December 2017 – Cryptography & Payments

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has-main-navigation jetpack-reblog-enabled"> <div id="page" class="site"> <a class="skip-link screen-reader-text" href="#content">Skip to content</a> <header id="masthead" class="site-header header_classes has-title-and-tagline has-menu" role="banner"> <div class="site-branding"> <p class="site-title"><a href="https://arthurvandermerwe.com/" rel="home">Cryptography & Payments</a></p> <p class="site-description"> Arthur Van Der Merwe </p> </div> <nav id="site-navigation" class="primary-navigation" role="navigation" aria-label="Main"> <button id="primary-close-menu" class="button close"> <span class="dropdown-icon close">Close <svg class="svg-icon" width="24" height="24" aria-hidden="true" role="img" focusable="false" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M12 10.9394L5.53033 4.46973L4.46967 5.53039L10.9393 12.0001L4.46967 18.4697L5.53033 19.5304L12 13.0607L18.4697 19.5304L19.5303 18.4697L13.0607 12.0001L19.5303 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class="page-title"><span class="archive-prefix">Monthly Archives: </span><span class="page-description">December 2017</span></h1> </header> <article id="post-249" class="post-249 post type-post status-publish format-standard hentry category-cryptography-2 tag-bi-linear-maps tag-cryptography entry"> <header class="entry-header default-max-width"> <h2 class="entry-title"><a href="https://arthurvandermerwe.com/2017/12/29/from-bi-linear-maps-to-searchable-encryption/" rel="bookmark">From Bi-Linear Maps to Searchable Encryption</a></h2> </header> <div class="entry-content"> <h2>Pairings-Based Cryptography</h2> <h3>Introduction</h3> <p>Theoretical research into pairings-based cryptography has been a well-researched area over the last few years, this cryptography scheme is based on the mapping of two cryptographical groups which allows for a new cryptographical scheme based on a trapdoor permutation between the groups with some interesting complexity properties.</p> <p>These two groups are called a Gap Groups in many instances, where the Decisional Diffie-Helman problem is easy, but the Computational Diffie-Helman still is hard to solve. Weil and Tate pairings are used in implementations but requires complex mathematics, this is why in this section we will use a slightly more abstract means to explain bilinear maps..</p> <h3>Bilinear Maps</h3> <p>Mostly all constructions of pairings-based cryptosystems use bilinear maps, for this we consider two groups <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> or a prime order <img src="https://s0.wp.com/latex.php?latex=q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="q" class="latex" />. We can denote <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> using additive notation and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> using multiplicative notation, even as the group operations of <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> are very different, <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> can also be written as a multiplicative group operation in some literature.</p> <p>If we consider two generators of group <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> as <img src="https://s0.wp.com/latex.php?latex=%5C+P&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C+P&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C+P&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\ P" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q" class="latex" />, we can write:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=aP%5C+%3D%5C+P%2BP%2B%5C+..P%5C+%5C%7D%5C+a&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=aP%5C+%3D%5C+P%2BP%2B%5C+..P%5C+%5C%7D%5C+a&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=aP%5C+%3D%5C+P%2BP%2B%5C+..P%5C+%5C%7D%5C+a&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="aP\ =\ P+P+\ ..P\ \}\ a" class="latex" /> times</p> <p>Using this we can also consider a map <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" /> as follows: <img src="https://s0.wp.com/latex.php?latex=%7Be%3A%5C+G%7D_1%5C+%5Ctimes+G_1%5C+%5Cto+%5C+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7Be%3A%5C+G%7D_1%5C+%5Ctimes+G_1%5C+%5Cto+%5C+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7Be%3A%5C+G%7D_1%5C+%5Ctimes+G_1%5C+%5Cto+%5C+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="{e:\ G}_1\ \times G_1\ \to \ \ G_2" class="latex" /></p> <p>This type of bilinear map has a main group <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and a shadow group <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> where we map two group elements in the first group to the second group would need have properties between them in order for it to be useful, Bilinearity, non-degenerate and computable.</p> <p><span style="text-decoration:underline;">Bilinearity:</span></p> <p>For Group <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> using generators <img src="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=%5C+Q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5C+Q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5C+Q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\ Q" class="latex" /> we can define a map to <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />, where the additive operation in group <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> equals the multiplicative operation in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\forall P,\ Q\in G_1, \forall a,\ b\ \in {\mathbb{Z}}^*_q," class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28aP%2C%5C+bQ%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28aP%2C%5C+bQ%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28aP%2C%5C+bQ%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Map:e\left(aP,\ bQ\right)=e(P,\ Q)^{ab}" class="latex" /></p> <p>If G1 and G2 where both multiplicative groups then the Bilinearity property would be the following:</p> <ul> <li><img src="https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C%5C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C%5C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cforall+P%2C%5C+Q%5Cin+G_1%2C%5C+%5Cforall+a%2C%5C+b%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q%2C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\forall P,\ Q\in G_1,\ \forall a,\ b\ \in {\mathbb{Z}}^*_q," class="latex" /></li> <li><img src="https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28P%5Ea%2C%5C+Q%5Eb%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28P%5Ea%2C%5C+Q%5Eb%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Map%3Ae%5Cleft%28P%5Ea%2C%5C+Q%5Eb%5Cright%29%3De%28P%2C%5C+Q%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Map:e\left(P^a,\ Q^b\right)=e(P,\ Q)^{ab}" class="latex" /></li> </ul> <p>This has an interesting property whereby it beaks the decisional Diffie-Helman problem, but this will be discussed in more details later.</p> <p><span style="text-decoration:underline;">Non-Degeneracy:</span></p> <p>If all the elements map to the identity of the group then if would not have any additional computational aspects to explore. It is therefore important not to create a map with the identity of either of the groups.<br /> <img src="https://s0.wp.com/latex.php?latex=%5Cforall+P%5C+%5Cin+G_1%2C%5C+P%5C+%5Cneq+0%5C+%5C+%5Cleft%5Clangle+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cright%5Crangle+%3DG_2%5C+%28e%5Cleft%28P%2C%5C+P%5Cright%29%5C+generates%5C+G_2%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cforall+P%5C+%5Cin+G_1%2C%5C+P%5C+%5Cneq+0%5C+%5C+%5Cleft%5Clangle+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cright%5Crangle+%3DG_2%5C+%28e%5Cleft%28P%2C%5C+P%5Cright%29%5C+generates%5C+G_2%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cforall+P%5C+%5Cin+G_1%2C%5C+P%5C+%5Cneq+0%5C+%5C+%5Cleft%5Clangle+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cright%5Crangle+%3DG_2%5C+%28e%5Cleft%28P%2C%5C+P%5Cright%29%5C+generates%5C+G_2%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\forall P\ \in G_1,\ P\ \neq 0\ \ \left\langle e\left(P,\ P\right)\right\rangle =G_2\ (e\left(P,\ P\right)\ generates\ G_2)" class="latex" /><br /> Such that: <img src="https://s0.wp.com/latex.php?latex=P%5C+%5Cneq+0%5C+%5CRightarrow+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cneq+1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P%5C+%5Cneq+0%5C+%5CRightarrow+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cneq+1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P%5C+%5Cneq+0%5C+%5CRightarrow+e%5Cleft%28P%2C%5C+P%5Cright%29%5Cneq+1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P\ \neq 0\ \Rightarrow e\left(P,\ P\right)\neq 1" class="latex" /></p> <p><span style="text-decoration:underline;">Computability:</span> <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" /> should be efficiently computable, there are some constructions of maps that are hard to compute.</p> <p>The construction of these bilinear pairs has been proven by Wei and Tate pairings, where <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> is a typical elliptic curve group, and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> is a finite field. These have proven to provide complex problems across these groups to construct cryptographical schemes.</p> <h3>Complex Problems</h3> <p>For the usage of bilinear maps in cryptographical schemes, we define a one-way function using two problems, the Decisional Diffie-Helman problem and the discrete log problem.</p> <p><strong>Theorem 1: The Discrete Log Problem in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> is no harder than the Discrete Log Problem in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />.</strong></p> <p>Proof 1: If we use our additive notation and consider that <img src="https://s0.wp.com/latex.php?latex=Q%5C+%3D%5C+aP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q%5C+%3D%5C+aP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q%5C+%3D%5C+aP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q\ =\ aP" class="latex" />, we then need to solve <img src="https://s0.wp.com/latex.php?latex=a&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="a" class="latex" />, which is random, for a given <img src="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P" class="latex" /> and a random <img src="https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=e%5Cleft%28P%2C%5C+Q%5Cright%29%3De%5Cleft%28P%2C%5C+aP%5Cright%29%3De%28P%2C%5C+P%29%5Ea&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%5Cleft%28P%2C%5C+Q%5Cright%29%3De%5Cleft%28P%2C%5C+aP%5Cright%29%3De%28P%2C%5C+P%29%5Ea&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%5Cleft%28P%2C%5C+Q%5Cright%29%3De%5Cleft%28P%2C%5C+aP%5Cright%29%3De%28P%2C%5C+P%29%5Ea&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e\left(P,\ Q\right)=e\left(P,\ aP\right)=e(P,\ P)^a" class="latex" /></p> <p>With this we can effectively reduce the Discrete Log Problem in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> to the Discrete Log Problem in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />, if we are given <img src="https://s0.wp.com/latex.php?latex=P%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P\in G_1" class="latex" /> and a random <img src="https://s0.wp.com/latex.php?latex=Q%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q\in G_1" class="latex" /> then the mapping of <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" /> is easily computable by calculating <img src="https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="{{log}_P (Q)}" class="latex" /> as:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=P%5E%60%3De%28P%2C%5C+P%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P%5E%60%3De%28P%2C%5C+P%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P%5E%60%3De%28P%2C%5C+P%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P^`=e(P,\ P)" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=Q%5E%60%3De%28P%2C%5C+Q%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q%5E%60%3De%28P%2C%5C+Q%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q%5E%60%3De%28P%2C%5C+Q%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q^`=e(P,\ Q)" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=a%3D%5C+%7B%7Blog%7D_%7BP%5E%60%7D+%5Cleft%28Q%5E%60%5Cright%29%5C+in%5C+%5C+G_2%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a%3D%5C+%7B%7Blog%7D_%7BP%5E%60%7D+%5Cleft%28Q%5E%60%5Cright%29%5C+in%5C+%5C+G_2%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a%3D%5C+%7B%7Blog%7D_%7BP%5E%60%7D+%5Cleft%28Q%5E%60%5Cright%29%5C+in%5C+%5C+G_2%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="a=\ {{log}_{P^`} \left(Q^`\right)\ in\ \ G_2\ }" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=a%3D%7B%7Blog%7D_P+%28Q%29%5C+%7D%5C+in%5C+G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a%3D%7B%7Blog%7D_P+%28Q%29%5C+%7D%5C+in%5C+G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a%3D%7B%7Blog%7D_P+%28Q%29%5C+%7D%5C+in%5C+G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="a={{log}_P (Q)\ }\ in\ G_1" class="latex" /></p> <p style="text-align:left;">With this we can see that the difficulty of solving the discrete log problem in both groups are the same, since the computation of <img src="https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%7B%7Blog%7D_P+%28Q%29%5C+%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="{{log}_P (Q)\ }" class="latex" /> have the same complexity in both groups.</p> <p><strong>Theorem 2: The Decisional Diffie-Helman is easy in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" />.</strong></p> <p>Proof 2: Solving the Decisional Diffie-Helman problem in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> requires distinguishing between:</p> <p><img src="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+cP+%5Crangle+%5C+with%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+cP+%5Crangle+%5C+with%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+cP+%5Crangle+%5C+with%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\langle P,\ aP,\ bP,\ cP \rangle \ with\ a,\ b,\ c\ \in {\mathbb{Z}}^*_q " class="latex" /> and<br /> <img src="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP+%5Crangle+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP+%5Crangle+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP+%5Crangle+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\langle P,\ aP,\ bP,\ abP \rangle \in {\mathbb{Z}}^*_q " class="latex" /><br /> If we can define <img src="https://s0.wp.com/latex.php?latex=P%2CA%2CB&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P%2CA%2CB&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P%2CA%2CB&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P,A,B" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="C" class="latex" /> as the distinguishers four values, then the distinguisher function is as follows:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=v_1%3DMap%3Ae%28A%2C%5C+B%29%5C+and%5C+v_2%3DMap%3Ae%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=v_1%3DMap%3Ae%28A%2C%5C+B%29%5C+and%5C+v_2%3DMap%3Ae%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=v_1%3DMap%3Ae%28A%2C%5C+B%29%5C+and%5C+v_2%3DMap%3Ae%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="v_1=Map:e(A,\ B)\ and\ v_2=Map:e(P,\ C)" class="latex" /></p> <p>If we have that <img src="https://s0.wp.com/latex.php?latex=v_1%3Dv_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=v_1%3Dv_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=v_1%3Dv_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="v_1=v_2" class="latex" />, then the tuple is of type <img src="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP%5Crangle+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP%5Crangle+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Clangle+P%2C%5C+aP%2C%5C+bP%2C%5C+abP%5Crangle+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\langle P,\ aP,\ bP,\ abP\rangle " class="latex" /></p> <p>From this we can take <img src="https://s0.wp.com/latex.php?latex=C%3D%5C+abP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C%3D%5C+abP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C%3D%5C+abP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="C=\ abP" class="latex" /> from (Theorem 1)</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=e%28A%2C%5C+B%29%3De%28aP%2C%5C+bP%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28A%2C%5C+B%29%3De%28aP%2C%5C+bP%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28A%2C%5C+B%29%3De%28aP%2C%5C+bP%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(A,\ B)=e(aP,\ bP)" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+P%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+P%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+P%29%5E%7Bab%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="=e(P,\ P)^{ab}" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+abP%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+abP%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+abP%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="=e(P,\ abP)" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="=e(P,\ C)" class="latex" /></p> <p>Since the map <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" /> is non-degenerate we can set <img src="https://s0.wp.com/latex.php?latex=e%5Cleft%28A%2C%5C+B%5Cright%29%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%5Cleft%28A%2C%5C+B%5Cright%29%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%5Cleft%28A%2C%5C+B%5Cright%29%3De%28P%2C%5C+C%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e\left(A,\ B\right)=e(P,\ C)" class="latex" /> equivalent to <img src="https://s0.wp.com/latex.php?latex=c%5C+%3Dab&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%5C+%3Dab&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c%5C+%3Dab&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="c\ =ab" class="latex" />. The distinguisher has thus a significant advantage given the mapping <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" /> to decide the Decisional Diffie-Helman problem.</p> <p><strong>Theorem 3 The Bilinear Diffie-Helman Problem is easy in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> but difficult in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /></strong></p> <p>Fact: If we are given two groups <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> with a map between them as <img src="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e" class="latex" />, there are no polynomial time algorithm that can compute <img src="https://s0.wp.com/latex.php?latex=%5Cleft%28P%2C%5C+aP%2C%5C+bP%2C%5C+cP%5Cright%29for%5C+%5C+some%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cleft%28P%2C%5C+aP%2C%5C+bP%2C%5C+cP%5Cright%29for%5C+%5C+some%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cleft%28P%2C%5C+aP%2C%5C+bP%2C%5C+cP%5Cright%29for%5C+%5C+some%5C+a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\left(P,\ aP,\ bP,\ cP\right)for\ \ some\ a,\ b,\ c\ \in \ {\mathbb{Z}}^*_q" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> given <img src="https://s0.wp.com/latex.php?latex=e%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(P,\ P)^{abc}" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />. With this we can construct the following properties between the groups as the following hard problems:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(aP,\ bP)^c\ in\ G_1=e(P,\ P)^{abc}\ in\ G_2" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(aP,\ cP)^b\ in\ G_1=e(P,\ P)^{abc}\ in\ G_2" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%5C+in%5C+G_1%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+in%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(bP,\ cP)^a\ in\ G_1=e(P,\ P)^{abc}\ in\ G_2" class="latex" /></p> <p>Using these theories, we can now construct cryptosystems based on these hard problems found in these groups.</p> <h3>Cryptography Schemes</h3> <p>Using these complexity problems, there has been an abundance of cryptosystems developed over the years, where the two most notable are the 3-party key agreement scheme, identity based encryption and searchable encryption.</p> <h4>The 3-party Diffie-Helman key agreement scheme</h4> <p>Joux introduced in 2000 a three-party key agreement scheme using bilinear maps utilizing the Bilinear Diffie-Helman problem for the construction.</p> <p>If we have two groups <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> with <img src="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P" class="latex" /> as a generator of <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" />, and three parties <img src="https://s0.wp.com/latex.php?latex=A%2C%5C+B%2C%5C+C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A%2C%5C+B%2C%5C+C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A%2C%5C+B%2C%5C+C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A,\ B,\ C" class="latex" /> that have respective secrets <img src="https://s0.wp.com/latex.php?latex=a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=a%2C%5C+b%2C%5C+c%5C+%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="a,\ b,\ c\ \in \ {\mathbb{Z}}^*_q" class="latex" /> we can construct a key agreement scheme where each party shares a secret key as follows:<br /> <img src="https://s0.wp.com/latex.php?latex=A%5C+%5Clongrightarrow+B%2C%5C+C%3AaP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A%5C+%5Clongrightarrow+B%2C%5C+C%3AaP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A%5C+%5Clongrightarrow+B%2C%5C+C%3AaP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A\ \longrightarrow B,\ C:aP" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=B%5C+%5Clongrightarrow+A%2C%5C+C%3AbP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=B%5C+%5Clongrightarrow+A%2C%5C+C%3AbP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=B%5C+%5Clongrightarrow+A%2C%5C+C%3AbP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="B\ \longrightarrow A,\ C:bP" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=C%5C+%5Clongrightarrow+A%2C%5C+B%3AcP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C%5C+%5Clongrightarrow+A%2C%5C+B%3AcP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C%5C+%5Clongrightarrow+A%2C%5C+B%3AcP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="C\ \longrightarrow A,\ B:cP" class="latex" /></p> <p>Using the Bilinear Diffie-Helman Problem we can define the following:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=A&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A" class="latex" /> computes <img src="https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28bP%2C%5C+cP%29%5Ea%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(bP,\ cP)^a=e(P,\ P)^{abc}" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=B&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=B&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=B&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="B" class="latex" /> computes <img src="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+cP%29%5Eb%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(aP,\ cP)^b=e(P,\ P)^{abc}" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="C" class="latex" /> computes <img src="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28aP%2C%5C+bP%29%5Ec%3De%28P%2C%5C+P%29%5E%7Babc%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(aP,\ bP)^c=e(P,\ P)^{abc}" class="latex" /></p> <p>All parties now have the same shared key <img src="https://s0.wp.com/latex.php?latex=K%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=K%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=K%3De%28P%2C%5C+P%29%5E%7Babc%7D%5C+%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="K=e(P,\ P)^{abc}\ \in G_2" class="latex" /> that can be used as an input to a symmetric encryption scheme.</p> <h4>Identity Based Encryption</h4> <p>The idea of using private information, like an email address, as a public key has been long debated and researched, whereby the corresponding private key can be delivered to the rightful owner. The role of the key generator must be to verify the private information before distributing the private key to the owner, although a public key infrastructure would solve this problem, there were substantial research into this area to move away from a trusted third party, and having the identity as part of the encryption.</p> <p>In Dan Boneh’s and Franklin’s paper an Identity based encryption scheme was created to remove the public key infrastructure with the use of bilinear maps and the bilinear Diffie-Helman problem, incorporating a random oracle model. This protocol consists out of five phases:</p> <p><strong>Setup</strong></p> <ul> <li>Defining two groups <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> with a bilinear map <img src="https://s0.wp.com/latex.php?latex=e%3AG_%7B1%5C+%7D%5Ctimes+G_1%5C+%5Cto+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%3AG_%7B1%5C+%7D%5Ctimes+G_1%5C+%5Cto+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%3AG_%7B1%5C+%7D%5Ctimes+G_1%5C+%5Cto+%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e:G_{1\ }\times G_1\ \to \ G_2" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P" class="latex" /> as a generator</li> <li>A System wide secret key <img src="https://s0.wp.com/latex.php?latex=s%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=s%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=s%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="s\in \ {\mathbb{Z}}^*_q" class="latex" /></li> <li>A corresponding system wide public key <img src="https://s0.wp.com/latex.php?latex=P_%7Bpub%7D%3DsP&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=P_%7Bpub%7D%3DsP&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=P_%7Bpub%7D%3DsP&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="P_{pub}=sP" class="latex" />, which are not distributed</li> <li>Public hash function <img src="https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5C+%5Cto+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5C+%5Cto+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5C+%5Cto+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_1:\{0,\ 1{\}}^*\ \to G^*_1" class="latex" />, a random oracle</li> <li>Public hash function <img src="https://s0.wp.com/latex.php?latex=H_2%3AG_2%5C+%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_2%3AG_2%5C+%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_2%3AG_2%5C+%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_2:G_2\ \to \{0,\ 1{\}}^n" class="latex" /> for some fixed <img src="https://s0.wp.com/latex.php?latex=n&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=n&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=n&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="n" class="latex" />, the second random oracle.</li> <li>The message space <img src="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BM%7D%3D%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cmathcal%7BM%7D%3D%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cmathcal%7BM%7D%3D%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\mathcal{M}=\{0,\ 1{\}}^n" class="latex" /></li> <li>The cypher space <img src="https://s0.wp.com/latex.php?latex=C%3DG%5E%2A_1%5C+%5Ctimes+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=C%3DG%5E%2A_1%5C+%5Ctimes+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=C%3DG%5E%2A_1%5C+%5Ctimes+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5En&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="C=G^*_1\ \times \{0,\ 1{\}}^n" class="latex" /></li> </ul> <p>To create a private key for a corresponding participant for <img src="https://s0.wp.com/latex.php?latex=ID%5Cin+%5C%7B0%2C1%7B%5C%7D%7D%5E%2A&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=ID%5Cin+%5C%7B0%2C1%7B%5C%7D%7D%5E%2A&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=ID%5Cin+%5C%7B0%2C1%7B%5C%7D%7D%5E%2A&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="ID\in \{0,1{\}}^*" class="latex" /> the system computes:</p> <p><img src="https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%28ID%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%28ID%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%28ID%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q_{ID}=H_1(ID)" class="latex" /> and<br /> <img src="https://s0.wp.com/latex.php?latex=d_%7BID%7D%3D%5C+%7BsQ%7D_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=d_%7BID%7D%3D%5C+%7BsQ%7D_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=d_%7BID%7D%3D%5C+%7BsQ%7D_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="d_{ID}=\ {sQ}_{ID}" class="latex" /> which is the private key that can be distributes to the user.</p> <p><strong>Encryption:</strong><br /> If we are now given a message <img src="https://s0.wp.com/latex.php?latex=m%5C+%5Cin+%5Cmathcal%7BM%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%5C+%5Cin+%5Cmathcal%7BM%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%5C+%5Cin+%5Cmathcal%7BM%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m\ \in \mathcal{M}" class="latex" /> we can compute the cyphertext as follows:</p> <ul> <li><img src="https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%5Cleft%28ID%5Cright%29%5Cin+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%5Cleft%28ID%5Cright%29%5Cin+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Q_%7BID%7D%3DH_1%5Cleft%28ID%5Cright%29%5Cin+G%5E%2A_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Q_{ID}=H_1\left(ID\right)\in G^*_1" class="latex" /></li> <li>We then choose a random <img src="https://s0.wp.com/latex.php?latex=r%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=r%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=r%5Cin+%5C+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_q&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="r\in \ {\mathbb{Z}}^*_q" class="latex" /></li> <li>We can now compute <img src="https://s0.wp.com/latex.php?latex=g_%7BID%7D%3De%5Cleft%28Q_%7BID%7D%2C%5C+P_%7Bpub%7D%5Cright%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=g_%7BID%7D%3De%5Cleft%28Q_%7BID%7D%2C%5C+P_%7Bpub%7D%5Cright%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=g_%7BID%7D%3De%5Cleft%28Q_%7BID%7D%2C%5C+P_%7Bpub%7D%5Cright%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="g_{ID}=e\left(Q_{ID},\ P_{pub}\right)\in G_2" class="latex" /></li> <li>And create the cyphertext: <img src="https://s0.wp.com/latex.php?latex=c%3D%28rP%2C%5C+m%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%3D%28rP%2C%5C+m%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c%3D%28rP%2C%5C+m%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="c=(rP,\ m\oplus H_2(g^r_{ID}))" class="latex" /></li> </ul> <p><strong>Decryption:</strong></p> <p>When the user receives the cyphertext, he has <img src="https://s0.wp.com/latex.php?latex=c%3D%28u%2C%5C+v%29%5Cin+C&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=c%3D%28u%2C%5C+v%29%5Cin+C&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=c%3D%28u%2C%5C+v%29%5Cin+C&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="c=(u,\ v)\in C" class="latex" /> and can decrypt it using his corresponding private key <img src="https://s0.wp.com/latex.php?latex=d_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=d_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=d_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="d_{ID}" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_2" class="latex" /></p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(d_{ID},\ u))" class="latex" /></p> <p>The main reason that both encryption and decryption works are because of the properties of pairings and the mask generated by <img src="https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_2" class="latex" /> that is xor’ed with the plaintext. We can prove the correctness by using simple substitution from the parameters above:</p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28d_%7BID%7D%2C%5C+u%29%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(d_{ID},\ u))" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28sH_1%28ID%29%5C+%2C%5C+rP%29%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28sH_1%28ID%29%5C+%2C%5C+rP%29%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28sH_1%28ID%29%5C+%2C%5C+rP%29%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(sH_1(ID)\ ,\ rP))" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28H_1%28ID%29%5C+%2C%5C+P%29%29%5E%7Brs%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28H_1%28ID%29%5C+%2C%5C+P%29%29%5E%7Brs%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28H_1%28ID%29%5C+%2C%5C+P%29%29%5E%7Brs%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(H_1(ID)\ ,\ P))^{rs}" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+sP%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+sP%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+sP%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(Q_{ID}\ ,\ sP))^r" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+P_%7Bpub%7D%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+P_%7Bpub%7D%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28e%28Q_%7BID%7D%5C+%2C%5C+P_%7Bpub%7D%29%29%5Er&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(e(Q_{ID}\ ,\ P_{pub}))^r" class="latex" /><br /> <img src="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28g%5Er_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28g%5Er_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3Dv%5Coplus+H_2%28g%5Er_%7BID%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=v\oplus H_2(g^r_{ID}" class="latex" />)<br /> <img src="https://s0.wp.com/latex.php?latex=m%3D%28m%5Coplus+H_2+%28g%5Er_%7BID%7D%29%29%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=m%3D%28m%5Coplus+H_2+%28g%5Er_%7BID%7D%29%29%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=m%3D%28m%5Coplus+H_2+%28g%5Er_%7BID%7D%29%29%5Coplus+H_2%28g%5Er_%7BID%7D%29%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="m=(m\oplus H_2 (g^r_{ID}))\oplus H_2(g^r_{ID}))" class="latex" /></p> <p style="text-align:center;"><img src="https://s0.wp.com/latex.php?latex=%3Dm&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%3Dm&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%3Dm&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="=m" class="latex" /></p> <p>This scheme provides us a way to use the identity as a parameter within the encryption and decryption without the use of a third party. The usage of identity is important, as this can bind the encryption and decryption to a owner of the keys.</p> <h2>Searchable Encryption</h2> <p>Searchable encryption schemes are a well-studied topic, and there have been several constructions using order revealing and order preserving schemes. For the a simplified construction a protocol I have chosen to use an order revealing encryption schemes based on bilinear maps, this construction is proven to be secure against adaptively chosen keyword attacks assuming the bilinear Diffie-Helman problem is intractable using the random oracle model.</p> <p>To use this construction , we will look at the following scenario:</p> <p>For this scenario, we need to define four entities that will be involved in the scheme:</p> <ul> <li>Users (1..n): responsible for the creation of messages that are sent to a trusted party for routing. These messages are sent and received via a secure channel to the messaging server.</li> <li>Third Party / Message Server: The messaging platform, that routes messages to users, and that can test weather a certain list of keywords are present in the message.</li> <li>Legal Authority (1..n): The party interested in searching the message data.</li> <li>Trusted Third Party: a Party responsible for securing the private key</li> </ul> <p>Suppose a Legal authority needs to be alerted when certain keywords are transmitted to a messaging server. For example, a user sends a message to another user that he is planning a bombing, the “bombing” needs to create an alert on the messaging server, and the legal authority needs to be sent a encryption of the message thread.</p> <p>If the messages between the users are encrypted using semantic means, then the messaging server cannot make any alerting decisions as it cannot decrypt the messages. Our goal here is to ensure that the messaging server provide a way to test whether a keyword has been transmitted between the users, without revealing the content of the messages. This can only be achievable by the legal authority providing a list of keywords to the message server that can be used, as well messaging server needs to have access to both the encryption and decryption key of the user’s messages.</p> <p>To do so, a user encrypts messages between users and messaging server using a standard public key cryptosystem and saves it in his database. The messages are appended with a Public-Key Encryption with keyword Search (PKS) of each keyword. For example, User Steve sends Peter a message <img src="https://s0.wp.com/latex.php?latex=M&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=M&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=M&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="M" class="latex" /> with words <img src="https://s0.wp.com/latex.php?latex=W_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W_1" class="latex" />, <img src="https://s0.wp.com/latex.php?latex=W_2..W_m&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W_2..W_m&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W_2..W_m&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W_2..W_m" class="latex" />, then the trusted messaging server create:<br /> <img src="https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D%28Message%29%5C+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_1%29%5Cparallel+%5Cdots+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_m%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D%28Message%29%5C+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_1%29%5Cparallel+%5Cdots+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_m%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D%28Message%29%5C+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_1%29%5Cparallel+%5Cdots+%5Cparallel+PKS%28A_%7Bpub%7D%2C%5C+W_m%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="E_{A_{pub}}(Message)\ \parallel PKS(A_{pub},\ W_1)\parallel \dots \parallel PKS(A_{pub},\ W_m)" class="latex" /><br /> Where <img src="https://s0.wp.com/latex.php?latex=%5Cparallel+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Cparallel+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Cparallel+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\parallel " class="latex" /> denotes concatenation and <img src="https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=E_%7BA_%7Bpub%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="E_{A_{pub}}" class="latex" /> is the public key of the legal authority (Alice). The reason for this form of encryption is so that the legal authority can provide a trapdoor <img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" /> to the messaging server to test whether a certain keyword has been used. Given a searchable encryption for a keyword <img src="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+%5C+W%5E%60%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+%5C+W%5E%60%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+%5C+W%5E%60%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="PKS(A_{pub},\ \ W^`)" class="latex" /> and a trapdoor <img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" /> the messaging server can determine is <img src="https://s0.wp.com/latex.php?latex=W%3DW%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W%3DW%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W%3DW%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W=W^`" class="latex" /> , if it’s the case that <img src="https://s0.wp.com/latex.php?latex=W%5Cneq+W%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W%5Cneq+W%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W%5Cneq+W%5E%60&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W\neq W^`" class="latex" /> then the messaging server does not learn any information about the word. It’s also quite interesting to note that this is not a very communitive scheme, as the searchable encryption (PKS) is constructed only using the public key of the legal authority.</p> <h3>Definitions</h3> <p>Throughout this section we will refer to a negligible function as <img src="https://s0.wp.com/latex.php?latex=f%5Cmathrm%7B%3A%7D%5Cmathrm%7B%5C+%7D%5Cmathbb%7BR%7D%5Cto+%5Cmathrm%7B%5B%7D0%2C%5C+1%5D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=f%5Cmathrm%7B%3A%7D%5Cmathrm%7B%5C+%7D%5Cmathbb%7BR%7D%5Cto+%5Cmathrm%7B%5B%7D0%2C%5C+1%5D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=f%5Cmathrm%7B%3A%7D%5Cmathrm%7B%5C+%7D%5Cmathbb%7BR%7D%5Cto+%5Cmathrm%7B%5B%7D0%2C%5C+1%5D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="f\mathrm{:}\mathrm{\ }\mathbb{R}\to \mathrm{[}0,\ 1]" class="latex" /> where <img src="https://s0.wp.com/latex.php?latex=f%5Cleft%28s%5Cright%29%3C1%2Fg%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=f%5Cleft%28s%5Cright%29%3C1%2Fg%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=f%5Cleft%28s%5Cright%29%3C1%2Fg%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="f\left(s\right)<1/g(s)" class="latex" /> for any polynomial <img src="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="g" class="latex" /> and sufficiently large <img src="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="s" class="latex" />. I will start by defining a searchable public key encryption scheme (PKS) where the public key refers to the cyphertext created by the messaging server using the public key of the legal authority , and the searchable encryption scheme (PKS) does not reveal any information about the message.</p> <p>Our goal is to enable the legal authority to send a short secret key (<img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" />) for a specific word to the messaging server, so that the messaging server can locate all messages that have this keyword without revealing the word <img src="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W" class="latex" />. The secret key (<img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" />) produced by the legal authority is based on the private key, and the messaging server send the message containing the words back to the legal authority, encrypted using the corresponding public key.</p> <p><strong>Definition 1.1: The following polynomial time randomized algorithms are part of a non-interactive searchable encryption scheme (PKS)</strong>.</p> <ul> <li><img src="https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="KeyGen(s)" class="latex" />: For a security parameter <img src="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="s" class="latex" /> a corresponding public/private key pair is generated (<img src="https://s0.wp.com/latex.php?latex=A_%7Bpriv%7D%2C%5C+A_%7Bpub%7D%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A_%7Bpriv%7D%2C%5C+A_%7Bpub%7D%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A_%7Bpriv%7D%2C%5C+A_%7Bpub%7D%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A_{priv},\ A_{pub})" class="latex" /> by the legal authority and the public key is sent to the messaging server.</li> <li><img src="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="PKS(A_{pub},\ W)" class="latex" />: For a word <img src="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W" class="latex" /> in the message, a searchable encryption (PKS) is generated using the public key <img src="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A_%7Bpub%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A_{pub}" class="latex" /> of the legal authority. We will denote the <img src="https://s0.wp.com/latex.php?latex=PKS&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=PKS&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=PKS&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="PKS" class="latex" /> function as <img src="https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="S" class="latex" /></li> <li><img src="https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Trapdoor(A_{priv},\ W)" class="latex" />: Given the private key of the legal authority, a certain word <img src="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W" class="latex" /> produces a trapdoor <img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" />.</li> <li><img src="https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Test(A_{pub},\ S,\ T_W)" class="latex" />: Given the public key of the legal authority <img src="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%2C%5C+%5C+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%2C%5C+%5C+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%2C%5C+%5C+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A_{pub},\ \ " class="latex" />and a searchable encryption <img src="https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=S&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="S" class="latex" /> on the messaging server, a trapdoor <img src="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w" class="latex" /> outputs `yes’ if <img src="https://s0.wp.com/latex.php?latex=W%3DW%27&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W%3DW%27&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W%3DW%27&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W=W'" class="latex" /></li> </ul> <p>The legal authority will run the <img src="https://s0.wp.com/latex.php?latex=KeyGen&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=KeyGen&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=KeyGen&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="KeyGen" class="latex" /> algorithm and generate its public/ private key pairs, and then use the <img src="https://s0.wp.com/latex.php?latex=Trapdoor&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Trapdoor&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Trapdoor&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Trapdoor" class="latex" /> function to generate a series of trapdoors for words <img src="https://s0.wp.com/latex.php?latex=W_1..%5C+W_i&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W_1..%5C+W_i&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W_1..%5C+W_i&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W_1..\ W_i" class="latex" /> that it wants to search for. The messaging server will then use the <img src="https://s0.wp.com/latex.php?latex=Test&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Test&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Test&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Test" class="latex" /> function to determine whether a given message has a keyword <img src="https://s0.wp.com/latex.php?latex=W_i&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W_i&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W_i&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W_i" class="latex" />.</p> <h3>Construction</h3> <p>For the definition above I will provide an efficient construction using bilinear maps based on a variant of the Decision Diffie-Hellman assumption with identity based encryption</p> <p>We will use two groups <img src="https://s0.wp.com/latex.php?latex=G_1%2C%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1%2C%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1%2C%5C+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1,\ G_2" class="latex" /> of prime order <img src="https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="p" class="latex" /> and a bilinear map <img src="https://s0.wp.com/latex.php?latex=e%3AG_1%5Ctimes+G_1%5Cto+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%3AG_1%5Ctimes+G_1%5Cto+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%3AG_1%5Ctimes+G_1%5Cto+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e:G_1\times G_1\to G_2" class="latex" /> between the two groups. This map satisfies the following three properties where the size of <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /> is determined by a security parameter:</p> <ul> <li><strong>Computable:</strong> If you are given two elements in <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> as <img src="https://s0.wp.com/latex.php?latex=g%2C%5C+h&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=g%2C%5C+h&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=g%2C%5C+h&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="g,\ h" class="latex" /> then there exists a polynomial time algorithm to compute the map <img src="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+h%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+h%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28g%2C%5C+h%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(g,\ h)" class="latex" /> in <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /></li> <li><strong>Bilinear:</strong> for all integers in the prime order, we have a map <img src="https://s0.wp.com/latex.php?latex=e%28g%5Ex%2C%5C+g%5Ey%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28g%5Ex%2C%5C+g%5Ey%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28g%5Ex%2C%5C+g%5Ey%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(g^x,\ g^y)" class="latex" /> = <img src="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29%5E%7Bxy%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29%5E%7Bxy%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29%5E%7Bxy%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(g,\ g)^{xy}" class="latex" /></li> <li><strong>Non-degenerate:</strong> if <img src="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="g" class="latex" /> is a generator of <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" /> then the map <img src="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=e%28g%2C%5C+g%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="e(g,\ g)" class="latex" /> is a generator of <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" /></li> </ul> <p>From this we can build a non-interactive searchable encryption scheme based on bilinear maps. For this we will need two hash function, or random oracles in each group as:</p> <p><img src="https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5Cto+G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5Cto+G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_1%3A%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%2A%5Cto+G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_1:\{0,\ 1{\}}^*\to G_1" class="latex" /> and <img src="https://s0.wp.com/latex.php?latex=H_2%3AG_2%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%7B%7Blog+p%5C+%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_2%3AG_2%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%7B%7Blog+p%5C+%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_2%3AG_2%5Cto+%5C%7B0%2C%5C+1%7B%5C%7D%7D%5E%7B%7Blog+p%5C+%7D%7D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_2:G_2\to \{0,\ 1{\}}^{{log p\ }}" class="latex" /></p> <p>Based on definition 1.1 we will construct the scheme using the same model based on the Dan Boneh Searchable Encryption Scheme:</p> <ul> <li><img src="https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=KeyGen%28s%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="KeyGen(s)" class="latex" />: The security parameter <img src="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=s&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="s" class="latex" /> determines the size of the prime order <img src="https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=p&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="p" class="latex" /> of the groups <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" />and <img src="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_2" class="latex" />. The legal authority then also selects a random <img src="https://s0.wp.com/latex.php?latex=%5Calpha+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Calpha+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Calpha+%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\alpha \in {\mathbb{Z}}^*_p" class="latex" /> and a generator <img src="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=g&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="g" class="latex" /> of <img src="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="G_1" class="latex" />. The Output is a public key <img src="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%3D%5Bg%2C%5C+h%3Dg%5E%7B%5Calpha+%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%3D%5Bg%2C%5C+h%3Dg%5E%7B%5Calpha+%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=A_%7Bpub%7D%3D%5Bg%2C%5C+h%3Dg%5E%7B%5Calpha+%7D%5D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="A_{pub}=[g,\ h=g^{\alpha }]" class="latex" /> and a private key <img src="https://s0.wp.com/latex.php?latex=%5Calpha+&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=%5Calpha+&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=%5Calpha+&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="\alpha " class="latex" />. The public key is then distributed to the messaging server.</li> <li><img src="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="PKS(A_{pub},\ W)" class="latex" />: Using the public key and a word <img src="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=W&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="W" class="latex" />, the messaging server computes a bilinear map <img src="https://s0.wp.com/latex.php?latex=t%5C+%3De%28H_1%28W%29%2C%5C+h%5Er%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=t%5C+%3De%28H_1%28W%29%2C%5C+h%5Er%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=t%5C+%3De%28H_1%28W%29%2C%5C+h%5Er%29%5Cin+G_2&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="t\ =e(H_1(W),\ h^r)\in G_2" class="latex" /> using the random oracle and a random <img src="https://s0.wp.com/latex.php?latex=r%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=r%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=r%5Cin+%7B%5Cmathbb%7BZ%7D%7D%5E%2A_p&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="r\in {\mathbb{Z}}^*_p" class="latex" />. Then outputs a searchable encryption <img src="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29%3D%5Bg%5Er%2C%5C+H_2%28t%29%5D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29%3D%5Bg%5Er%2C%5C+H_2%28t%29%5D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=PKS%28A_%7Bpub%7D%2C%5C+W%29%3D%5Bg%5Er%2C%5C+H_2%28t%29%5D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="PKS(A_{pub},\ W)=[g^r,\ H_2(t)]" class="latex" />.</li> <li><img src="https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Trapdoor%28A_%7Bpriv%7D%2C%5C+W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Trapdoor(A_{priv},\ W)" class="latex" />: The legal authority uses the random oracle and its private key to generate a trapdoor <img src="https://s0.wp.com/latex.php?latex=T_w%3DH_1%28W%29%5E%7B%5Calpha+%7D%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=T_w%3DH_1%28W%29%5E%7B%5Calpha+%7D%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=T_w%3DH_1%28W%29%5E%7B%5Calpha+%7D%5Cin+G_1&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="T_w=H_1(W)^{\alpha }\in G_1" class="latex" /></li> <li><img src="https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=Test%28A_%7Bpub%7D%2C%5C+S%2C%5C+T_W%29&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="Test(A_{pub},\ S,\ T_W)" class="latex" />: When the messaging server receives a Test function from the legal authority as <img src="https://s0.wp.com/latex.php?latex=S%3D%5BA%2C%5C+B%5D&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=S%3D%5BA%2C%5C+B%5D&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=S%3D%5BA%2C%5C+B%5D&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="S=[A,\ B]" class="latex" /> it can test if <img src="https://s0.wp.com/latex.php?latex=H_2%28e%28T_w%2C%5C+A%29%29%3DB&bg=FFFFFF&fg=000&s=0&c=20201002" srcset="https://s0.wp.com/latex.php?latex=H_2%28e%28T_w%2C%5C+A%29%29%3DB&bg=FFFFFF&fg=000&s=0&c=20201002 1x, https://s0.wp.com/latex.php?latex=H_2%28e%28T_w%2C%5C+A%29%29%3DB&bg=FFFFFF&fg=000&s=0&c=20201002&zoom=4.5 4x" alt="H_2(e(T_w,\ A))=B" class="latex" /></li> </ul> <p>The construction of the scheme can be viewed as a derivative of Identity Based Encryption with a limited number of identities. Using this scheme, the messaging server needs to have the ability to create an index of the words that’s exchanged between the users of the system that can be tested. Unfortunately, this construction has several issues relating to the sharing of the creation of the trapdoor function. None the less, the use of bi-linear maps and hash functions allows us to identify encrypted words without revealing what they actually are.</p> <p> </p> <div id="atatags-702045271-67420daf8d9c5"></div> <script> __ATA.cmd.push(function() { __ATA.initDynamicSlot({ id: 'atatags-702045271-67420daf8d9c5', location: 120, formFactor: '001', label: { text: 'Advertisements', }, creative: { reportAd: { text: 'Report this ad', }, privacySettings: { text: 'Privacy', } } }); }); </script> </div> <footer class="entry-footer default-max-width"> <span class="byline"><svg class="svg-icon" width="16" height="16" aria-hidden="true" role="img" focusable="false" viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><path fill-rule="evenodd" clip-rule="evenodd" d="M15 7.5C15 9.15685 13.6569 10.5 12 10.5C10.3431 10.5 9 9.15685 9 7.5C9 5.84315 10.3431 4.5 12 4.5C13.6569 4.5 15 5.84315 15 7.5ZM16.5 7.5C16.5 9.98528 14.4853 12 12 12C9.51472 12 7.5 9.98528 7.5 7.5C7.5 5.01472 9.51472 3 12 3C14.4853 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December 2017 – Cryptography & Payments