Pseudorandom Generators from the Second Fourier Level and Applications to AC0 with Parity Gates

We study entropy flattening: Given a circuit CX implicitly describing an n-bit source X (namely, X is the output of CX on a uniform random input), construct another circuit CY describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have CY evaluate CX altogether Θ(n2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit CY for entropy flattening that repeatedly queries CX as an oracle requires ω(n2) queries.Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [12, 22, 13, 6, 11, 10, 7, 24]. It is also used in reductions between problems complete for statistical zero-knowledge [19, 23, 4, 25]. The Θ(n2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

The relationship between efficient verification and quantum computing is one of the most important and least well-understood questions in the theory of computation. In particular, is there a problem that can be solved efficiently on a quantum computer that cannot be verified? In this thesis we give evidence that BQP ⊄ PH, relating the classes of languages decidable with a quantum computer to a generalization of NP. In so doing we connect a question in pseudorandomness, first studied [BSW03] to the problem of finding an oracle relative to which BQP ⊄ PH. The primary technical challenge is to construct a unitary matrix, realized by an efficient quantum circuit and whose rows are supported on nearly disjoint subsets. Using this matrix and assuming the validity of the aforementioned question in pseudorandomness, we show an instantiation of the Nisan-Wigderson pseudorandom generator that can be broken with quantum computers, but not with the relevant mode of classical computation.

Long period pseudo-random sequence plays an important role in modern information processing systems. Base on residue number system (RNS) and permutation polynomials over finite fields, a pseudorandom sequence generation scheme is proposed in this paper. It extends several short period random sequences to a long period pseudo-random sequence by using RNS. The short period random sequences are generated parallel by the iterations of permutation polynomials over finite fields. Due to the small dynamic range of each iterative calculation, the bit width in hardware implementation is reduced. As a result, we can use full look-up table (LUT) architecture to achieve high-speed sequence output. The methods to find proper permutation polynomials to generate long period sequences and the optimization algorithm of Chinese remainder theorem (CRT) mapping are also proposed in this paper. The period of generated pseudorandom sequence can exceed 2100 easily based on common used field programmable gate array (FPGA) chips. Meanwhile, this scheme has extensive freedom in choosing permutation polynomials. For example, 10905 permutation polynomials meet the long period requirement over the finite field F q with q ≢ 1(mod 3) and q ⩽ 503. The hardware implementation architecture is simple and multiplier free. Using Xilinx XC7020 FPGA chip, we implement a sequence generator with the period over 250, which only costs 20 18kb-BRAMs (block RAM) and a small amount of logics. And the speed can reach 449.236 Mbps. The National Institute of Standards and Technology (NIST) test results show that the sequence has good random properties.

Cryptographic Primitives that Resist Backdooring and Subversion

This article explores randomness and how general purpose computers generate it. It also discusses the security requirements that a pseudo-random number generator must satisfy for use in cryptographic applications. And also special attention is paid to the known and new methods of testing random bit sequences. Analyzing the effectiveness of pseudo-random sequence generators is a pressing problem for smart cards when using more advanced methods of encryption and information protection. The available methods show low flexibility and versatility in the means of finding hidden patterns in data. To solve this problem, it is proposed to use algorithms based on multivariate statistics. These algorithms combine all the advantages of statistical methods and are the only alternative for analyzing short and medium-length sequences. The paper considers the scheme of operation of pseudo-random number generators in limited devices. The main requirements for modern smart cards are highlighted. A criterion for checking the randomness of bit sequences of small length (up to 100 bits) is proposed. This approach is appropriate for testing a lightweight pseudo-random number generator in devices with certain resource constraints. The paper presents the compatible distributions of the number of 2-strings and the number of 3-strings of a fixed form of a random bit sequence, which make it possible to carry out a statistical analysis of local sections of this sequence. A possible application of the obtained formulas can be to test the hypothesis of the randomness of the arrangement of zeros and ones in a (0, 1) -sequence of finite length. Research has shown that even with limited resources and a limited entropy environment like a smart card, good quality pseudo-random sequences can be created that can satisfy all the requirements for pseudo-random number generators, even those used for general purpose computers. In the work, the set of statistical tests was expanded to include other tests that are not included in the statistical set of NIST tests, and to analyze the work of the proposed algorithms. The paper presents algorithms for testing a pseudo-random sequence using multivariate statistics to illustrate their possible application in a smart card environment.

Pseudorandom generator as the heart of security system is a potential target by an adversary to weaken or derail the system performance. One possible active attack on a pseudorandom generator is an insertion attack on a random sequence generated by the pseudo random number generator, in order to reduce or even omit the randomness property of the output sequence. In this paper, authors tried to measure the effect of the insertion attack on stream cipher algorithms, Dragon and Rabbit. The insertion attack performed in five levels of insertion block (32-bit, 64-bit, 128-bit, 256-bit and 512-bit), by inserting 1 bit-through 3-bits in a random manner. The bits inserted are in random form and extreme form. To measure the insertion attack effects, the sequences before and after insertion attack are evaluated by performing statistical distance tests and entropy measures, under the advantage parameter e = 0.0001. From the statistical distance tests, it is found that the insertion attack with random bits does not effect the randomness property of Dragon and Rabbit under advantage of e = 0.0001, but it does under e = 0.00001 by indication that more than 60% the value of maximum statistical distance exceeded 0.0001. Meanwhile, from the entropy difference tests we found that more than 20% the maximum entropy difference has already exceeded the value e = 0.0001, which proved that the insertion attack using random bits does affect the randomness property of Dragon and Rabbit extreme under advantage e = 0.0001. On the other hand, it was shown that the insertion attack using extreme bits affects the randomness property of Dragon and Rabbit significantly under advantage of e = 0.0001 where more than 95% the test values exceeded the advantage value. This results proved that the insertion attack is a potential treats that should not be disobeyed.

Image steganography conceals secret information in a cover image .It is used for secure data transfer. Various LSB (least significant bits) approaches have been used over the years in steganography. This paper proposes a novel technique to improve the conventional LSB technique for image steganography by using pseudo random number generation using Henon chaotic map. The random numbers are used to encrypt the hide image which is embedded in the cover picture. This encryption using pseudo random generator provides sufficient security to the payload as the same set of random numbers cannot be regenerated without knowing the exact random generator function and thus the secret data cannot be retrieved easily. The proposed technique has been tested successfully on various .jpg files and corresponding PSNR values have been calculated.

We give an explicit construction of pseudorandom generators against low degree polynomials over finite fields. We show that the sum of 2d small-biased generators with error e2O(d) is a pseudorandom generator against degree d polynomials with error e. This gives a generator with seed length 2O(d) log(n/e). Our construction follows the recent breakthrough result of Bogadnov and Viola. Their work shows that the sum of d small-biased generators is a pseudo-random generator against degree d polynomials, assuming the Inverse Gowers Conjecture. However, this conjecture is only proven for d=2,3. The main advantage of our work is that it does not rely on any unproven conjectures.

A function f: Fn2 → {0, 1} is odd-cycle-free if there are no x1,..., xk e Fn2 with k an odd integer such that f(x1) =... = f(xk) = 1 and x1 +... + xk = 0. We show that one can distinguish odd-cycle-free functions from those e-far from being odd-cycle-free by making poly(1/e) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.The first problem we study is directly reducing testing linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have been shown previously (Kral-Serra-Vena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linear-invariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question still remains open.

Random numbers are a fundamental resource in science and engineering, and are also widely used in many fields, such as simulation, cryptography, lotteries, fundamental science. There are algorithmically generated numbers that look like random numbers but are not truly random. These pseudo random numbers are generated from a computer, using deterministic algorithms. Because, we cannot use pseudo random generators in situations, where true randomness is necessary, we use true random number generators. In this case, we use unpredictable random events as a random source. Quantum random number generators (QRNG) output true random numbers based on inherent randomness in quantum measurements. Our goal is to generate fast random numbers at a lower cost. At the same time, a high level of randomness is essential. We present the improved novel quantum random number generator, which is based the on time of arrival QRNG. It is rather efficient, as it uses the simple version of the detectors with rather few requirements. The novel QRNG produces more than one random bit per each photon detection.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

Simulation is an interesting alternative to solve Markovian models. However, when compared to analytical and numerical solutions it suffers from a lack of precision in the results due to the very nature of simulation, which is the choice of samples through pseudorandom generation. This paper proposes a different way to simulate Markovian models by using a Bootstrap-based statistical method to minimize the effect of sample choices. The effectiveness of the proposed method, called Bootstrap simulation, is compared to the numerical solution results for a set of examples described using Stochastic Automata Networks modeling formalism.

Security and privacy problems in communication systems and social media platforms where digital image/video are shared almost always, have attracted researchers interest on information security. Starting from this point of view, a novel one-dimensional chaotic maps based pseudo random bit generator is proposed to make a significant contribution to the literature about protection of personal data in any network. Electronic circuit realizations of Logistic and Tent maps as entropy source are designed on Orcad-Pspice environment and state variables are inputted to novel post-processor algorithm. The rest of main blocks of proposed pseudo random bit generator design are built up fixed-point binary conversion algorithm, XOR processor and H function post-processor. The generated pseudo random bit series are tested by using NIST 800.22 statistical test suite and applied to color image encryption in order to show the effectiveness of proposed design. Cryptanalysis processes such as histogram, NPCR-UACI and correlation analysis are demonstrated. Analysis results show that the proposed design can be used successfully in many secure communication and media transmission applications.

We consider the problem of constructing hitting set generators for sparse multivariate polynomials over any finite fields. Hitting set generators, just as pseudorandom generators, play a fundamental role in the study of derandomization. Pseudorandom generators with a logarithmic seed length are only known for polynomials of a constant degree \cite{Lov09, Vio09}. On the other hand, hitting set generators with a logarithmic seed length are known for polynomials of larger degrees, but only over fields which are much larger than the degrees \cite{KS01, Bog05}. Our main result is the construction of a hitting set generator with a seed length of O(\log s), which works for s-term polynomials of any degrees over any finite fields of constant size. This gives the first optimal hitting set generator which allows the fields to be smaller than the degrees of polynomials. For larger fields, of non-constant size, we provide another hitting set generator with a seed length of O(\log (sd)), which works for s-term polynomials of any degree d, as long as d is slightly smaller than the field size.

Transmission of images on the network is considered insecure which has security-related issues. In this paper, to provide security to digital images, an encryption system that uses four chaotic maps for binary key sequence generation is proposed. The system consists of stages namely, confusion, generation of random binary chaotic key sequence, pseudo-random number generation (RN), and diffusion. Keys for encryption are chosen randomly based on a pseudo random generator from the selected chaotic maps by the linear feedback shift register (LFSR). The algorithm achieves good results in terms of NPCR, UACI, and entropy values. The developed cryptosystem resists differential attacks, is sensitive to minor alterations in the keys, and has a large key space.