Cayley–Purser algorithm: Difference between revisions

Revision as of 15:01, 16 July 2019

Revision as of 14:43, 3 October 2019

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== History ==

During a work-experience placement with [[Baltimore Technologies]], Flannery was shown an unpublished paper by [[Michael Purser]] which outlined a new [[public-key]] cryptographic scheme using [[non-commutative]] multiplication. She was asked to write an implementation of this scheme in [[Mathematica]].

Before this placement, Flannery had attended the 1998 [[Young Scientist and Technology Exhibition|ESAT Young Scientist and Technology Exhibition]] with a project describing already existing cryptographic techniques from the [[Caesar cipher]] to [[RSA (algorithm)|RSA]]. This had won her the Intel Student Award which included the opportunity to compete in the 1998 [[Intel International Science and Engineering Fair]] in the United States. Feeling that she needed some original work to add to her exhibition project, Flannery asked Michael Purser for permission to include work based on his cryptographic scheme.

On advice from her mathematician father, Flannery decided to use [[Matrix (mathematics)|matrices]] to implement Purser's scheme as [[matrix multiplication]] has the necessary property of being non-commutative. As the resulting algorithm would depend on multiplication it would be a great deal faster than the [[RSA (algorithm)|RSA]] algorithm which uses an [[exponent]]ial step. For her Intel Science Fair project Flannery prepared a demonstration where the same plaintext was enciphered using both RSA and her new Cayley–Purser algorithm and it did indeed show a significant time improvement.

Returning to the ESAT Young Scientist and Technology Exhibition in 1999, Flannery formalised Cayley-Purser's runtime and analyzed a variety of known attacks, none of which were determined to be effective.

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=== Key generation ===

Like [[RSA (algorithm)|RSA]], Cayley-Purser begins by generating two large primes ''p'' and ''q'' and their product ''n'', a [[semiprime]]. Next, consider [[general linear group|GL]](2,''n''), the [[general linear group]] of 2×2 matrices with integer elements and [[modular arithmetic]] mod ''n''. For example, if ''n''=5, we could write:

:<math>\left[\begin{matrix}0 & 1 \\ 2 & 3\end{matrix}\right] +