Original Massey-Omura (cont.) • Alive calculates MEA (in GF(2m))=M1 • Bob receives this value M1 and calculates M1EB(in GF(2m))=M2 • Alice receives this value M2 and calculates M2DA (in GF(2m))=M3 This decrypts her part of the encryption. • Bob receives this value M3 and calculates M3DB (in GF(2m))=M4 This decrypts his part of the encryption. • M4 = M, therefore Bob has Alice’s message.

11. Variations of Massey-Omura • Elliptic curve version where multiplication of a constant (secret keys) times a point (the encoded message) takes the place of exponentiation. • L(D,N) is a LUCAS group where N is a large prime. M is encoded as a point in L(D,N) and the order of L(D,N) is used as the modulus. M is then raised to the power of the secret keys (as in normal Massey-Omura) based on the rules of exponentiation of LUCAS groups.

12. Conclusions • Massey-Omura is a good way of making the No-Key algorithm work mathematically and practically. • No-Key systems are Zero-Knowledge, which means they are just as secure as whatever encryption algorithm is used. • Even so No-Key seems are not used in practice very much because of the impracticality of having to transfer the entire message three times.

13. Questions? • Questions anyone?

14. References • US Patent #4,567,600 submitted by James L. Massey (Swiss) & Jimmy K Omura(USA) on September 14, 1982. • Boise State University Mathematics Department • Dr. J. von zur Gathen und Dr. J. Teich