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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Homomorphic signatures for network coding ： ウィキペディア英語版 Homomorphic signatures for network coding Network coding has been shown to optimally use bandwidth in a network, maximizing information flow but the scheme is very inherently vulnerable to pollution attacks by malicious nodes in the network. A node injecting garbage can quickly affect many receivers. The pollution of network packets spreads quickly since the output of (even an) honest node is corrupted if at least one of the incoming packets is corrupted. An attacker can easily corrupt a packet even if it is encrypted by either forging the signature or by producing a collision under the hash function. This will give an attacker access to the packets and the ability to corrupt them. Denis Charles, Kamal Jain and Kristin Lauter designed a new homomorphic encryption signature scheme for use with network coding to prevent pollution attacks.〔http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.60.4738&rep=rep1&type=pdf〕 The homomorphic property of the signatures allows nodes to sign any linear combination of the incoming packets without contacting the signing authority. In this scheme it is computationally infeasible for a node to sign a linear combination of the packets without disclosing what linear combination was used in the generation of the packet. Furthermore, we can prove that the signature scheme is secure under well known cryptographic assumptions of the hardness of the discrete logarithm problem and the computational Elliptic curve Diffie–Hellman. ==Network coding== Let $G\; =\; (V,\; E)$ be a directed graph where $V$ is a set, whose elements are called vertices or nodes, and $E$ is a set of ordered pairs of vertices, called arcs, directed edges, or arrows. A source $s\; \backslash in\; V$ wants to transmit a file $D$ to a set $T\; \backslash subseteq\; V$ of the vertices. One chooses a vector space $W/\backslash mathbb\_p$(say of dimension $d$), where $p$ is a prime, and views the data to be transmitted as a bunch of vectors $w\_1\; ,\backslash ldots\; ,\; w\_k\; \backslash in\; W$. The source then creates the augmented vectors $v\_1,\backslash ldots\; ,\; v\_k$ by setting $v\_i\; =\; (0,\; \backslash ldots\; ,\; 0,\; 1,\; \backslash ldots\; ,\; 0,\; w\_,\; \backslash ldots\; ,\; w)$ where $w\_$ is the $j$-th coordinate of the vector $w\_i$. There are $(i-1)$ zeros before the first '1' appears in $v\_i$. One can assume without loss of generality that the vectors $v\_i$ are linearly independent. We denote the linear subspace (of $\backslash mathbb\_p^$ ) spanned by these vectors by $V$ . Each outgoing edge $e\; \backslash in\; E$ computes a linear combination, $y(e)$, of the vectors entering the vertex $v\; =\; in(e)$ where the edge originates, that is to say : $y(e)\; =\; \backslash sum\_(m\_e(f)y(f))$ where $m\_e(f)\; \backslash in\; \backslash mathbb\_p$. We consider the source as having $k$ input edges carrying the $k$ vectors $w\_i$. By induction, one has that the vector $y(e)$ on any edge is a linear combination $y(e)\; =\; \backslash sum\_(g\_i(e)v\_i)$ and is a vector in $V$ . The k-dimensional vector $g(e)\; =\; (g\_1(e),\; \backslash ldots\; ,\; g\_k(e))$ is simply the first ''k'' coordinates of the vector $y(e)$. We call the matrix whose rows are the vectors $g(e\_1),\; \backslash ldots\; ,\; g(e\_k)$, where $e\_i$ are the incoming edges for a vertex $t\; \backslash in\; T$, the global encoding matrix for $t$ and denote it as $G\_t$. In practice the encoding vectors are chosen at random so the matrix $G\_t$ is invertible with high probability. Thus any receiver, on receiving $y\_1,\; \backslash ldots\; ,\; y\_k$ can find $w\_1,\backslash ldots\; ,w\_k$ by solving : $\backslash begin\; y\text{'}\backslash \backslash \; y\_2\text{'}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; y\_k\text{'}\; \backslash end\; =\; G\_t\; \backslash begin\; w\_1\backslash \backslash \; w\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; w\_k\; \backslash end$ where the $y\_i\text{'}$ are the vectors formed by removing the first $k$ coordinates of the vector $y\_i$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Homomorphic signatures for network coding」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.