The conditional result we saw last time does much more than that. If we have a language in the uniform class E not in the non-uniform class SIZE(s(n)) then a PRG exists. I.e., there is a way to systematically replace random bits with pseudorandom bits such that no BPP algorithm will (substantially) notice the difference. Thus, not only it shows that all languages in BPP are in fact in P (or wha...

In the first part, we introduce randomized algorithms as a new notion of efficient algorithms for decision problems. We classify randomized algorithms according to their error probabilities, and define appropriate complexity classes. (RP, coRP, ZPP, BPP, PP). We discuss which classes are realistic proposals for design of probabilistic algorithms. We cover the implementation of randomized algori...

Pour des données axiales, le test d’uniformité plus classique sur la sphère unité de Rp est Bingham (Ann. Statist. 2 (1974) 1201–1225). Dans cet article, nous étudions puissance ce dans scénarios asymptotiques grande dimension pour lesquels p=pn diverge vers l’infini avec taille d’échantillon n. Nous considérons une classe semiparamétrique contre-hypothèses incluant les distributions Watson et ...

the non-uniformity can almost be removed with a local change of variables from c(s, t) to b(x, t). So, in numerical computations non-uniformity need not imply any reduction from optimal accuracy. A computational grid that was uniformly spaced with respect to s becomes non-uniform with respect to the intrinsic xcoordinate. Suitable non-uniform grid, optimal and near-optimal compact finitediffere...

We argue that it is unnatural and undesirable to use the non-uniform model of complexity for practice-oriented security reductions in cryptography.

We revisit security proofs for various cryptographic primitives in the auxiliary-input randomoracle model (AI-ROM), in which an attacker A can compute arbitrary S bits of leakage about the random oracle O before attacking the system and then use additional T oracle queries to O during the attack. This model has natural applications in settings where traditional randomoracle proofs are not usefu...