Security of 2t-root identification and signatures, Proceedings CRYPTO'96, Springer LNCS 1109, (1996), pp. 143{156 page 148, section 3, line 5 of the proof of Theorem 3. Correction.

Korrektur zu: C.P. Schnorr: Security of 2t-Root Identification and Signatures, Proceedings CRYPTO'96, Springer LNCS 1109, (1996), pp. 143-156 page 148, section 3, line 5 of the proof of Theorem 3. Die Korrektur wurde präsentiert als: "Factoring N via proper 2 t-Roots of 1 mod N" at Eurocrypt '97 rump session

Satisfiability is quasilinear complete in NQL

Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.2

New practical algorithms for the approximate shortest lattice vector

We present a practical algorithm that given an LLL-reduced lattice basis of dimension n, runs in time O(n3(k=6)k=4+n4) and approximates the length of the shortest, non-zero lattice vector to within a factor (k=6)n=(2k). This result is based on reasonable heuristics. Compared to previous practical algorithms the new method reduces the proven approximation factor achievable in a given time to less than its fourthth root. We also present a sieve algorithm inspired by Ajtai, Kumar, Sivakumar [AKS01]

The process complexity and effective random tests

We propose a variant of the Kolmogorov concept of complexity which yields a common theory of finite and infinite random sequences. The process complexity does not oscillate. We establish some concepts of effective tests which are proved to be equivalent

Security of almost ALL discrete log bits

Let G be a finite cyclic group with generator \alpha and with an encoding so that multiplication is computable in polynomial time. We study the security of bits of the discrete log x when given \exp_{\alpha}(x), assuming that the exponentiation function \exp_{\alpha}(x) = \alpha^x is one-way. We reduce he general problem to the case that G has odd order q. If G has odd order q the security of the least-significant bits of x and of the most significant bits of the rational number \frac{x}{q} \in [0,1) follows from the work of Peralta [P85] and Long and Wigderson [LW88]. We generalize these bits and study the security of consecutive shift bits lsb(2^{-i}x mod q) for i=k+1,...,k+j. When we restrict \exp_{\alpha} to arguments x such that some sequence of j consecutive shift bits of x is constant (i.e., not depending on x) we call it a 2^{-j}-fraction of \exp_{\alpha}. For groups of odd group order q we show that every two 2^{-j}-fractions of \exp_{\alpha} are equally one-way by a polynomial time transformation: Either they are all one-way or none of them. Our key theorem shows that arbitrary j consecutive shift bits of x are simultaneously secure when given \exp_{\alpha}(x) iff the 2^{-j}-fractions of \exp_{\alpha} are one-way. In particular this applies to the j least-significant bits of x and to the j most-significant bits of \frac{x}{q} \in [0,1). For one-way \exp_{\alpha} the individual bits of x are secure when given \exp_{\alpha}(x) by the method of Hastad, N\"aslund [HN98]. For groups of even order 2^{s}q we show that the j least-significant bits of \lfloor x/2^s\rfloor, as well as the j most-significant bits of \frac{x}{q} \in [0,1), are simultaneously secure iff the 2^{-j}-fractions of \exp_{\alpha'} are one-way for \alpha' := \alpha^{2^s}. We use and extend the models of generic algorithms of Nechaev (1994) and Shoup (1997). We determine the generic complexity of inverting fractions of \exp_{\alpha} for the case that \alpha has prime order q. As a consequence, arbitrary segments of (1-\varepsilon)\lg q consecutive shift bits of random x are for constant \varepsilon >0 simultaneously secure against generic attacks. Every generic algorithm using $t$ generic steps (group operations) for distinguishing bit strings of j consecutive shift bits of x from random bit strings has at most advantage O((\lg q) j\sqrt{t} (2^j/q)^{\frac14})

Assessing hepatic metabolic changes during progressive colonization of germ-free mouse by 1H NMR spectroscopy

It is well known that gut bacteria contribute significantly to the host homeostasis, providing a range of benefits such as immune protection and vitamin synthesis. They also supply the host with a considerable amount of nutrients, making this ecosystem an essential metabolic organ. In the context of increasing evidence of the link between the gut flora and the metabolic syndrome, understanding the metabolic interaction between the host and its gut microbiota is becoming an important challenge of modern biology.1-4 Colonization (also referred to as normalization process) designates the establishment of micro-organisms in a former germ-free animal. While it is a natural process occurring at birth, it is also used in adult germ-free animals to control the gut floral ecosystem and further determine its impact on the host metabolism. A common procedure to control the colonization process is to use the gavage method with a single or a mixture of micro-organisms. This method results in a very quick colonization and presents the disadvantage of being extremely stressful5. It is therefore useful to minimize the stress and to obtain a slower colonization process to observe gradually the impact of bacterial establishment on the host metabolism. In this manuscript, we describe a procedure to assess the modification of hepatic metabolism during a gradual colonization process using a non-destructive metabolic profiling technique. We propose to monitor gut microbial colonization by assessing the gut microbial metabolic activity reflected by the urinary excretion of microbial co-metabolites by 1H NMR-based metabolic profiling. This allows an appreciation of the stability of gut microbial activity beyond the stable establishment of the gut microbial ecosystem usually assessed by monitoring fecal bacteria by DGGE (denaturing gradient gel electrophoresis).6 The colonization takes place in a conventional open environment and is initiated by a dirty litter soiled by conventional animals, which will serve as controls. Rodents being coprophagous animals, this ensures a homogenous colonization as previously described.7 Hepatic metabolic profiling is measured directly from an intact liver biopsy using 1H High Resolution Magic Angle Spinning NMR spectroscopy. This semi-quantitative technique offers a quick way to assess, without damaging the cell structure, the major metabolites such as triglycerides, glucose and glycogen in order to further estimate the complex interaction between the colonization process and the hepatic metabolism7-10. This method can also be applied to any tissue biopsy11,12