Simultaneous Communication Protocols with Quantum and Classical Messages

We study the simultaneous message passing (SMP) model of communication complexity, for the case where one party is quantum and the other is classical. We show that in an SMP protocol that computes some function with the first party sending q qubits and the second sending c classical bits, the quantum message can be replaced by a randomized message of O(qc) classical bits, as well as by a deterministic message of O(q c log q) classical bits. Our proofs rely heavily on earlier results due to Scott Aaronson. In particular, our results imply that quantum-classical protocols need to send Omega(sqrt{n/log n}) bits/qubits to compute Equality on n-bit strings, and hence are not significantly better than classical-classical protocols (and are much worse than quantum-quantum protocols such as quantum fingerprinting). This essentially answers a recent question of Wim van Dam. Our results also imply, more generally, that there are no superpolynomial separations between quantum-classical and classical-classical SMP protocols for functional problems. This contrasts with the situation for relational problems, where exponential gaps between quantum-classical and classical-classical SMP protocols are known. We show that this surprising situation cannot arise in purely classical models: there, an exponential separation for a relational problem can be converted into an exponential separation for a functional problem.Comment: 11 pages LaTeX. 2nd version: author added and some changes to the writin

Tight hardness of the non-commutative Grothendieck problem

We prove that for any $\varepsilon > 0$ it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor $1/2 + \varepsilon$, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC'13). Our proof uses an embedding of $\ell_2$ into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates

Tight hardness of the non-commutative Grothendieck problem

We prove that for any ε > 0 it is NP-hard to approximate the non-commutative Grothendieck problem to within a factor 1=2+ε, which matches the approximation ratio of the algorithm of Naor, Regev, and Vidick (STOC’13). Our proof uses an embedding of ℓ2 into the space of matrices endowed with the trace norm with the property that the image of standard basis vectors is longer than that of unit vectors with no large coordinates. We also observe that one can obtain a tight NP-hardness result for the commutative Little Grothendieck problem; previously, this was only known based on the Unique Games Conjecture (Khot and Naor, Mathematika 2009)

Recovering short generators of principal ideals in cyclotomic rings

Abstract: A handful of recent cryptographic proposals rely on the conjectured hardness of the following problem in the ring of integers of a cyclotomic number field: given a basis of a principal ideal that is guaranteed to have a ``rather short'' generator, find such a generator. Recently, Bernstein and Campbell-Groves-Shepherd sketched potential attacks against this problem; most notably, the latter authors claimed a \emph{polynomial-time quantum} algorithm. (Alternatively, replacing the quantum component with an algorithm of Biasse and Fieker would yield a \emph{classical subexponential-time} algorithm.) A key claim of Campbell \etal\ is that one step of their algorithm---namely, decoding the \emph{log-unit} lattice of the ring to recover a short generator from an arbitrary one---is classically efficient (whereas the standard approach on general lattices takes exponential time). However, very few convincing details were provided to substantiate this claim. In this work, we clarify the situation by giving a rigorous proof that the log-unit lattice is indeed efficiently decodable, for any cyclotomic of prime-power index. Combining this with the quantum algorithm from a recent work of Biasse and Song confirms the main claim of Campbell \etal\xspace Our proof consists of two main technical contributions: the first is a geometrical analysis, using tools from analytic number theory, of the standard generators of the group of cyclotomic units. The second shows that for a wide class of typical distributions of the short generator, a standard lattice-decoding algorithm can recover it, given any generator. By extending our geometrical analysis, as a second main contribution we obtain an efficient algorithm that, given any generator of a principal ideal (in a prime-power cyclotomic), finds a 2^O~(n^1/2) -approximate shortest vector in the ideal. Combining this with the result of Biasse and Song yields a quantum polynomial-time algorithm for the 2^O~(n^1/2)-approximate Shortest Vector Problem on principal ideal lattices

On Public Key Encryption from Noisy Codewords

Several well-known public key encryption schemes, including those of Alekhnovich (FOCS 2003), Regev (STOC 2005), and Gentry, Peikert and Vaikuntanathan (STOC 2008), rely on the conjectured intractability of inverting noisy linear encodings. These schemes are limited in that they either require the underlying field to grow with the security parameter, or alternatively they can work over the binary field but have a low noise entropy that gives rise to sub-exponential attacks. Motivated by the goal of efficient public key cryptography, we study the possibility of obtaining improved security over the binary field by using different noise distributions. Inspired by an abstract encryption scheme of Micciancio (PKC 2010), we consider an abstract encryption scheme that unifies all the three schemes mentioned above and allows for arbitrary choices of the underlying field and noise distributions. Our main result establishes an unexpected connection between the power of such encryption schemes and additive combinatorics. Concretely, we show that under the ``approximate duality conjecture from additive combinatorics (Ben-Sasson and Zewi, STOC 2011), every instance of the abstract encryption scheme over the binary field can be attacked in time $2^{O(\sqrt{n})}$, where $n$ is the maximum of the ciphertext size and the public key size (and where the latter excludes public randomness used for specifying the code). On the flip side, counter examples to the above conjecture (if false) may lead to candidate public key encryption schemes with improved security guarantees. We also show, using a simple argument that relies on agnostic learning of parities (Kalai, Mansour and Verbin, STOC 2008), that any such encryption scheme can be {\em unconditionally} attacked in time $2^{O(n/\log n)}$, where $n$ is the ciphertext size. Combining this attack with the security proof of Regev\u27s cryptosystem, we immediately obtain an algorithm that solves the {\em learning parity with noise (LPN)} problem in time $2^{O(n/\log \log n)}$ using only $n^{1+\epsilon}$ samples, reproducing the result of Lyubashevsky (Random 2005) in a conceptually different way. Finally, we study the possibility of instantiating the abstract encryption scheme over constant-size rings to yield encryption schemes with no decryption error. We show that over the binary field decryption errors are inherent. On the positive side, building on the construction of matching vector families (Grolmusz, Combinatorica 2000; Efremenko, STOC 2009; Dvir, Gopalan and Yekhanin, FOCS 2010), we suggest plausible candidates for secure instances of the framework over constant-size rings that can offer perfectly correct decryption

Richard Stallmanin tekijänoikeuskritiikin yhteiskuntafilosofinen analyysi

Käsittelen työssäni Richard Stallmanin vapaiden ohjelmistojen filosofiaa ja siihen sisältyvää tekijänoikeuskritiikkiä. Analyysin taustana ovat tekijänoikeuden perustasta esitetyt teoriat ja liberalistinen yhteiskuntafilosofia sekä kommunitaristien siihen kohdistama kritiikki. Tekijänoikeuden oikeutusta koskevat kolme keskeistä teoriaa ovat työteoria, persoonallisuusteoria ja utilitaristinen teoria. John Locken luonnonoikeusfilosofiaan nojaavassa työteoriassa katsotaan, että jostakin resurssista tulee tekijänsä omaisuutta, kun siihen sekoittuu tekijän työpanos. Persoonallisuusteoria perustuu Immanuel Kantin ja G. W. F. Hegelin näkemykseen omaisuudesta osana henkilön persoonallisuutta, jonka vapautta on suojattava. Utilitaristinen teoria katsoo tekijänoikeuden olevan perusteltua siihen liittyvän kannustimen vuoksi: enemmän teoksia syntyy, mikäli niiden tekijöille annetaan teokseen erityinen yksinoikeus. Stallman kyseenalaistaa tietokoneohjelmiin saatavan tekijänoikeuden oikeutuksen viittaamalla käyttäjien oikeuksiin. Stallmanin mukaan tietokoneohjelman käyttäjälle kuuluvat neljä vapautta: 1) vapaus käyttää ohjelmaa rajoituksetta, 2) vapaus muuttaa ohjelmaa tarpeiden mukaan, 3) vapaus levittää ohjelmaa muille, 4) vapaus parannella ohjelmaa ja jakaa se muiden kanssa. Näitä vapauksia Stallman perustelee viittaamalla yhteisöön. Hän ottaa moraalifilosofiseksi lähtökohdaksi vasta­vuoroisuuden periaatteen, jonka mukaan yhteisön jäsenten tulee auttaa toinen toisiaan esimerkiksi jakamalla tietokoneohjelmia. Tekijänoikeuden asettamat rajoitukset ovat esteitä tämäntyyppiselle solidaarisuudelle ja yhteisössä sekä yhteiskunnassa vallitsevalle avunannon ilmapiirille. Yhteiskuntafilosofiselta perusluonteeltaan Stallmanin filosofia edustaa liberalismia. Väite käyttäjien yksilöllisistä oikeuksista perustuu liberalistiseen ihmiskuvaan. Se on myös universalistinen: Stallman pitää käyttäjien oikeuksia joko luonnollisina oikeuksina tai ihmisoikeuksina. Toisaalta teoriassa on keskeisellä sijalla ajatus yhteisön eheydestä ja yhteisöllisen hyvän edistämisestä, mikä tuo siihen kommunitaristisia piirteitä. Toteuttamalla ohjelmistojen vapauden elämäntapaa yhteisö toteuttaa tietynlaista hyvän elämän politiikkaa. Teoriaa ei kuitenkaan voi kokonaisuudessaan pitää kommunitaristisena siihen sisältyvän vahvan universalismin vuoksi. Vapautta korostavan luon­teensa ja negatiivisen omaisuuskäsityksensä vuoksi osuvampi luonnehdinta Stallmanin filosofialle on digitaalinen anarkismi. Asiasanat: tekijänoikeus, liberalismi, kommunitarismi, Richard Stallman, vapaat ohjelmistot, avoin lähdekoodi, tietokoneohjelmisto