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1930-5346
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1930-5338
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Advances in Mathematics of Communications
May 2007 , Volume 1 , Issue 2
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2007, 1(2): 173-195
doi: 10.3934/amc.2007.1.173
+[Abstract](1905)
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Abstract:
Let $A(n, d)$ denote the maximum number of codewords in a binary code of length n and minimum Hamming distance $d$. Upper and lower bounds on $A(n, d)$ have been a subject for extensive research. In this paper we examine upper bounds on $A(n, d)$ as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on $A(n, d)$ as well as on the sizes of constant weight codes are also discussed.
Let $A(n, d)$ denote the maximum number of codewords in a binary code of length n and minimum Hamming distance $d$. Upper and lower bounds on $A(n, d)$ have been a subject for extensive research. In this paper we examine upper bounds on $A(n, d)$ as a special case of bounds on the size of subsets in metric association scheme. We will first obtain general bounds on the size of such subsets, apply these bounds to the binary Hamming scheme, and use linear programming to further improve the bounds. We show that the sphere packing bound and the Johnson bound as well as other bounds are special cases of one of the bounds obtained from association schemes. Specific bounds on $A(n, d)$ as well as on the sizes of constant weight codes are also discussed.
2007, 1(2): 197-221
doi: 10.3934/amc.2007.1.197
+[Abstract](2474)
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Abstract:
We present public-key cryptographic protocols for key exchange, digital signatures, and encryption whose security is based on the presumed intractability of solving the principal ideal problem, or equivalently, the distance problem, in the real model of a hyperelliptic curve. Our protocols represent a significant improvement over existing protocols using real hyperelliptic curves. Theoretical analysis and numerical experiments indicate that they are comparable to the imaginary model in terms of efficiency, and hold much more promise for practical applications than previously believed.
We present public-key cryptographic protocols for key exchange, digital signatures, and encryption whose security is based on the presumed intractability of solving the principal ideal problem, or equivalently, the distance problem, in the real model of a hyperelliptic curve. Our protocols represent a significant improvement over existing protocols using real hyperelliptic curves. Theoretical analysis and numerical experiments indicate that they are comparable to the imaginary model in terms of efficiency, and hold much more promise for practical applications than previously believed.
2007, 1(2): 223-238
doi: 10.3934/amc.2007.1.223
+[Abstract](1833)
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Abstract:
In this paper, we consider double circulant and quasi-twisted selfdual codes over $\mathbb F_5$ and $\mathbb F_7$. We determine the highest minimum weights for such codes of lengths up to 34 for $\mathbb F_5$ and up to 28 for $\mathbb F_7$, and classify the codes with these minimum weights. In particular, we give a double circulant self-dual [32, 16] code over $\mathbb F_5$ which has a higher minimum weight than the previously best known linear code with these parameters. In addition, a self-dual code over $\mathbb F_7$ is presented which has a higher minimum weight than the previously best known self-dual code for length 28.
In this paper, we consider double circulant and quasi-twisted selfdual codes over $\mathbb F_5$ and $\mathbb F_7$. We determine the highest minimum weights for such codes of lengths up to 34 for $\mathbb F_5$ and up to 28 for $\mathbb F_7$, and classify the codes with these minimum weights. In particular, we give a double circulant self-dual [32, 16] code over $\mathbb F_5$ which has a higher minimum weight than the previously best known linear code with these parameters. In addition, a self-dual code over $\mathbb F_7$ is presented which has a higher minimum weight than the previously best known self-dual code for length 28.
2007, 1(2): 239-242
doi: 10.3934/amc.2007.1.239
+[Abstract](2207)
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Abstract:
Recently Terence Tao approached Szemerédi's Regularity Lemma from the perspectives of Probability Theory and of Information Theory instead of Graph Theory and found a stronger variant of this lemma, which involves a new parameter. To pass from an entropy formulation to an expectation formulation he found the following: Let $Y$ , and $X,X'$ be discrete random variables taking values in $mathcal Y$ and $mathcal X$, respectively, where $mathcal Y \subset$ [ −1, 1 ], and with $X' = f(X)$ for a (deterministic) function $f$. Then we have
$ \E(|\E(Y|X')-\E(Y|X)|)\leq2I(X\wedge Y|X')^{\frac12}.$
We show that the constant $2$ can be improved to $(2 \l n2)^{\frac{1}{2}}$ and that this is the best possible constant.
Recently Terence Tao approached Szemerédi's Regularity Lemma from the perspectives of Probability Theory and of Information Theory instead of Graph Theory and found a stronger variant of this lemma, which involves a new parameter. To pass from an entropy formulation to an expectation formulation he found the following: Let $Y$ , and $X,X'$ be discrete random variables taking values in $mathcal Y$ and $mathcal X$, respectively, where $mathcal Y \subset$ [ −1, 1 ], and with $X' = f(X)$ for a (deterministic) function $f$. Then we have
$ \E(|\E(Y|X')-\E(Y|X)|)\leq2I(X\wedge Y|X')^{\frac12}.$
We show that the constant $2$ can be improved to $(2 \l n2)^{\frac{1}{2}}$ and that this is the best possible constant.
2007, 1(2): 243-250
doi: 10.3934/amc.2007.1.243
+[Abstract](2504)
+[PDF](125.4KB)
Abstract:
We use elementary facts about quadratic forms in characteristic 2 to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. We prove that the Gold functions yield bent functions when restricted to certain hyperplanes. We also use the sign information to determine the dual bent function.
We use elementary facts about quadratic forms in characteristic 2 to evaluate the sign of some Walsh transforms in terms of a Jacobi symbol. These results are applied to the Walsh transforms of the Gold and Kasami-Welch functions. We prove that the Gold functions yield bent functions when restricted to certain hyperplanes. We also use the sign information to determine the dual bent function.
2007, 1(2): 251-256
doi: 10.3934/amc.2007.1.251
+[Abstract](2262)
+[PDF](114.1KB)
Abstract:
In this note, we study the covering radii of extremal doubly even self-dual codes. We give slightly improved lower bounds on the covering radii of extremal doubly even self-dual codes of lengths 64, 80 and 96. The covering radii of some known extremal doubly even self-dual [64, 32, 12] codes are determined.
In this note, we study the covering radii of extremal doubly even self-dual codes. We give slightly improved lower bounds on the covering radii of extremal doubly even self-dual codes of lengths 64, 80 and 96. The covering radii of some known extremal doubly even self-dual [64, 32, 12] codes are determined.
2007, 1(2): 257-260
doi: 10.3934/amc.2007.1.257
+[Abstract](1886)
+[PDF](91.5KB)
Abstract:
The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices $\mathbb Z$, $A_2$ and (presumably) $D_4, E_8$ and the Leech lattice in being simultaneously optimal with respect to all three criteria.
The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices $\mathbb Z$, $A_2$ and (presumably) $D_4, E_8$ and the Leech lattice in being simultaneously optimal with respect to all three criteria.
2007, 1(2): 261-267
doi: 10.3934/amc.2007.1.261
+[Abstract](2313)
+[PDF](111.1KB)
Abstract:
An extremal singly even self-dual [88, 44, 16] code is constructed for the first time. Some optimal (extremal) singly even self-dual codes with weight enumerators which were not known to be attainable are also found for lengths 68 and 92.
An extremal singly even self-dual [88, 44, 16] code is constructed for the first time. Some optimal (extremal) singly even self-dual codes with weight enumerators which were not known to be attainable are also found for lengths 68 and 92.
2007, 1(2): 269-280
doi: 10.3934/amc.2007.1.269
+[Abstract](1893)
+[PDF](160.4KB)
Abstract:
Rivest proposed the idea of a chaffing-and-winnowing scheme, in which confidentiality is achieved through the use of an authentication code. Thus it would still be possible to have confidential communications even if conventional encryption schemes were outlawed. Hanaoka et al. constructed unconditionally secure chaffing-and-winnowing schemes which achieve perfect secrecy in the sense of Shannon. Their schemes are constructed from unconditionally secure authentication codes.
In this paper, we construct unconditionally secure chaffing-and-winnowing schemes from unconditionally secure authentication codes in which the authentication tags are very short. This could be a desirable feature, because certain types of unconditionally secure authentication codes can provide perfect secrecy if the length of an authentication tag is at least as long as the length of the plaintext. The use of such a code might be prohibited if encryption schemes are made illegal, so it is of interest to construct chaffing-and-winnowing schemes based on ''short'' authentication tags.
Rivest proposed the idea of a chaffing-and-winnowing scheme, in which confidentiality is achieved through the use of an authentication code. Thus it would still be possible to have confidential communications even if conventional encryption schemes were outlawed. Hanaoka et al. constructed unconditionally secure chaffing-and-winnowing schemes which achieve perfect secrecy in the sense of Shannon. Their schemes are constructed from unconditionally secure authentication codes.
In this paper, we construct unconditionally secure chaffing-and-winnowing schemes from unconditionally secure authentication codes in which the authentication tags are very short. This could be a desirable feature, because certain types of unconditionally secure authentication codes can provide perfect secrecy if the length of an authentication tag is at least as long as the length of the plaintext. The use of such a code might be prohibited if encryption schemes are made illegal, so it is of interest to construct chaffing-and-winnowing schemes based on ''short'' authentication tags.
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