A Commitment Scheme with Output Locality-3 Fit for the IoT Device

Low output locality is a property of functions, in which every output bit depends on a small number of input bits. In IoT devices with only a fragile CPU, it is important for many IoT devices to cooperate to execute a single function. In such IoT’s collaborative work, a feature of low output locality is very useful. This is why it is desirable to reconstruct cryptographic primitives with low output locality. However, until now, commitment with a constant low output locality has been constructed by using strong randomness extractors from a nonconstant-output-locality collision-resistant hash function. In this paper, we construct a commitment scheme with output locality-3 from a constant-output-locality collision-resistant hash function for the first time. We prove the computational hiding property of our commitment by the decisional M , δ -bSVP assumption and prove the computational binding property by the M , δ -bSVP assumption, respectively. Furthermore, we prove that the M , δ -bSVP assumption can be reduced to the decisional M , δ -bSVP assumption. We also give a parameter suggestion for our commitment scheme with the 128 bit security.

Sunflower Theorems in Monotone Circuit Complexity

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a "small" circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erd\H{o}s and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called "robust sunflowers", to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossman's bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain a significant progress on the sunflower conjecture. In this work, we will show how the recent progress in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. Our exposition is self-contained. These results were obtained in a collaboration with Benjamin Rossman and Mrinal Kumar.

Two-sources randomness extractors in finite fields and in elliptic curves

We propose two-sources randomness extractors over finite fields and on elliptic curves that can extract from two sources of information without consideration of other assumptions that the starting algorithmic assumptions with a competitive level of security. These functions have several applications. We propose here a description of a version of a Diffie-Hellman key exchange protocol and key extraction. Nous proposons des extracteurs d'aléas 2-sources sur les corps finis et sur les courbes elliptiques capables d'extraire à partir de plusieurs sources d'informations sans considération d'autres hypothèses que les hypothèses algorithmiques de départ avec un niveau de sécurité compétitif. Ces fonctions possèdent plusieurs applications. Nous proposons ici une version du protocole d'échange de clé Diffie-Hellman incluant la phase d'extraction.

NON-MALLEABLE RANDOMNESS EXTRACTORS

Ekstraktory losowości należą do jednego z głównych nurtów badań współ- czesnej kryptografii teoretycznej. Zadaniem tych deterministycznych funkcji jest przekształcenie źródeł słabej losowości w takie, których rozkład jest bliski rozkładowi jednostajnemu. W pracy przedstawiona jest teorioliczbowa konstrukcja ekstraktora o pewnych szczególnych własnościach – ekstraktora niekowalnego. Wynik ten stanowi udoskonalenie warunkowego rezultatu Y. Dodisa i in. opublikowanego na prestiżowej konferencji FOCS’11.