Posts Tagged: cryptography

  • web

    Lists 2010 : StackExchange sites

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    One of the trends of 2010 was the proliferation of StackExchange sites. I guess by now most of us visit MathOverflow along with the arXiv daily. But, there are plenty of other StackExchange sites around that may be of interest to the mathematics-community : Mathematics somewhat less high-brow than MathO. Physics still in the beta-phase… Read more »

  • games, number theory

    On2 : transfinite number hacking

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    Surely Georg Cantor’s transfinite ordinal numbers do not have a real-life importance? Well, think again.

  • stories

    return of the cat ceilidh

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    I couldn’t believe my eyes. I was watching an episode of numb3rs, ‘undercurrents’ to be precise, and there it was, circled in the middle of the blackboard, CEILIDH, together with some of the key-exchange maps around it… Only, the plot doesn’t involve any tori-crypto… okay, there is an I-Ching-coded-tattoo which turns out to be a… Read more »

  • stories

    now what?

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    You may not have noticed, but the really hard work was done behind the scenes, resurrecting about 300 old posts (some of them hidden by giving them ‘private’-status). Ive only deleted about 10 posts with little or no content and am sorry I’ve self-destructed about 20-30 hectic posts over the years by pressing the ‘delete… Read more »

  • featured

    the crypto lattice

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    Last time we have seen that tori are dual (via their group of characters) to lattices with a Galois action. In particular, the Weil descent torus $R_n=R^1_{\mathbb{F}_{p^n}/\mathbb{F}_p} \mathbb{G}_m $ corresponds to the permutation lattices $R_n^* = \mathbb{Z}[x]/(x^n-1) $. The action of the generator $\sigma $ (the Frobenius) of the Galois group $Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) $ acts on… Read more »

  • featured

    Weil descent

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    A classic Andre Weil-tale is his narrow escape from being shot as a Russian spy The war was a disaster for Weil who was a conscientious objector and so wished to avoid military service. He fled to Finland, to visit Rolf Nevanlinna, as soon as war was declared. This was an attempt to avoid being… Read more »

  • featured

    key-compression

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    The main application of tori to cryptography is to exchange keys more efficiently while preserving the same security standards. In the Diffie-Hellman key-exchange one interchanges elements of the finite field $\mathbb{F}_q $ where $q=p^N $ is a prime-power of a large prime number $p $. If we call an element of the prime field $\mathbb{F}_p… Read more »

  • featured

    tori & crypto : Diffie-Hellman or GCHQ?

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    Boris Kunyavskii arXived the paper Algebraic tori – thirty years after dedicated to the 80th anniversary of V. E. Voskresenskii. The goal is to give an overview of results of V. E. Voskresenskii on arithmetic and birational properties of algebraic tori which culminated in his monograph “Algebraic Tori” published in Russian 30 years ago. As… Read more »

  • stories

    daddy wasn’t impressed

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    A first year-first semester course on group theory has its hilarious moments. Whereas they can relate the two other pure math courses (linear algebra and analysis) _somewhat_ to what they’ve learned before, with group theory they appear to enter an entirely new and strange world. So, it is best to give them concrete examples :… Read more »

  • stories

    problema bovinum

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    Suppose for a moment that some librarian at the Bodleian Library announces that (s)he discovered an old encrypted book attributed to Isaac Newton. After a few months of failed attempts, the code is finally cracked and turns out to use a Public Key system based on the product of two gigantic prime numbers, $2^{32582657}-1 $… Read more »

  • featured

    micro-sudoku

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    One cannot fight fashion… Following ones own research interest is a pretty frustrating activity. Not only does it take forever to get a paper refereed but then you have to motivate why you do these things and what their relevance is to other subjects. On the other hand, following fashion seems to be motivation enough… Read more »