Designs, Codes and Cryptography最新文献

Sequences with low aperiodic autocorrelation are used in communications and remote sensing for synchronization and ranging. The autocorrelation demerit factor of a sequence is the sum of the squared magnitudes of its autocorrelation values at every nonzero shift when we normalize the sequence to have unit Euclidean length. The merit factor, introduced by Golay, is the reciprocal of the demerit factor. We consider the uniform probability measure on the (2^ell ) binary sequences of length (ell ) and investigate the distribution of the demerit factors of these sequences. Sarwate and Jedwab have respectively calculated the mean and variance of this distribution. We develop new combinatorial techniques to calculate the pth central moment of the demerit factor for binary sequences of length (ell ). These techniques prove that for (pge 2) and (ell ge 4), all the central moments are strictly positive. For any given p, one may use the technique to obtain an exact formula for the pth central moment of the demerit factor as a function of the length (ell ). Jedwab’s formula for variance is confirmed by our technique with a short calculation, and we go beyond previous results by also deriving an exact formula for the skewness. A computer-assisted application of our method also obtains exact formulas for the kurtosis, which we report here, as well as the fifth central moment.

A storage code is an assignment of symbols to the vertices of a connected graph G(V, E) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in G. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on ({mathbb Z}) and ({mathbb Z}^2) (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.

In this paper, we investigate how to construct the required sequences to be used as pilot signals for packet detection in physical-layer security. Our construction starts from the frequency domain, where a set of orthogonal frequencies cover an entire given bandwidth. The construction is a generalized construction from Milewski’s construction, where it takes the inverse discrete Fourier transform of the given frequency domain sequences. In this paper, we call a set of the q sequences of length (ell q) with an equal distanced, nonzero frequency response in the frequency domain a frequency distance sequence set (FDSS) and a sequence interleaved from this set an FDSS interleaved sequence. By applying frequency and time domain relations, we show that such a set is mutually orthogonal, and is a complementary sequence set if and only if the seed sequence is perfect (i.e., zero autocorrelation at all out-of-phase shift). The FDSS interleaved sequence is perfect if and only if the seed sequence is perfect. We apply the proposed sequences to real world experiments as pilot sequences for coarse synchronization. In our experiments, we selected Frank–Zadoff–Chu sequences and Golay pair sequences in our construction for use with an ADALM-Pluto SDR from Analog Devices and simulations, and we show the pilot detection rate under different noisy channel conditions, when compared to alternative pilot selections. The false negative detection rate of our pilot decreases to zero when the SNR is 20 dB. In contrast, a general OFDM QPSK pilot has a false-negative detection rate near 70% at the same SNR. In general, our pilot sequence consistently has a lower false-negative rate to the OFDM QPSK pilot, which failed to detect most packets in the ADALM-Pluto SDR environment.

Proxy re-encryption (PRE) is a cryptosystem that realizes efficient encrypted data sharing by allowing a third party proxy to transform a ciphertext intended for a delegator (i.e., Alice) to a ciphertext intended for a delegatee (i.e., Bob). Attribute-based proxy re-encrypftion (AB-PRE) generalizes PRE to the attribute-based scenarios, enabling fine-grained access control on ciphertexts. However, the existing AB-PRE schemes do not adequately address the following problems: (1) the risk of decryption key leakage, and (2) the need of time-based delegation. To resolve these problems, we introduce a primitive called time-based attribute-based proxy re-encryption (TB-AB-PRE) with decryption key update. TB-AB-PRE associates keys with the current time information and supports efficient periodical decryption key update for each time transition. This property guarantees that a compromise of a decryption key for some time does not breach the security of ciphertexts from the others. Leveraging this time-based property, the proposed TB-AB-PRE elegantly achieves time-based delegation which enables Alice to decide which ciphertexts can be transformed and their decryptable timeframe after being transformed. The proposed construction is proven to be secure against honest re-encryption attacks with decryption key exposure resistance, under the learning with errors assumption.

For an odd prime power q, let (mathbb {F}_{q^2}=mathbb {F}_q(alpha )), (alpha ^2=tin mathbb {F}_q) be the quadratic extension of the finite field (mathbb {F}_q). In this paper, we consider the irreducible polynomials (F(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-cdots -c_{2}^qx^2+c_{1}^qx-1) over (mathbb {F}_{q^2}), where k is an odd integer and the coefficients (c_i) are in the form (c_i=a_i+b_ialpha ) with at least one (b_ine 0). For a given such irreducible polynomial F(x) over (mathbb {F}_{q^2}), we provide an algorithm to construct an irreducible polynomial (G(x)=x^k-A_1x^{k-1}+A_2x^{k-2}-cdots -A_{k-2}x^2+A_{k-1}x-A_k) over (mathbb {F}_q), where the (A_i)’s are explicitly given in terms of the (c_i)’s. This gives a bijective correspondence between irreducible polynomials over (mathbb {F}_{q^2}) and (mathbb {F}_q). This fact generalizes many recent results on this subject in the literature.

Circulant Column Parity Mixers (CCPMs) are a particular type of linear maps, used as the mixing layer in permutation-based cryptographic primitives like Keccak-f (SHA3) and Xoodoo. Although being successfully applied, not much is known regarding their algebraic properties. They are limited to invertibility of CCPMs, and that the set of invertible CCPMs forms a group. A possible explanation is due to the complexity of describing CCPMs in terms of linear algebra. In this paper, we introduce a new approach to studying CCPMs using module theory from commutative algebra. We show that many interesting algebraic properties can be deduced using this approach, and that known results regarding CCPMs resurface as trivial consequences of module theoretic concepts. We also show how this approach can be used to study the linear layer of Xoodoo, and other linear maps with a similar structure which we call DCD-compositions. Using this approach, we prove that every DCD-composition where the underlying vector space with the same dimension as that of Xoodoo has a low order. This provides a solid mathematical explanation for the low order of the linear layer of Xoodoo, which equals 32. We design a DCD-composition using this module-theoretic approach, but with a higher order using a different dimension.

The Gray map converts a symbol in (mathbb {Z}_4) to a pair of binary symbols. Therefore, under the Gray map, a linear function from (mathbb {Z}_4^n) to (mathbb {Z}_4) gives rise to a pair of boolean functions from (mathbb {F}_2^{2n}) to (mathbb {F}_2). This paper studies such boolean functions. We state and prove a condition for the nonlinearity of such functions and derive closed-form expressions for them. Further, results related to the mutual information between random variables that satisfy such expressions have been derived. These results are then used to construct a couple of nonlinear boolean secret sharing schemes. These schemes are then analyzed for their closeness to ‘perfectness’ and their ability to resist ‘Tompa–Woll’-like attacks.

Let ({mathcal {K}}) be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space ({mathbb {R}}^n), there is a well-studied notion of “ultrametric orthogonality” in ({mathcal {K}}^n). In this paper, motivated by a question of Erdős in the real case, given integers (k ge ell ge 2), we investigate the maximum size of a subset (S subseteq {mathcal {K}}^n {setminus }{textbf{0}}) satisfying the following property: for any (E subseteq S) of size k, there exists (F subseteq E) of size (ell ) such that any two distinct vectors in F are orthogonal. Other variants of this property are also studied.

We consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.

Locally repairable codes (LRCs) have attracted a lot of attention due to their applications in distributed storage systems. In this paper, we provide new constructions of optimal ((2, delta ))-LRCs over (mathbb {F}_q) with flexible parameters. Firstly, employing techniques from finite geometry, we introduce a simple yet useful condition to ensure that a punctured simplex code becomes a ((2, delta ))-LRC. It is worth noting that this condition only imposes a requirement on the size of the puncturing set. Secondly, utilizing character sums over finite fields and Krawtchouk polynomials, we determine the parameters of more punctured simplex codes with puncturing sets of new structures. Several infinite families of LRCs with new parameters are derived. All of our new LRCs are optimal with respect to the generalized Cadambe–Mazumdar bound and some of them are also Griesmer codes or distance-optimal codes.