Chris Christensen

Northern Kentucky University

Lester Hill Revisited

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.)

Hunter CollegeHill 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.

David Kahn The CodebreakersDavid 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.

National Crypt0logic Museum

1929 “Cryptography in an algebraic alphabet”

1931 “Concerning certain linear transformation apparatus of cryptography”

The American Mathematical Monthly

Monoalphabetic substitution

Pre-HillPolygraphic substitution

“Cryptography in an algebraic alphabet”

Encryption of norse using 2x2matrix

How do we decrypt DSDOKK?

Calculation of the key inverse

What is the condition on the key?

Must be able to divide by determinant

Integers mod 26 under multiplication

Key inverse

Integers mod 26 under multiplication

Key inverse

Hill cipher

Hoe large is the 4x4 keyspace?

Reference

“The keyspace of the Hill cipher” by Overby, Traves, and Wojdylo in Cryptologia, 2005.

How large is the keyspace?

4x4 key

Involutory key?

Encryption by polynomials

Coordinate functions

What’s wrong with the Hill cipher?

It’s LINEAR.

What’s wrong?

Solution is NP-hard.

Multivariate quadratic polynomials

Lester Hill’s message protector

Input data from a check

Amount $128

Check number 586

Date December 26, 1928

Data from a check

Amount $1281 28Check number 5865 86Date December 26,

192826 28

56 99

01 12

72 64

Input

Data from a chart

Transformation

Transformation

Transformation

Input

Output

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

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

A checking scheme

“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.”

Lester Hill to Lloyd Wilson November 21, 1925

“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.”

984600007405000090

Checking procedure

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.

The sender and the receiver

All error detecting codes require some repetition of message information.

The goal is to minimize the amount of repetition.

Error detecting codes

Error detecting codes

The history of error detecting codes is not clear.

Claude Shannon (1948)

Richard Hamming (1948)

Marcel Golay (1949)

HistoryError correcting codes

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.

How much did Hill understand?

“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.”

Hill to Wilson

Secret communications.

Integrity.

Authentication and non-repudiation.

What do cryptographers do?