On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$

High Speed Under-Sampling Frequency Measurements on FPGA

A Sampling rate is less than Nyquist rate in some applications because of hardware limitations. Consequently, extensive researches have been conducted on frequency detection from sub-sampled signals. Previous studies on under-sampling frequency measurements have mostly discussed under-sampling frequency detection in theory and suggested possible methods for fast under-sampling frequencies detection. This study examined few suggested methods on Field Programmable Gate Array (FPGA) for fast under-sampling frequencies measurement. Implementation of the suggested methods on FPGA has issues that make them improper for fast data processing. This study tastes and discusses different methods for frequency detection including Least Squares (LS), Direct State Space (DSS), Goertzel filter, Sliding DFT, Phase changes of Fast Furrier Transform (FFT), peak amplitude of FFT to conclude which one from these methods are suitable for fast under-sampling frequencies detection on FPGA. Moreover, our proposed approach for sub-sampling detection from real waveform has less complextity than previous approaches from complex waveform

Robust Multidimentional Chinese Remainder Theorem for Integer Vector Reconstruction

The problem of robustly reconstructing an integer vector from its erroneous remainders appears in many applications in the field of multidimensional (MD) signal processing. To address this problem, a robust MD Chinese remainder theorem (CRT) was recently proposed for a special class of moduli, where the remaining integer matrices left-divided by a greatest common left divisor (gcld) of all the moduli are pairwise commutative and coprime. The strict constraint on the moduli limits the usefulness of the robust MD-CRT in practice. In this paper, we investigate the robust MD-CRT for a general set of moduli. We first introduce a necessary and sufficient condition on the difference between paired remainder errors, followed by a simple sufficient condition on the remainder error bound, for the robust MD-CRT for general moduli, where the conditions are associated with (the minimum distances of) these lattices generated by gcld's of paired moduli, and a closed-form reconstruction algorithm is presented. We then generalize the above results of the robust MD-CRT from integer vectors/matrices to real ones. Finally, we validate the robust MD-CRT for general moduli by employing numerical simulations, and apply it to MD sinusoidal frequency estimation based on multiple sub-Nyquist samplers.Comment: 12 pages, 5 figur

TOPICS IN COMPUTATIONAL NUMBER THEORY AND CRYPTANALYSIS - On Simultaneous Chinese Remaindering, Primes, the MiNTRU Assumption, and Functional Encryption

This thesis reports on four independent projects that lie in the intersection of mathematics, computer science, and cryptology: Simultaneous Chinese Remaindering: The classical Chinese Remainder Problem asks to find all integer solutions to a given system of congruences where each congruence is defined by one modulus and one remainder. The Simultaneous Chinese Remainder Problem is a direct generalization of its classical counterpart where for each modulus the single remainder is replaced by a non-empty set of remainders. The solutions of a Simultaneous Chinese Remainder Problem instance are completely defined by a set of minimal positive solutions, called primitive solutions, which are upper bounded by the lowest common multiple of the considered moduli. However, contrary to its classical counterpart, which has at most one primitive solution, the Simultaneous Chinese Remainder Problem may have an exponential number of primitive solutions, so that any general-purpose solving algorithm requires exponential time. Furthermore, through a direct reduction from the 3-SAT problem, we prove first that deciding whether a solution exists is NP-complete, and second that if the existence of solutions is guaranteed, then deciding whether a solution of a particular size exists is also NP-complete. Despite these discouraging results, we studied methods to find the minimal solution to Simultaneous Chinese Remainder Problem instances and we discovered some interesting statistical properties. A Conjecture On Primes In Arithmetic Progressions And Geometric Intervals: Dirichlet’s theorem on primes in arithmetic progressions states that for any positive integer q and any coprime integer a, there are infinitely many primes in the arithmetic progression a + nq (n ∈ N), however, it does not indicate where those primes can be found. Linnik’s theorem predicts that the first such prime p0 can be found in the interval [0;q^L] where L denotes an absolute and explicitly computable constant. Albeit only L = 5 has been proven, it is widely believed that L ≤ 2. We generalize Linnik’s theorem by conjecturing that for any integers q ≥ 2, 1 ≤ a ≤ q − 1 with gcd(q, a) = 1, and t ≥ 1, there exists a prime p such that p ∈ [q^t;q^(t+1)] and p ≡ a mod q. Subsequently, we prove the conjecture for all sufficiently large exponent t, we computationally verify it for all sufficiently small modulus q, and we investigate its relation to other mathematical results such as Carmichael’s totient function conjecture. On The (M)iNTRU Assumption Over Finite Rings: The inhomogeneous NTRU (iNTRU) assumption is a recent computational hardness assumption, which claims that first adding a random low norm error vector to a known gadget vector and then multiplying the result with a secret vector is sufficient to obfuscate the considered secret vector. The matrix inhomogeneous NTRU (MiNTRU) assumption essentially replaces vectors with matrices. Albeit those assumptions strongly remind the well-known learning-with-errors (LWE) assumption, their hardness has not been studied in full detail yet. We provide an elementary analysis of the corresponding decision assumptions and break them in their basis case using an elementary q-ary lattice reduction attack. Concretely, we restrict our study to vectors over finite integer rings, which leads to a problem that we call (M)iNTRU. Starting from a challenge vector, we construct a particular q-ary lattice that contains an unusually short vector whenever the challenge vector follows the (M)iNTRU distribution. Thereby, elementary lattice reduction allows us to distinguish a random challenge vector from a synthetically constructed one. A Conditional Attack Against Functional Encryption Schemes: Functional encryption emerged as an ambitious cryptographic paradigm supporting function evaluations over encrypted data revealing the result in plain. Therein, the result consists either in a valid output or a special error symbol. We develop a conditional selective chosen-plaintext attack against the indistinguishability security notion of functional encryption. Intuitively, indistinguishability in the public-key setting is based on the premise that no adversary can distinguish between the encryptions of two known plaintext messages. As functional encryption allows us to evaluate functions over encrypted messages, the adversary is restricted to evaluations resulting in the same output only. To ensure consistency with other primitives, the decryption procedure of a functional encryption scheme is allowed to fail and output an error. We observe that an adversary may exploit the special role of these errors to craft challenge messages that can be used to win the indistinguishability game. Indeed, the adversary can choose the messages such that their functional evaluation leads to the common error symbol, but their intermediate computation values differ. A formal decomposition of the underlying functionality into a mathematical function and an error trigger reveals this dichotomy. Finally, we outline the impact of this observation on multiple DDH-based inner-product functional encryption schemes when we restrict them to bounded-norm evaluations only

Information-theoretic investigation of multi-unit activity properties under different stimulus conditions in mouse primary visual cortex

Primary visual cortex (V1) is the first cortical processing level receiving topographically mapped inputs from the retina, relayed through thalamus. Electrophysiological studies discovered its important role in early sensory processing particularly in edge detection in single cells. To this end, little is investigated how these activities relate on a population level. Orientation tuning in mouse V1 has long been reported as salt-and pepper organised, lacking apparent structure as was found in e.g. cat or primates. This is a novel synthesis of specially designed in-vivo electrophysiological experiments aiming to make certain information-theoretic data analysis approaches viable. Sophisticated state-of-the-art data analysis techniques are applied to answer questions about stimulus information in mouse V1. Multi-unit electrophysiological experiments were devised, performed and evaluated in the anaesthetised and in left hemisphere V1 of the awake behaving, head-fixed mouse. A detailed laboratory and computational analysis is presented validating the use of Multi-Unit-Activity (MUA) and information-theoretic measures. Our results indicate left forward drifting gratings (moving from the temporal to nasal visual field) elicit consistently highest neuronal responses across cortical layers and columns, challenging the common understanding of random organisation. These directional biasses of MUA were also observable on the population level. In addition to individual multi-unit analyses, population responses in terms of binary word distributions appear more similar between spontaneous activity and responses to natural movies than either/both to moving gratings, suggesting that mouse V1 processes natural scenes differently from sinusoidal drifting gratings. Response pattern distributions for different gratings emerge to be spatially but not orientationally clustered. Further computational analysis suggests population firing rates can partially account for these differences. Electrophysiological experiments in the awake behaving mouse indicate V1 to contain information about behavioural outcome in a GO/NOGO task. This, along with other statistical measures is examined with statistical models such as the population tracking model, which suggest that population interactions are required to explain these observations.Open Acces