![Page 1: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/1.jpg)
Chris ChristensenNorthern Kentucky University
Algebraic Cryptology from an Historical Viewpoint
![Page 2: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/2.jpg)
Cryptography
CryptologyCryptanalysis
![Page 3: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/3.jpg)
Algebraic cryptology
CryptographyPolynomialsFinite rings and fields
CryptanalysisSolving systems of multivariate polynomial equations
![Page 4: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/4.jpg)
Julius Caesar (100 – 44 BC)
![Page 5: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/5.jpg)
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
![Page 6: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/6.jpg)
Lester S. Hill (1891 – 1961)Monthly articles:
“Cryptography in an algebraic alphabet.” 1929.
“Concerning certain linear transformations apparatus of cryptography.” 1931.
![Page 7: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/7.jpg)
Hill’s cipher
![Page 8: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/8.jpg)
Hill’s cipher is algebraic
![Page 9: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/9.jpg)
1986 Fell-Diffie “Analysis of a public key approach based upon polynomial substitution”
![Page 10: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/10.jpg)
1983 Matsumoto and Imai
1999 Tzuong-Tsieng Moh
Multivariate Public Key Cryptosystems, Ding, Gower, and Schmidt.
Algebraic cryptography
![Page 11: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/11.jpg)
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
![Page 12: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/12.jpg)
Hill cipher, again
![Page 13: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/13.jpg)
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
![Page 14: Algebraic Cryptology from an Historical Viewpoint](https://reader035.vdocument.in/reader035/viewer/2022062410/5681614d550346895dd0d005/html5/thumbnails/14.jpg)
LN WQ JW