RNS-to-Binary Converters for Two Four-Moduli Sets {2n−1,2n,2n+1,2n+1−1} and {2n−1,2n,2n+1,2n+1+1}

In this paper, reverse converters for two recently proposed four-moduli sets {2ⁿ - 1,2ⁿ,2ⁿ + 1,2ⁿ+1 - 1} and {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} are described. The reverse conversion in the three-moduli set {2ⁿ - 1,2ⁿ,2ⁿ + 1} has been optimized in literature. Hence, the proposed converters are based on two new moduli sets {(2ⁿ(2²ⁿ-1)),2ⁿ⁺¹-1} and {(2ⁿ(2²ⁿ-1)), 2ⁿ⁺¹+1} and use mixed radix conversion. The resulting designs do not require any ROM. Both are similar in their architecture except that the converter for the moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} is slightly complicated due to the difficulty in performing reduction modulo (2ⁿ⁺¹+1) as compared with modulo (2ⁿ⁺¹-1). The proposed conversion techniques are compared with earlier realizations described in literature with regard to conversion time as well as area requirements.     

On the binary quadratic residue system with noncoprime moduli

The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2ⁿ+1 (n is even) have been considered. An efficient such set is the three-moduli set (2^2k-2+1.2^2k+1.2^2k+2+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2ⁿ+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2ⁿ +1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system     

On the norm and covering radius of the first-order Reed-Mullercodes

Let ρ(1,m) and N(1,m) be the covering radius and norm of the first-order Reed-Muller code R(1,m), respectively. It is known that ρ(1,2k+1)⩽lower bound [2^2k-2^(2k-1/2)] and N(1,2k+1)⩽2 lower bound [2^2k-2^(2k-1/2)] (k>0). We prove that ρ(1,2k+1)⩽2 lower bound [2^2k-1-2^(2k-3/2)] and N(1,2k+1)⩽4 lower bound [2^2k-1-2^(2k-3/2)] (k>0). We also discuss the connections of the two new bounds with other coding theoretic problems     

Fast algorithms for determining the linear complexity of sequences over GF(p/sup m/) with period 2/sup t/n

We prove a result which reduces the computation of the linear complexity of a sequence over GF(p^m) (p is an odd prime) with period 2n (n is a positive integer such that there exists an element bisinGF(p^m), bⁿ=-1) to the computation of the linear complexities of two sequences with period n. By combining with some known algorithms such as the Berlekamp-Massey algorithm and the Games-Chan algorithm we can determine the linear complexity of any sequence over GF(p^m) with period 2^tn (such that 2 ^t|p^m-1 and gcd(n,p^m-1)=1) more efficiently     

Linear complexity of de Bruijn sequences-old and new results

The linear complexity of a de Bruijn sequence is the degree of the shortest linear recursion which generates the sequence. It is well known that the complexity of a binary de Bruijn sequence of length 2ⁿ is bounded below by 2^n-1+n and above by 2^n-1 for n⩾3. We briefly survey the known knowledge in this area. Some new results are also presented, in particular, it is shown that for each interval of length 2^{[log n]+1} in the above range, there exist binary de Bruijn sequences of length 2ⁿ with linear complexity in the interval     

A New Family of Ternary Almost Perfect Nonlinear Mappings

A mapping f(x) from GF(pⁿ) to GF(pⁿ) is differentially k-uniform if k is the maximum number of solutions x isin GF(pⁿ) of f(x+a) - f(x) = b, where a, b isin GF(pⁿ) and a ne 0. A 2-uniform mapping is called almost perfect nonlinear (APN). This correspondence describes new families of ternary APN mappings over GF(3ⁿ), n>3 odd, of the form f(x) = ux^d + x^{d 2} where d₁ = (3^n-1)/2 - 1 and d₂ = 3ⁿ - 2.     

RNS-To-Binary Converter for a New Three-Moduli Set {2n+1−1,2n,2n−1}

In this brief, the design of residue number system (RNS) to binary converters for a new powers-of-two related three-moduli set {2ⁿ⁺¹ - 1, 2ⁿ, 2ⁿ - 1} is considered. This moduli set uses moduli of uniform word length (n to n + 1 bits). It is derived from a previously investigated four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2^{n +1} - 1}. Three RNS-to-binary converters are proposed for this moduli set: one using mixed radix conversion and the other two using Chinese remainder theorem. Detailed architectures of the three converters as well as comparison with some earlier proposed converters for three-moduli sets with uniform word length and the four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ⁺¹ - 1} are presented.     

A Residue-to-Binary Converter for a New Five-Moduli Set

The efficiency of the residue number system (RNS) depends not only on the residue-to-binary converters but also the operand sizes and the modulus in each residue channel. Due to their special number theoretic properties, RNS with a moduli set consisting of moduli in the form of 2 ⁿplusmn1 is more attractive than those with other forms of moduli. In this paper, a new five-moduli set RNS {2ⁿ-1,2ⁿ,2ⁿ+1,2ⁿ+1-1,2 ^n-1-1} for even n is proposed. The new moduli set has a dynamic range of (5n-1) bits. It incorporates two additional moduli to the celebrated three-moduli set, {2ⁿ-1,2ⁿ,2ⁿ +1} with VLSI efficient implementations for both the binary-to-residue conversion and the residue arithmetic units. This extension increases the parallelism and reduces the size of each residue channel for a given dynamic range. The proposed residue-to-binary converter relies on the properties of an efficient residue-to-binary conversion algorithm for {2ⁿ-1,2ⁿ,2ⁿ+1,2ⁿ+1-1} and the mixed-radix conversion (MRC) technique for the two-moduli set RNS. The hardware implementation of the proposed residue-to-binary converter employs adders as the primitive operators. Besides, it can be easily pipelined to attain a high throughput rate     

对联接杂凑函数的“特洛伊”消息攻击

“特洛伊”消息攻击是Andreeva等针对MD结构杂凑函数提出的一种攻击方法,首次将其应用于不同于MD结构的一类杂凑函数,即联接杂凑。结合联接杂凑的特点,综合利用Joux的多碰撞和深度为n?l的“钻石树”结构多碰撞,构造出了2n-bit联接杂凑函数的长度为 块的“特洛伊”消息,并据此首次提出了对其的固定前缀“特洛伊”消息攻击,其存储复杂性为 块消息,时间复杂性为 次压缩函数运算,远低于理想的时间复杂性 。     

Large families of quaternary sequences with low correlation

A family of quaternary (Z₄-alphabet) sequences of length L=2^r-1, size M⩾L²+3L+2, and maximum nontrivial correlation parameter C_max⩽2√(L+1)+1 is presented. The sequence family always contains the four-phase family 𝒜. When r is odd, it includes the family of binary Gold sequences. The sequence family is easily generated using two shift registers, one binary, the other quaternary. The distribution of correlation values is provided. The construction can be extended to produce a chain of sequence families, with each family in the chain containing the preceding family. This gives the design flexibility with respect to the number of intermittent users that can be supported, in a code-division multiple-access cellular radio system. When r is odd, the sequence families in the chain correspond to shortened Z₄-linear versions of the Delsarte-Goethals codes     

A non-Shannon-type conditional inequality of information quantities

Given n discrete random variables Ω={X₁,…,X_n}, associated with any subset α of {1,2,…,n}, there is a joint entropy H(X_α) where X_α={X_i: i∈α}. This can be viewed as a function defined on 2^{1,2,…,n} taking values in [0, +∞). We call this function the entropy function of Ω. The nonnegativity of the joint entropies implies that this function is nonnegative; the nonnegativity of the conditional joint entropies implies that this function is nondecreasing; and the nonnegativity of the conditional mutual information implies that this function is two-alternative. These properties are the so-called basic information inequalities of Shannon's information measures. An entropy function can be viewed as a 2ⁿ -1-dimensional vector where the coordinates are indexed by the subsets of the ground set {1,2,…,n}. As introduced by Yeng (see ibid., vol.43, no.6, p.1923-34, 1997) Γ_n stands for the cone in IR(2ⁿ-1) consisting of all vectors which have all these properties. Let Γ_n* be the set of all 2ⁿ -1-dimensional vectors which correspond to the entropy functions of some sets of n discrete random variables. A fundamental information-theoretic problem is whether or not Γ¯_n*=Γ_n. Here Γ¯_n * stands for the closure of the set Γ_n*. We show that Γ¯_n* is a convex cone, Γ₂*=Γ₂, Γ₃*≠Γ₃, but Γ¯₃ *=Γ₃. For four random variables, we have discovered a conditional inequality which is not implied by the basic information inequalities of the same set of random variables. This lends an evidence to the plausible conjecture that Γ¯_n*≠Γ_n for n>3     

A new random walk model for PCS networks

This paper proposes a new approach to simplify the two-dimensional random walk models capturing the movement of mobile users in personal communications services (PCS) networks. Analytical models are proposed for the new random walks. For a PCS network with hexagonal configuration, our approach reduces the states of the two-dimensional random walk from (3n²+3n-5) to n(n+1)/2, where n is the layers of a cluster. For a mesh configuration, our approach reduces the states from (2n2-2n+1) to (n²+2n+4)/4 if n is even and to (n ²+2n+5)/4 if n is odd. Simulation experiments are conducted to validate the analytical models. The results indicate that the errors between the analytical and simulation models are within 1%. Three applications (i.e., microcell/macrocell configuration, distance-based location update, and GPRS mobility management for data routing) are used to show how our new model can be used to investigate the performance of PCS networks     

4-phase sequences with near-optimum correlation properties

Two families of four-phase sequences are constructed using irreducible polynomials over Z₄. Family A has period L =2^r-1. size L+2. and maximum nontrivial correlation magnitude C_max⩽1+√(L+1), where r is a positive integer. Family B has period L=2(2^r-1). size (L+2)/4. and C_max for complex-valued sequences. Of particular interest, family A has the same size and period as the family of binary Gold sequences. but its maximum nontrivial correlation is smaller by a factor of √2. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, family A is thus superior to the best possible binary design of the same family size. Unlike the Gold design, families A and B are asymptotically optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are easily, implemented using shift registers. The exact distribution of correlation values is given for both families     

Yet another result on multi-log2N networks

One-to-many connection (i.e., multicast) is an important communication primitive used in parallel processing and high-speed switching in order to simultaneously send data from an input to more than one output. We prove that for even (respectively, odd) n, a multi-log₂N network is strictly nonblocking for a one-to-many connection traffic if it is designed by vertically stacking at least (δn)/4+1((δ/2)(n-1)+1) planes of a log₂N network together, where N=2ⁿ, δ=2^[n/2], and [x] denotes the greatest integer less than or equal to x. We thus give answer to the open problem and introduce yet another strictly nonblocking multicast network. The characterized network has self-routing capability, regular topology, O(2log₂N+2log₂(log₂N)) stages, and fewer crosspoints than the Clos network for N⩾512. We then extend multi log₂N multicast networks to the fanout restricted nonblocking networks. It turns out that the multi-log₂N network nonblocking in a strict-sense for a one-to-one connection traffic is also wide-sense nonblocking for a multicast traffic in which the fanout of any connection does not exceed δ, provided that for even (respectively, odd) n, the fanout capability of each log₂N network is restricted to stage (n/2)(((n-1)/2)+1) through n-1     

Ternary m-sequences with three-valued cross-correlation function:new decimations of Welch and Niho type

We show that the cross correlation between two ternary m-sequences of period 3ⁿ-1 that differ by the decimation d=2·3^m+1, where n=2m+1, takes on three different values. We conjecture the same result for the decimation d=2·3 ^r+1, where n is odd and r is defined by the condition 4r+1≡0 mod n. These two new cases form in a sense ternary counterparts of two previously confirmed binary cases, the conjectures of Welch and Niho (1972)     

Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums Sigma_xisinF(q)zeta_p ^Tr(af(x)+bx)(a,bisinF_q,q=p^m,pges3) for all perfect nonlinear functions f which is a Dembowski-Ostrom polynomial or p = 3,f=^x(3(k)+1)/2 where k is odd and (k,m)=1. As applications, we determine (1) the correlation distribution of the m-sequence {a_lambda=Tr(gamma^lambda)}(lambda=0,1,...) and the sequence {b_lambda=Tr(f(gamma^lambda))}(lambda=0,1,...) over F_p where gamma is a primitive element of F_q and (2) the weight distributions of the linear codes over F_p defined by f.     

Optimal biphase sequences with large linear complexity derived fromsequences over Z4

New families of biphase sequences of size 2^r-1+1, r being a positive integer, are derived from families of interleaved maximal-length sequences over Z₄ of period 2(2^r-1). These sequences have applications in code-division spread-spectrum multiuser communication systems. The families satisfy the Sidelnikov bound with equality on &thetas;_max, which denotes the maximum magnitude of the periodic cross-correlation and out-of-phase autocorrelation values. One of the families satisfies the Welch bound on &thetas;_max with equality. The linear complexity and the period of all sequences are equal to r(r+3)/2 and 2(2 ^r-1), respectively, with an exception of the single m-sequence which has linear complexity r and period 2^r-1. Sequence imbalance and correlation distributions are also computed     

Minimum bias multiple taper spectral estimation

Two families of orthonormal tapers are proposed for multitaper spectral analysis: minimum bias tapers, and sinusoidal tapers {υ ^{(k/)}, where υ}sub n//sup (k/)=√(2/(N+1))sin(πkn/N+1), and N is the number of points. The resulting sinusoidal multitaper spectral estimate is Sˆ(f)=(1/2K(N+1))Σ_j=1^K |y(f+j/(2N+2))-y(f-j/(2N+2))|², where y(f) is the Fourier transform of the stationary time series, S(f) is the spectral density, and K is the number of tapers. For fixed j, the sinusoidal tapers converge to the minimum bias tapers like 1/N. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the jth taper is simply 1/N centered about the frequencies (±j)/(2N+2). Thus, the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, ∫_-w^w|V(f)|²df of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian (1978) tapers. In contrast, the Slepian tapers can have the local bias, ∫_-½^½f ²|V(f)|²df, much larger than of the minimum bias tapers and the sinusoidal tapers     

Saturable absorbers based on impurity and defect centers incrystals

Saturation of near-infrared absorption and transmission dynamics are investigated in tetravalent-chromium-doped Gd₃Sc₂ Ga₃O₁₂, Gd₃Sc₂Al₃ O₁₂, and Mg₂SiO₄ crystals, as well as in reduced SrTiO₃ using 20 ps 1.08 μm laser pulses. An absorption cross section of (5±0.5)×10^-18 cm² in garnets and (2.3±0.3)×10^-18 cm² in forsterite is estimated for the ³A ₂-³T₂ transition of tetrahedral Cr⁴⁺. Q-switched and ultra-short pulses are realized in neodymium lasers using chromium-doped crystals as the saturable absorbers. Saturation of free-carrier absorption with ultra-short relaxation time is observed in SrTiO₃ at 10⁸-10 ¹⁰ W/cm² pump intensities, while at 10¹⁰-10¹¹ W/cm² three-photon interband transitions predominate. The free-carrier absorption cross section is estimated to be (2.7±0.3)×10^-18 cm²     

Theoretical analysis of word-level switching activity in thepresence of glitching and correlation

This paper presents a novel analytical approach to compute the switching activity in digital circuits at the word level in the presence of glitching and correlation. The proposed approach makes use of signal statistics such as mean, variance, and autocorrelation. It is shown that the switching activity α_f at the output node f of any arbitrary circuit in the presence of glitching and correlation is computed as α_f=Σ_i=1^S-1α(f _i,i+1)=Σ_i=1^{S- 1}p(f_i+1)(1-p(f_i))(1-ρ(f_i,i+1 )) (1) where ρ(f_i,i+1)=ρ(f_i,i+1)=(E[f_i(Sn)f _i+1(Sn)]- p(f_i)p(f_i+1))/(√(p(f _i)-p(f_i)²)(p(f_i+1)- p(f_i+1²))) (2). S number of time slots in a cycle; ρ(f_i,+1) time-slot autocorrelation coefficient; E[x]=expected value of x; p_x=probability of the signal x being “one”. The switching activity analysis of a signal at the word level is computed by summing the activities of all the individual bits constituting the signal. It is also shown that if the correlation coefficient of the higher order bits of a normally distributed signal x is ρ(x_c), then the bit P₀ where the correlation begins and the correlation coefficient is related hy ρ(x_c)=erfc{(2(P₀-1)-1)/(√2σ_x )} where erfc(x)=complementary error function; σ_x=variance of x. The proposed approach can estimate the switching activity in less than a second which is orders of magnitude faster than simulation-based approaches. Simulation results show that the errors using the proposed approach are about 6.1% on an average and that the approach is well suited even for highly correlated speech and music signals