On the time required for group multiplication

Group multiplication of logical elements by networks having limited inputs and unit delay in computing output functio

Relationship of the nutrition of Streptococcus lactis to bacteriophage proliferation

A chemically defined medium made by adding sodium acetate and Tween 80 (a source of oleic acid) to the medium of Niven (1944) permitted the growth of all strains of the lactic group of streptococci which dial not grow on the unsupplemented medium. The addition of sodium acetate and Tween 80 was necessary for the growth of 22 strains of S. cremoris and 9 of 31 strains of S. lactis. Reticulogen, a commercial liver extract, could be substituted in somewhat smaller quantities for sodium acetate and Tween 80 and also permitted rapid growth of one strain of S. cremoris which did not show detectable growth until after 24 hours in the medium supplemented with sodium acetate and Tween 80;Using ammonia formation from arginine and growth at 40°C as the basis for separation, all strains of the lactic group which had been in the laboratory for considerable time were found to be S. cremoris, while all recently isolated strains were found to be S. lactis;With two S. lactis-bacteriophage combinations, multiplication of bacteriophages and organisms were affected similarly by the omission of individual components from the unsupplemented synthetic medium of Niven (1944). Bacteriophage multiplication seems to be closely associated with organism multiplication for these two combinations;When calcium was available in the medium, eight bacteriophage strains tested were found to multiply on their susceptible host cells in a completely synthetic medium which did not permit these bacteriopage strain to proliferate without the addition of CaCl2·2HOH. When calcium was available in the medium and these bacteriophages multiplied, the close relationship between bacteriophage multiplication and organism multiplication seemed evident for these strains also. Calcium was rendered unavailable in a medium containing 0.1 percent CaCl2·2HOH by either autoclaving the entire medium or by increasing the K2HPO4 content above 0.1 percent;A close relationship seems to exist between the nutrition or organisms of the S. lactis group and multiplication of their homologous bacteriophages. However, calcium, while of no detectable importance to organism growth, seems to be required for the multiplication of many bacteriophages

EFFICIENT FLOATING POINT FAST FOURIER TRANSFORM BUTTERFLY ARCHITECTURE USING BINARY SIGNED DIGIT MULTIPLIER AND ADDERS

Fast Fourier transform (FFT) is one of the most important tools in digital signal processing as well as communication system because transforming time domain to S-plane is very convenient using FFT. As FFT uses various techniques to convert a signal from time domain to S-domain and inverse, out of which butterfly technique is the one on which paper is focused on. Butterfly technique uses additions and multiplications of operands to get the required output. Floating point (FP) is used as operands due to their flexibility. As the computations involving FP has less speed, we have used binary signed digit (BSD). BSD will take the less time for addition and subtraction. Three bit BSD adder and FP adder together will make a fused dot product add (FDPA) unit. In FDPA, unit addition and subtraction will be one group and multiplication will be one group and then their respective results will be fused. Modified booth encoding and decoding algorithm are used here to make the complex multiplication with ease.Â

Deterministic algorithms for skewed matrix products

Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em deterministic} matrix multiplication algorithms. Motivated by applications in data mining, we first consider the case of real-valued, nonnegative $n$-by-$n$ input matrices $A$ and $B$, and show how to obtain a deterministic approximation of the weights of individual entries, as well as the entrywise $p$-norm, of the product $AB$. The algorithm is simple, space efficient and runs in one pass over the input matrices. For a user defined $b \in (0, n^2)$ the algorithm runs in time $O(nb + n\cdot\text{Sort}(n))$ and space $O(n + b)$ and returns an approximation of the entries of $AB$ within an additive factor of $\|AB\|_{E1}/b$, where $\|C\|_{E1} = \sum_{i, j} |C_{ij}|$ is the entrywise 1-norm of a matrix $C$ and $\text{Sort}(n)$ is the time required to sort $n$ real numbers in linear space. Building upon a result by Berinde et al. we show that for skewed matrix products (a common situation in many real-life applications) the algorithm is more efficient and achieves better approximation guarantees than previously known randomized algorithms. When the input matrices are not restricted to nonnegative entries, we present a new deterministic group testing algorithm detecting nonzero entries in the matrix product with large absolute value. The algorithm is clearly outperformed by randomized matrix multiplication algorithms, but as a byproduct we obtain the first $O(n^{2 + \varepsilon})$-time deterministic algorithm for matrix products with $O(\sqrt{n})$ nonzero entries

Cut and join operator ring in Aristotelian tensor model

Recent advancement of rainbow tensor models based on their superintegrability (manifesting itself as the existence of an explicit expression for a generic Gaussian correlator) has allowed us to bypass the long-standing problem seen as the lack of eigenvalue/determinant representation needed to establish the KP/Toda integrability. As the mandatory next step, we discuss in this paper how to provide an adequate designation to each of the connected gauge-invariant operators that form a double coset, which is required to cleverly formulate a tree-algebra generalization of the Virasoro constraints. This problem goes beyond the enumeration problem per se tied to the permutation group, forcing us to introduce a few gauge fixing procedures to the coset. We point out that the permutation-based labeling, which has proven to be relevant for the Gaussian averages is, via interesting complexity, related to the one based on the keystone trees, whose algebra will provide the tensor counterpart of the Virasoro algebra for matrix models. Moreover, our simple analysis reveals the existence of nontrivial kernels and co-kernels for the cut operation and for the join operation respectively that prevent a straightforward construction of the non-perturbative RG-complete partition function and the identification of truly independent time variables. We demonstrate these problems by the simplest non-trivial Aristotelian RGB model with one complex rank-3 tensor, studying its ring of gauge-invariant operators, generated by the keystone triple with the help of four operations: addition, multiplication, cut and join.Comment: 55 page

Quantum resource estimates for computing elliptic curve discrete logarithms

We give precise quantum resource estimates for Shor's algorithm to compute discrete logarithms on elliptic curves over prime fields. The estimates are derived from a simulation of a Toffoli gate network for controlled elliptic curve point addition, implemented within the framework of the quantum computing software tool suite LIQ$Ui|\rangle$. We determine circuit implementations for reversible modular arithmetic, including modular addition, multiplication and inversion, as well as reversible elliptic curve point addition. We conclude that elliptic curve discrete logarithms on an elliptic curve defined over an $n$-bit prime field can be computed on a quantum computer with at most $9n + 2\lceil\log_2(n)\rceil+10$ qubits using a quantum circuit of at most $448 n^3 \log_2(n) + 4090 n^3$ Toffoli gates. We are able to classically simulate the Toffoli networks corresponding to the controlled elliptic curve point addition as the core piece of Shor's algorithm for the NIST standard curves P-192, P-224, P-256, P-384 and P-521. Our approach allows gate-level comparisons to recent resource estimates for Shor's factoring algorithm. The results also support estimates given earlier by Proos and Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA.Comment: 24 pages, 2 tables, 11 figures. v2: typos fixed and reference added. ASIACRYPT 201