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
An old result of Masuda (1955), using an even older lemma by Speiser (1919), asserts than whenever the character-lattice
of the torus
(recall from last time that the field on the right-hand side is the field of fractions of the
The basic observation made by Rubin and Silverberg was that the known results on crypto-compression could be reformulated in the language of algebraic tori as : the tori
Recall that as a group, the
(again the action of the Frobenius is given by multiplication with
What have mathematicians proved on
which, sadly, is only of cryptographic-use if
At Crypto 2004, Marten van Dijk and David Woodruff were able to use an explicit form of Voskresenskii stable rationality result to get an asymptotic optimal crypto-compression rate of
and the number of added parameters (32) is way too big to be of use.
But then, one can use century-old results on cyclotomic polynomials to get a much better bound, as was shown in the paper Practical cryptography in high dimensional tori by the collective group of all people working (openly) on tori-cryptography. The idea is that whenever q is a prime and a is an integer not divisible by q, then on the level of cyclotomic polynomials we have the identity
On the level of tori this equality implies (via the character-lattices) an ismorphism (with same assumptions)
whenever aq is not divisible by p. Apply this to the special case when
and because we know that
which can be used to get better crypto-compression than the CEILIDH-system!
This concludes what I know of the OPEN state of affairs in tori-cryptography. I’m sure ‘people in hiding’ know a lot more at the moment and, if not, I have a couple of ideas I’d love to check out. So, when I seem to have disappeared, you know what happened…
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