Virtual FPLLL Days 6 aka Bounded Distance Development

The sixth FPLLL Days will be held on 19 and 20 November 2020. For obvious reasons they will be held online. (Who knows, we might end up liking the format enough to do more of these online in a post-COVID world, too).

As with previous incarnations of FPLLL Days, everyone who wishes to contribute to open-source lattice-reduction software is welcome to attend. In particular, you do not have to work on FPLLL or any of its sibling projects like FPyLLL or G6K. Work on whatever advances the state of the art or seems useful to you personally. That said, we do have the ambition to suggest projects from the universe of FPLLL and I personally hope that some people will dig in with me to do the sort of plumbing that keeps projects like these running. To give you an idea of what people worked on in the past, you can find the list of project of the previous FPLLL Days on the wiki and a report on the PROMETHEUS blog.

Format is yet to be determined. We are going to coordinate using Zulip. I think it would also be useful to have a brief conference call, roll call style, on the first day to break the ice and to make it easier for people to reach out to each other for help during the event. This is why we’re asking people to indicate their timezone on the Wiki. If you have ideas for making this event a success, please let us know.

PS: For the record: Joe threatened a riot if I didn’t call it “Bounded Distance Development”, so credit goes to him for the name.

Revisiting the Expected Cost of Solving uSVP and Applications to LWE

Our — together with Florian Göpfert, Fernando Virdia and Thomas Wunderer — paper Revisiting the Expected Cost of Solving uSVP and Applications to LWE is now available on ePrint. Here’s the abstract:

Reducing the Learning with Errors problem (LWE) to the Unique-SVP problem and then applying lattice reduction is a commonly relied-upon strategy for estimating the cost of solving LWE-based constructions. In the literature, two different conditions are formulated under which this strategy is successful. One, widely used, going back to Gama & Nguyen’s work on predicting lattice reduction (Eurocrypt 2008) and the other recently outlined by Alkim et al. (USENIX 2016). Since these two estimates predict significantly different costs for solving LWE parameter sets from the literature, we revisit the Unique-SVP strategy. We present empirical evidence from lattice-reduction experiments exhibiting a behaviour in line with the latter estimate. However, we also observe that in some situations lattice-reduction behaves somewhat better than expected from Alkim et al.’s work and explain this behaviour under standard assumptions. Finally, we show that the security estimates of some LWE-based constructions from the literature need to be revised and give refined expected solving costs.

Our work is essentially concerned with spelling out in more detail and experimentally verifying a prediction made in the New Hope paper on when lattice reduction successfully recovers an unusually short vector.

Denoting by the unusually short vector in some lattice of dimension (say, derived from some LWE instance using Kannan’s embedding), the block size used for the BKZ algorithm and the root-Hermite factor for , then the New Hope paper predicts that can be found if

under the assumption that the Geometric Series Assumption holds (until a projection of the unusually short vector is found).

The rationale is that this condition ensures that the projection of orthogonally to the first (Gram-Schmidt) vectors (denoted as ) is shorter than the expectation for the -th Gram-Schmidt vector under the GSA and thus would be found by the SVP oracle when called on the last block of size . Hence, for any satisfying the above inequality, the actual behaviour would deviate from that predicted by the GSA. Finally, the argument can be completed by appealing to the intuition that a deviation from expected behaviour on random instances — such as the GSA — leads to a revelation of the underlying structural, secret information. In any event, such a deviation would already solve Decision-LWE.

In our work, we spell out this argument in more detail (e.g. how is recovered from ) and throw 23k core hours at the problem of checking if the predicted behaviour, e.g.

matches the observed behaviour, e.g.

Just like for the above plots, the general answer is a clear “yes”.

Pretty Pictures or GTFO

I forgot the most important bit. The behaviour of the BKZ algorithm on uSVP(-BDD) instances can be observed in this video.

You can observe the basis approaching the GSA until the SVP oracle finds the unusually short vector . From , is then immediately recovered using size reduction. The grey area is the currently worked on block. The notation in the legend isn’t consistent with the plots above or even internally ( v ), but the general idea should still be apparent. In case you’re wondering about the erratic behaviour of the tails (which occasionally goes all over the place), this is due to a bug in fpylll which has recently been fixed.

Adventures in Cython Templating

Fpylll makes heavy use to Cython to expose Fplll’s functionality to Python. Fplll, in turn, makes use of C++ templates. For example, double, long double, dd_real (http://crd.lbl.gov/~dhbailey/mpdist/) and mpfr_t (http://www.mpfr.org/) are supported as floating point types. While Cython supports C++ templates, we still have to generate code for all possible instantiations of the C++ templates for Python to use/call. The way I implemented these bindings is showing its limitations. For example, here’s how attribute access to the dimension of the Gram-Schmidt object looks like:

    @property
    def d(self):
        """
        Number of rows of ``B`` (dimension of the lattice).

        >>> from fpylll import IntegerMatrix, GSO, set_precision
        >>> A = IntegerMatrix(11, 11)
        >>> M = GSO.Mat(A)
        >>> M.d
        11

        """
        if self._type == gso_mpz_d:
            return self._core.mpz_d.d
        IF HAVE_LONG_DOUBLE:
            if self._type == gso_mpz_ld:
                return self._core.mpz_ld.d
        if self._type == gso_mpz_dpe:
            return self._core.mpz_dpe.d
        IF HAVE_QD:
            if self._type == gso_mpz_dd:
                return self._core.mpz_dd.d
            if self._type == gso_mpz_qd:
                return self._core.mpz_qd.d
        if self._type == gso_mpz_mpfr:
            return self._core.mpz_mpfr.d

        if self._type == gso_long_d:
            return self._core.long_d.d
        IF HAVE_LONG_DOUBLE:
            if self._type == gso_long_ld:
                return self._core.long_ld.d
        if self._type == gso_long_dpe:
            return self._core.long_dpe.d
        IF HAVE_QD:
            if self._type == gso_long_dd:
                return self._core.long_dd.d
            if self._type == gso_long_qd:
                return self._core.long_qd.d
        if self._type == gso_long_mpfr:
            return self._core.long_mpfr.d

        raise RuntimeError("MatGSO object '%s' has no core."%self)

In the code above uppercase IF and ELSE are compile-time conditionals, lowercase if and else are run-time checks. If we wanted to add Z_NR<double> to the list of supported integer types (yep, Fplll supports that), then the above Python approximation of a switch/case statement would grow by a factor 50%. The same would have to be repeated for every member function or attribute. There must be a more better way.

Fplll Days 3: July 6 – 14, Amsterdam

We’ll have an fplll coding sprint aka “FPLLL Days” in July. This time around, we plan a slightly modified format compared to previous instances. That is, in order to encourage new developers to get involved, we plan to have a 2 day tutorial session (shorter or longer depending on participants/interest) before the start of FPLLL Days proper.

fplll 5.1 and fpylll 0.2.4dev

New versions of fplll and fpylll were released today. I’ve reproduced release notes below for greater visibility. The biggest user-visible changes for fplll are probably that

• CVP enumeration is not experimental any more,
• support for external enumeration libraries (go write that GPU implementation of enumeration) was added and
• support for OSX was greatly improved.

On the fpylll side, the biggest user-visible changes are probably various API updates and a much nicer strategy/framework for gathering statistics about BKZ.

The next version of fplll will contain support for LLL reduction on Gram matrices.