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