Chris ChristensenNorthern Kentucky University

Algebraic Cryptology from an Historical Viewpoint

Cryptography

CryptologyCryptanalysis

Algebraic cryptology

CryptographyPolynomialsFinite rings and fields

CryptanalysisSolving systems of multivariate polynomial equations

Julius Caesar (100 – 44 BC)

A. A. Albert (1905 – 1972)

… we shall see that cryptography is more than a subject permitting mathematical formulation, for indeed it would not be an exaggeration to state that abstract cryptography is identical with abstract mathematics.

November 22, 1941

Lester S. Hill (1891 – 1961)Monthly articles:

“Cryptography in an algebraic alphabet.” 1929.

“Concerning certain linear transformations apparatus of cryptography.” 1931.

Hill’s cipher

Hill’s cipher is algebraic

1986 Fell-Diffie “Analysis of a public key approach based upon polynomial substitution”

1983 Matsumoto and Imai

1999 Tzuong-Tsieng Moh

Multivariate Public Key Cryptosystems, Ding, Gower, and Schmidt.

Algebraic cryptography

Claude Shannon (1916 – 2001)

Thus if we could show that solving a certain [crypt0]system requires as least as much work as solving a system of simultaneous equations in a large number of unknowns, of a complex type, then we would have a lower bound of sorts for [its security]. 1949

Hill cipher, again

1965 Bruno Buchberger, Grobner basis

1999 and 2002 Jean-Charles Faugere, F4 and F5

1999 Kipnis and Shamir, XL

2006 Ding, mutant XL

Algebraic Cryptanalysis, Gregory V. Bard

Algebraic cryptanalysis

LN WQ JW