Tag: rationality

Langlands versus Connes

Published October 26, 2010 by lieven

This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program.

Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative boundary to the Langlands program (at least for $G L_{1}$ and $G L_{2}$ over the rationals $Q$ ).

Recall that Langlands for $G L_{1}$ over the rationals is the correspondence, given by the Artin reciprocity law, between on the one hand the abelianized absolute Galois group

$G a l (\overset{―}{Q} / Q)^{a b} = G a l (Q (μ_{\infty}) / Q) ≃ {\hat{Z}}^{*}$

and on the other hand the connected components of the idele classes

$A_{Q}^{*} / Q^{*} = R_{+}^{*} \times {\hat{Z}}^{*}$

The locally compact Abelian group of idele classes can be viewed as the nice locus of the horrible quotient space of adele classes $A_{Q} / Q^{*}$ . There is a well-defined map

$A_{Q}^{'} / Q^{*} \to R_{+} (x_{\infty}, x_{2}, x_{3}, \dots) \mapsto | x_{\infty} | \prod | x_{p} |_{p}$

from the subset $A_{Q}^{'}$ consisting of adeles of which almost all terms belong to $Z_{p}^{*}$ . The inverse image of this map over $R_{+}^{*}$ are precisely the idele classes $A_{Q}^{*} / Q^{*}$ . In this way one can view the adele classes as a closure, or ‘compactification’, of the idele classes.

This is somewhat reminiscent of extending the nice action of the modular group on the upper-half plane to its badly behaved action on the boundary as in the Manin-Marcolli cave post.

The topological properties of the fiber over zero, and indeed of the total space of adele classes, are horrible in the sense that the discrete group $Q^{*}$ acts ergodically on it, due to the irrationality of $l o g (p_{1}) / l o g (p_{2})$ for primes $p_{i}$ . All this is explained well (in the semi-local case, that is using $A_{Q}^{'}$ above) in the Connes-Marcolli book (section 2.7).

In much the same spirit as non-free actions of reductive groups on algebraic varieties are best handled using stacks, such ergodic actions are best handled by the tools of noncommutative geometry. That is, one tries to get at the geometry of $A_{Q} / Q^{*}$ by studying an associated non-commutative algebra, the skew-ring extension of the group-ring of the adeles by the action of $Q^{*}$ on it. This algebra is known to be Morita equivalent to the Bost-Connes algebra which is the algebra featuring in Connes’ approach to the Riemann hypothesis.

It shouldn’t thus come as a major surprise that one is able to recover the other side of the Langlands correspondence, that is the Galois group $G a l (Q (μ_{\infty}) / Q)$ , from the Bost-Connes algebra as the symmetries of certain states.

In a similar vein one can read the Connes-Marcolli $G L_{2}$ -system (section 3.7 of their book) as an exploration of the noncommutative closure of the Langlands-space $G L_{2} (A_{Q}) / G L_{2} (Q)$ .

At the moment I’m running a master-seminar noncommutative geometry trying to explain this connection in detail. But, we’re still in the early phases, struggling with the topology of ideles and adeles, reciprocity laws, L-functions and the lot. Still, if someone is interested I might attempt to post some lecture notes here.

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the crypto lattice

Published January 12, 2008 by lieven

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 $R_{n} = R_{F_{p^{n}} / F_{p}}^{1} G_{m}$ corresponds to the permutation lattices $R_{n}^{*} = Z [x] / (x^{n} - 1)$ . The action of the generator $σ$ (the Frobenius) of the Galois group $G a l (F_{p^{n}} / F_{p})$ acts on the lattice by multiplication with $x$ .

An old result of Masuda (1955), using an even older lemma by Speiser (1919), asserts than whenever the character-lattice $T^{*}$ of a torus $T$ is a permutation-lattice, the torus is rational, that is, the function-field
of the torus $F_{p} (T)$ is purely trancendental

$F_{p} (y_{1}, \dots, y_{d}) = F_{p} (T) = (F_{q^{n}} (T^{*}))^{G a l}$

(recall from last time that the field on the right-hand side is the field of fractions of the $G a l$ -invariants of the group-algebra of the free Abelian group $T^{*} = Z \oplus \dots \oplus Z$ where the rank is equal to the dimension $d$ of the torus).

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 $T_{2}$ (LUC-system) and $T_{6}$ (CEILIDH-system) are rational! So, what about the next cryptographic challenges? Are the tori $T_{30}$ , $T_{210}$ etc. also rational varieties?

Recall that as a group, the $F_{p}$ -points of the torus $T_{n}$ , is the subgroup of $F_{p^{n}}^{*}$ corresponding to the most crypto-challenging cyclic subgroup of order $Φ_{n} (p)$ where $Φ_{n} (x)$ is the n-th cyclotomic polynomial. The character-lattice of this crypto-torus $T_{n}$ we call the crypto-lattice and it is

$T_{n}^{*} = Z [x] / (Φ_{n} (x))$

(again the action of the Frobenius is given by multiplication with $x$ ) and hence has rank $ϕ (n)$ , explaining that the torus $T_{n}$ has dimension $ϕ (n)$ and hence that we can at best expect a compression from $n$ -pits to $ϕ (n)$ -pits. Note that the lattice $T_{n}^{*}$ is no longer a permutation lattice, so we cannot use the Masuda-Speiser result to prove rationality of $T_{n}$ .

What have mathematicians proved on $T_{n}$ before it became a hot topic? Well, there is an old conjecture by V. E. Voskresenskii asserting that all $T_{n}$ should be rational! Unfortunately, he could prove this only when $n$ is a prime power. Further, he proved that for all $n$ , the lattice $T_{n}$ is at least stably-rational meaning that it is rational upto adding free parameters, that is

$F_{p} (T_{n}) (z_{1}, \dots, z_{l}) = F_{p} (y_{1}, \dots, y_{d + l})$

which, sadly, is only of cryptographic-use if $l$ is small (see below). A true rationality result on $T_{n}$ was proved by A.A. Klyashko : $T_{n}$ is rational whenever $n = p^{a} . q^{b}$ a product of two prime powers.But then, $30 = 2 \times 3 \times 5$ the first unknown case…

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 $n / ϕ (n)$ , but their method was of little practical use in the $T_{30}$ , for what their method gave was a rational map

$T_{30} \times A_{F_{p}}^{32} \to A_{F_{p}}^{40}$

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

$Φ_{a q} (x) Φ_{a} (x) = Φ_{a} (x^{q})$

On the level of tori this equality implies (via the character-lattices) an ismorphism (with same assumptions)

$T_{a q} (F_{p}) \times T_{a} (F_{p}) ≃ (R_{F_{p^{q}} / F_{p}}^{1} T_{a}) (F_{p}) = T_{a} (F_{p^{q}})$

whenever aq is not divisible by p. Apply this to the special case when $q = 5, a = 6$ then we get

$T_{30} (F_{p}) \times T_{6} (F_{p}) ≃ R_{F_{p^{5}} / F_{p}}^{1} T_{6} (F_{p})$

and because we know that $T_{6}$ is a 2-dimensional rational torus we get, using Weil descent, a rational map

$T_{30} \times A_{F_{p}}^{2} \to A_{F_{p}}^{10}$

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…

tori & crypto : Diffie-Hellman or GCHQ?

Published January 3, 2008 by lieven

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 Ive worked on this stuff a long time ago I glanced through the paper and it contains a nice summary of the work of V.E. Voskresenskii, and later of Jean-Louis Colliot-Thelene, Jean-Jacques Sansuc and David Saltman. To my surprise I also made a guest-appearance and even seem to have a conjecture (??!!). Fortunately the ‘conjecture’ turned out to be correct as was proved by Nicole Lemire and Martin Lorenz. But a much bigger surprise (at least to me) is contained in the final section of the paper where applications of (stable) rationality of certain tori are given to primality testing and public key cryptography!

In [GPS]
the authors propose to use a similar idea of compression for using tori
in an even more recent cryptographic protocol (so-called pairing-based
cryptography). It is interesting to note that the efficiency (compression factor) of the above mentioned cryptosystems heavily depends on
rationality of tori under consideration (more precisely, on an explicit
rational parameterization of the underlying variety). As the tori used
by Rubin and Silverberg are known to be stably rational, the seemingly abstract question on rationality of a given stably rational torus
is moving to the area of applied mathematics. The first challenging
problem here is to obtain an explicit rational parameterization of the
8-dimensional torus $T_{30}$ , deïfined over a finite field k and splitting over
its cyclic extension L of degree 30.

This is a particular case of a problem posed by Voskresenskii [Vo77,
Problem 5.12] 30 years ago. Let us hope that we will not have to wait
another 30 years for answering this question on a degree 30 extension.

That’s all it takes to get me seriously side-tracked… so the last couple of hours I’ve been reading up on this connection between tori and cryptography. I will spend a couple of posts on these beautiful results. The latest seems to be that, while rationality of $T_{30}$ is still unknown, one can use an explicit stable-rationality description of it to get a better bound than the XTR-system (the system corresponding to the torus $T_{6}$ ) which in turn is better than the LUC-system (corresponding to $T_{2}$ ), which is turn is twice as efficient as the Diffie-Hellman key exchange system… So let us start gently with the latter one…

Whitfield Diffie (r.) and Martin Hellman (m.) published in 1976 their public key-exchange system. Take a large prime power $q = p^{N}$ , make it public and consider the finite field $F_{q}$ which is known to have a cyclic group of units $F_{q}^{*}$ of order $q - 1$ . Now, take $g$ to be an element in it of large order (preferable a generator but that isnt necessary) and also make this element public.

Now choose a random integer $a$ (your hidden secret) and compute the element $g^{a} \in F_{q}$ and publicize this element. Suppose someone else published his/her element $g^{b}$ constructed from his/her secret integer $b$ then both you and this other person can compute from the published data and their secret numbers the element (the shared key)

$g^{a b} = (g^{b})^{a} = (g^{a})^{b}$

(because you know $a$ and the published $g^{b}$ and your correspondent knows $b$ and the published $g^{a}$ ) but nobody else can compute it from the public-available data only because discrete logarithms cannot be feasibly computed in the group $F_{q}^{*}$ . Hellman suggests to call this system the Diffie-Hellman-Merkl key-exchange (via this link)

The first researchers to discover and publish the concepts of PKC were Whitfield Diffie and Martin Hellman from Stanford University, and Ralph Merkle from the University of California at Berkeley. As so often happens in the scientific world, the two groups were working independently on the same problem — Diffie and Hellman on public key cryptography and Merkle on public key distribution — when they became aware of each other’s work and realized there was synergy in their approaches. In Hellman’s words: “We each had a key part of the puzzle and while it’s true one of us first said X, and another of us first said Y, and so on, it was the combination and the back and forth between us that allowed the discovery.”

And that was the full story until 1997. In December, 1997, it was revealed that researchers at the GCHQ organization did some work in the early 1970’s in the field of “non-secret encryption”. The people involved are James Ellis, Clifford Cocks and Malcolm Williamson (r.).

Here is a note by Ellis on his recollection of the history of ‘Non-secret encryption” :

Cryptography is a most unusual science. Most professional scientists aim to be the first to publish their work,
because it is through dissemination that the work realises its value. In contrast, the fullest value of cryptography
is realised by minimising the information available to potential adversaries. Thus professional cryptographers
normally work in closed communities to provide sufficient professional interaction to ensure quality while
maintaining secrecy from outsiders. Revelation of these secrets is normally only sanctioned in the interests
of historical accuracy after it has been demonstrated clearly that no further benefit can be obtained from
continued secrecy.
In keeping with this tradition it is now appropriate to tell the story of the invention and development within
CESG of non-secret encryption (NSE) which was our original name for what is now called PKC. The task of writing
this paper has devolved on me because NSE was my idea and I can therefore describe these early developments from
personal experience. No techniques not already public knowledge, or specific applications of NSE will be mentioned…

The once secret notes of Williamson are also available. NON-SECRET ENCRYPTION USING A FINITE FIELD
by M J Williamson, 21 January 1974 and THOUGHTS ON CHEAPER NON-SECRET ENCRYPTION
M J Williamson, 10 August 1976.