ISSN:

1930-5346

eISSN:

1930-5338

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## Advances in Mathematics of Communications

February 2013 , Volume 7 , Issue 1

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2013, 7(1): i-i
doi: 10.3934/amc.2013.7.1i

*+*[Abstract](2280)*+*[PDF](97.8KB)**Abstract:**

Six years have passed since the founding of

*Advances in Mathematics of Communications*, a journal devoted to all mathematical aspects of information and communications technology. As expressed in the editorial of the inaugural volume, communications technology is omnipresent in contemporary life. However, its ubiquity sometimes obscures the fact that the foundations of communications technology are genuinely mathematical, as are its methods, both analytical and constructive.

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2013, 7(1): 1-38
doi: 10.3934/amc.2013.7.1

*+*[Abstract](3871)*+*[PDF](509.4KB)**Abstract:**

We take a critical look at security models that are often used to give "provable security" guarantees. We pay particular attention to digital signatures, symmetric-key encryption, and leakage resilience. We find that there has been a surprising amount of uncertainty about what the "right" definitions might be. Even when definitions have an appealing logical elegance and nicely reflect certain notions of security, they fail to take into account many types of attacks and do not provide a comprehensive model of adversarial behavior.

2013, 7(1): 39-56
doi: 10.3934/amc.2013.7.39

*+*[Abstract](3260)*+*[PDF](296.4KB)**Abstract:**

In a threshold scheme, the sensitivity of the secret as well as the number of players may fluctuate due to various reasons, e.g., mutual trust may vary or the structure of the players' organization might be changed. A possible solution to this problem is to modify the threshold and/or change the secret. Moreover, a common problem with almost all secret sharing schemes is that they are "one-time", meaning that the secret and shares are known to everyone after a public secret recovery process. This problem could be resolved if the dealer shares various secrets at the beginning, but a better solution is to dynamically generate new secrets in the absence of the dealer. These issues are our main motivation to revisit dynamic threshold schemes.

Therefore, we first provide the first comprehensive study of threshold modification techniques in both the passive and active adversary models. We first review an existing method for threshold modification based on resharing shares of a secret; this method is secure in the setting of a passive adversarial coalition. We then discuss two methods, termed

*public evaluation*(for threshold reduction) and

*zero addition*(for threshold increase) that can be used in both the passive and active adversarial setting. In the case of an active adversary, the techniques make use of

*verifiable secret sharing schemes*, whereas the schemes considered in the passive adversary model are all based on the

*Shamir scheme*. As an application, we discuss how the threshold and the secret can be changed multiple times to arbitrary values after the scheme's initialization.

2013, 7(1): 57-90
doi: 10.3934/amc.2013.7.57

*+*[Abstract](3166)*+*[PDF](532.0KB)**Abstract:**

Additive codes over $\mathbb{F}_4$ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic $p$ are connected to codes over $\mathbb{F}_{p^2}$ that are $\mathbb{F}_p$-linear in [30]. These codes that arise as connections with quantum codes are self-orthogonal under a particular inner product. We study a further generalization to codes termed $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are self-dual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case $q=3$ and $t=2$.

2013, 7(1): 91-101
doi: 10.3934/amc.2013.7.91

*+*[Abstract](2388)*+*[PDF](341.2KB)**Abstract:**

In this paper, three constructions of frequency hopping sequences (FHSs) are proposed using a new generalized cyclotomy with respect to $\textbf{Z}_{p^n}$, where $p$ is an odd prime and $n$ is a positive integer. Based on some basic properties of the new generalized cyclotomy, it is shown that all the constructed FHSs are optimal with respect to the well-known Lempel-Greenberger bound. Furthermore, these FHSs have new parameters which are not reported in the literature.

2013, 7(1): 103-111
doi: 10.3934/amc.2013.7.103

*+*[Abstract](3119)*+*[PDF](310.5KB)**Abstract:**

For symmetric pairings $e : \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$, Verheul proved that the existence of an efficiently-computable isomorphism $\phi : \mathbb{G}_T \rightarrow \mathbb{G}$ implies that the Diffie-Hellman problems in $\mathbb{G}$ and $\mathbb{G}_T$ can be efficiently solved. In this paper, we explore the implications of the existence of efficiently-computable isomorphisms $\phi_1 : \mathbb{G}_T \rightarrow \mathbb{G}_1$ and $\phi_2 : \mathbb{G}_T \rightarrow \mathbb{G}_2$ for asymmetric pairings $e : \mathbb{G}_1 \times \mathbb{G}_2 \rightarrow \mathbb{G}_T$. We also give a simplified proof of Verheul's theorem.

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