Lester Hill Revisited

1. Lester Hill Revisited Chris Christensen Northern Kentucky University

2. Lester S. Hill (1891 - 1961) • B.A. in mathematics from Columbia in 1911. Master’s degree 1913. Ph.D. from Yale in 1926. • 1916 joined US Navy Reserves and served in World War I as a LT (j.g.)

3. Hunter College • Hill joined the faculty at Hunter College in 1927. • Taught at the Army University in Biarritz, France in 1945. • Hill remained at Hunter until his retirement due to illness in 1960. • Hill died in 1961.

4. David Kahn The Codebreakers • David Kahn met with Hill’s widow after Hill’s death and collected papers of Hill’s that were “laying around the house.” • Those papers are now at the National Cryptologic Museum library.

5. National Crypt0logic Museum

6. The American Mathematical Monthly • 1929 “Cryptography in an algebraic alphabet” • 1931 “Concerning certain linear transformation apparatus of cryptography”

7. Pre-Hill Monoalphabetic substitution Polygraphic substitution

8. “Cryptography in an algebraic alphabet”

9. Encryption of norse using 2x2matrix

10. How do we decrypt DSDOKK?

11. Calculation of the key inverse

12. What is the condition on the key?

13. Must be able to divide by determinant

14. Integers mod 26 under multiplication

15. Key inverse

16. Integers mod 26 under multiplication

17. Key inverse

18. Hill cipher

19. Hoe large is the 4x4 keyspace?

20. How large is the keyspace? Reference 4x4 key “The keyspace of the Hill cipher” by Overby, Traves, and Wojdylo in Cryptologia, 2005.

21. Involutory key?

22. Encryption by polynomials

23. Coordinate functions

24. What’s wrong with the Hill cipher?

25. What’s wrong? It’s LINEAR.

26. Multivariate quadratic polynomials Solution is NP-hard.

27. Lester Hill’s message protector

28. Input data from a check Amount $128 Check number 586 Date December 26, 1928

29. Input Data from a check Data from a chart Amount $128 1 28 Check number 586 5 86 Date December 26, 1928 26 28 56 99 01 12 72 64

30. Transformation

31. Transformation

32. Transformation

33. Input

34. Output

35. Collisions must occur Input is 6 numbers between 00 and 100 56 99 01 12 72 64 101^6 = 1,061,520,150,601 Output is 3 numbers between 00 and 100 100 40 68 101^3 = 1,030,301

36. A checking scheme • In 1926 and 1927, while he was a Ph.D. student at Yale, Hill published three papers in Telegraph and Telephone Age which describe a checking scheme. • “He hoped to make some money from his checking scheme, which he was seeking to have patented. This did not go anywhere, but it sparked his interest in secret communications.” David Kahn

37. Lester Hill to Lloyd Wilson November 21, 1925 “Briefly stated, what I now have in mind – and have not noticed hitherto – is that, if my checking procedure were applied generally, it would be very easy to make the telephone (long distance) take over effectively, in a novel way, a goodly portion of the present domestic telegraph business.”

38. 984600007405000090 “We are not interested in the origin or significance of the component parts of the number, nor in the method of transmittal. Thus, 7405 might be a sum of money, and 000090 a combination of testing figures compounded from the initials to whom the money is being sent and from other elements; 98460 might refer to an entry in some code book or other volume, etc. The entire number may be sent as it stands, or by means of code and cipher. Our object here is merely to supply a check upon the accurate transmittal.”

39. Checking procedure

40. The sender and the receiver • The nine-digit message is checked by the sequence 97 90 39. • The sender send the message 984600007405000090 appended by the check 979039. • The receiver calculates the check string from the received message string and compares it to the received check string. • If the two check strings are the same, it is assumed that the message was transmitted without error.

41. Error detecting codes • All error detecting codes require some repetition of message information. • The goal is to minimize the amount of repetition.

42. History Error detecting codes Error correcting codes The history of error detecting codes is not clear. • Claude Shannon (1948) • Richard Hamming (1948) • Marcel Golay (1949)

43. How much did Hill understand? • It is not clear from Hill’s Telegraph and Telephone Age papers whether he understood that the method he was describing was matrix multiplication. • “The checking of the accuracy of transmittal of telegraphic communications by means of operations in finite fields” Undated; in the David Kahn collection.

44. Hill to Wilson “My correspondent will be absolutely sure that he has precisely the message which I sent him, or absolutely sure that a mistake is present . … And nobody in the world except my correspondent can possibly decipher the meaning of my message. Moreover, my correspondent will be deadly sure, if the message checks, that message was sent by me and nobody else in the world. If this message checks, … correspondent can accept it as having all of my authority behind it.”

45. What do cryptographers do? • Secret communications. • Integrity. • Authentication and non-repudiation.