Algebraic Cryptology from an Historical Viewpoint

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