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24 24" class="breadcrumbHomeIcon_OVgt"><path d="M10 19v-5h4v5c0 .55.45 1 1 1h3c.55 0 1-.45 1-1v-7h1.7c.46 0 .68-.57.33-.87L12.67 3.6c-.38-.34-.96-.34-1.34 0l-8.36 7.53c-.34.3-.13.87.33.87H5v7c0 .55.45 1 1 1h3c.55 0 1-.45 1-1z" fill="currentColor"></path></svg></a></li><li class="breadcrumbs__item"><span class="breadcrumbs__link">About Milagro</span><meta itemprop="position" content="1"></li><li itemscope="" itemprop="itemListElement" itemtype="https://schema.org/ListItem" class="breadcrumbs__item breadcrumbs__item--active"><span class="breadcrumbs__link" itemprop="name">Milagro Crypto</span><meta itemprop="position" content="2"></li></ul></nav><div class="tocCollapsible_ETCw theme-doc-toc-mobile tocMobile_ITEo"><button type="button" class="clean-btn tocCollapsibleButton_TO0P">On this page</button></div><div class="theme-doc-markdown markdown"><header><h1>Milagro Crypto</h1></header><p>One of the critical points about information security is to give access to resources only to authorized entities and deny access to unauthorized ones. |
| Preventing unauthorized access very often comes down to making it <strong><em>almost impossible</em></strong>, i.e., tough, expensive, complicated, and time-consuming for the unauthorized entities to get access to resources.</p><p>The same principles apply to cryptography. In most cases, a suitable encryption mechanism satisfies at least two basic requirements:</p><ol><li>It is possible to give easy access to encrypted, cryptographically protected content to authorized entities.</li><li>It is possible to make it extremely challenging for unauthorized entities to access encrypted (ditto) content.</li></ol><p>Using the above, we can define an operation: Encryption that is tough to reverse without possessing a particular parameter, for example, a decryption key. |
| The difference in computational complexity (between performing the multiplication and reversing the result to retrieve the multiplier) is one of the essential cornerstones of elliptic curve cryptography.</p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="pairing-based-cryptography">Pairing Based Cryptography<a class="hash-link" href="#pairing-based-cryptography" title="Direct link to heading"></a></h2><p>Using elliptic curves we can now define on some elliptic curve a bilinear function called a <strong><em>pairing</em></strong>, which enables a mapping from two points on the same curve (or points on two related curves) into a different mathematical structure called a finite field. The bilinearity of the pairing is the key characteristic that makes pairing interesting and widely used in cryptography.</p><p>A <strong><em>bilinear pairing</em></strong> <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em"></span><span class="mord mathnormal">e</span></span></span></span></span> maps a pair of points (hence the name pairing) on an elliptic curve <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.05764em">E</span></span></span></span></span>, defined over some field <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>q</mi></msub></mrow><annotation encoding="application/x-tex">F_{q}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em"><span></span></span></span></span></span></span></span></span></span></span> to an element of the multiplicative group of a finite extension of <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><msup><mi>q</mi><mi>k</mi></msup></msub></mrow><annotation encoding="application/x-tex">{F}_{q^k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0221em;vertical-align:-0.3387em"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em">F</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em"><span style="top:-2.4974em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.782em"><span style="top:-2.786em;margin-right:0.0714em"><span class="pstrut" style="height:2.5em"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em">k</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3387em"><span></span></span></span></span></span></span></span></span></span></span>.</p><div class="math math-display"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>m</mi><mi>A</mi><mo>+</mo><mi>B</mi><mo separator="true">,</mo><mi>n</mi><mi>P</mi><mo>+</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>P</mi><msup><mo stretchy="false">)</mo><mrow><mi>m</mi><mi>n</mi></mrow></msup><mi>e</mi><mo stretchy="false">(</mo><mi>B</mi><mo separator="true">,</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(mA+B, nP + Q) = e(A,P)^{mn} e(B, Q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em"></span><span class="mord mathnormal" style="margin-right:0.05017em">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">n</span><span class="mord mathnormal" style="margin-right:0.13889em">P</span><span class="mspace" style="margin-right:0.2222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal" style="margin-right:0.13889em">P</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em"><span style="top:-3.113em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">mn</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em"></span><span class="mord mathnormal">Q</span><span class="mclose">)</span></span></span></span></span></div><p>The elements <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.13889em">P</span></span></span></span></span> and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em"></span><span class="mord mathnormal">Q</span></span></span></span></span> lie in two different groups, respectively <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> and <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>. The choice of those two different group determines a different <strong><em>types</em></strong> of pairing.</p><p>Let <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em"></span><span class="mord mathnormal" style="margin-right:0.05764em">E</span></span></span></span></span> an ordinary elliptic curve, take <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mn>1</mn></msub><mo mathvariant="normal">≠</mo><msub><mi>G</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">G_{1} \neq G_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span>, and if there is not an efficiently computable isomorphism <span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msub><mi>G</mi><mn>1</mn></msub><mo>→</mo><msub><mi>G</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phi:G_{1}\to G_{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em"></span><span class="mord mathnormal">ϕ</span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em"><span class="pstrut" style="height:2.7em"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em"><span></span></span></span></span></span></span></span></span></span></span> then the pairing is said to be of <strong><em>Type<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">-3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em"></span><span class="mord">−</span><span class="mord">3</span></span></span></span></span></em></strong>.</p><p>Currently, most of the state-of-the-art implementations of pairings take place on ordinary curves that assume the <strong><em>Type<span class="math math-inline"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">-3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em"></span><span class="mord">−</span><span class="mord">3</span></span></span></span></span></em></strong> scenario for reasons of efficiency and secure implementation.</p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="identity-based-encryption">Identity Based Encryption<a class="hash-link" href="#identity-based-encryption" title="Direct link to heading"></a></h2><p>Pairing-based cryptography builds on elliptic curve-based cryptography, but the extra functionality of the pairing enables us to design schemes which would otherwise be impossible to realize, or would be prohibitively expensive to do using RSA-based cryptosystems.</p><p>Identity-based encryption (IBE) is one such scheme that has received a large of amount of attention from the crypto community, and where commercially available products have been on the market for some time and are in wide use today.</p><p>IBE is similar to classical asymmetric key cryptography, in that each user has a public key for encryption and a private key for decryption. But there are many differences:</p><ol><li>IBE allows public keys to be set to the value of a pre-existing identifier, such as an email address, while in PKI the public key does not contain the notion of an identity, and the association with an identifier is created by a certificate signed by a third party (Certification Authority).</li><li>Clients or individual users do not generate private keys, but must instead download them from a trusted third party known as the Trust Authority (TA).</li><li>In IBE, to encrypt messages, the sender must obtain public “system parameters” from the Trust Authority (TA). These system parameters are used in combination with the intended recipient’s identity string (e.g. email address) to generate an encrypted message.</li></ol><h2 class="anchor anchorWithStickyNavbar_LWe7" id="post-quantum-cryptography">Post-Quantum Cryptography<a class="hash-link" href="#post-quantum-cryptography" title="Direct link to heading"></a></h2><p>"Post-quantum cryptography (sometimes referred to as quantum-proof, quantum-safe or quantum-resistant) refers to cryptographic algorithms (usually public-key algorithms) that are thought to be secure against an attack by a quantum computer. As of 2018, this is not true for the most popular public-key algorithms, which can be efficiently broken by a sufficiently strong hypothetical quantum computer. </p><p>The problem with currently popular algorithms is that their security relies on one of three hard mathematical problems: the integer factorization problem, the discrete logarithm problem or the elliptic-curve discrete logarithm problem. All of these problems can be easily solved on a sufficiently powerful quantum computer running Shor's algorithm. </p><p>Even though current, publicly known, experimental quantum computers lack processing power to break any real cryptographic algorithm, many cryptographers are designing new algorithms to prepare for a time when quantum computing becomes a threat."<sup id="fnref-first"><a href="#fn-first" class="footnote-ref">first</a></sup></p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="zero-knowledge-proof">Zero Knowledge Proof<a class="hash-link" href="#zero-knowledge-proof" title="Direct link to heading"></a></h2><p>"In cryptography, a <strong>zero-knowledge proof</strong> or <strong>zero-knowledge protocol</strong> is a method by which one party (the <em>prover</em>) can prove to another party (the <em>verifier</em>) that a given statement is true, without conveying any information apart from the fact that the statement is indeed true.</p><p>If proving the statement requires knowledge of some secret information on the part of the prover, the definition implies that the verifier will not be able to prove the statement in turn to anyone else, since the verifier does not possess the secret information.</p><p>Notice that the statement being proved must include the assertion that the prover has such knowledge (otherwise, the statement would not be proved in zero-knowledge, since at the end of the protocol the verifier would gain the additional information that the prover has knowledge of the required secret information).</p><p>If the statement consists <em>only</em> of the fact that the prover possesses the secret information, it is a special case known as <em>zero-knowledge proof of knowledge</em>, and it nicely illustrates the essence of the notion of zero-knowledge proofs: proving that one has knowledge of certain information is trivial if one is allowed to simply reveal that information; the challenge is proving that one has such knowledge without revealing the secret information or anything else."<sup id="fnref-second"><a href="#fn-second" class="footnote-ref">second</a></sup></p><h2 class="anchor anchorWithStickyNavbar_LWe7" id="summary">Summary<a class="hash-link" href="#summary" title="Direct link to heading"></a></h2><p><strong>Elliptic curve cryptography</strong> is an attractive alternative to conventional public key cryptography for implementation on constrained devices, where the significant computational resources i.e. speed, memory are limited and low-power wireless communication protocols is essential. It attains the same security levels as traditional cryptosystems using smaller parameter sizes.</p><p><strong>Pairing-based cryptography</strong> builds on elliptic curve-based cryptography, but the extra functionality of the pairing enables us to design schemes which would otherwise be impossible to realize, or would be prohibitively expensive. Examples include identity-based encryption, group signatures and non-interactive zero-knowledge proofs.</p><p><strong>Identity-based encryption</strong> doesn’t require certificates and certificate authorities. A trusted third party generates all the private keys, but all the public keys can be derived knowing the identity of the public key owner, for example, an email address. That means that no certificate is needed to bind a public key to its owner. Typically, it is up to the application to verify the owner possesses access to the unique identity attribute during the enrollment process in which a client obtains a private key.</p><p><strong>Post-quantum cryptography</strong> Post-quantum cryptography are cryptosystems which can be run on a classical computer, but are secure even if an adversary possesses a quantum computer. For data that must be private, but has the potential to be long lived and publicly available, using post-quantum secure algorithms like AES 256-bit for encryption and SIKE for encapsulation of encryption keys is essential. </p><p><strong>Zero-knowledge proof</strong> is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true, without conveying any information apart from the fact that the statement is indeed true.</p><p>For an in-depth dive into the cryptographic protocols in use within the Milagro framework, see the next section <a href="/docs/milagro-protocols">Milagro Protocols</a>.</p><hr><div class="theme-admonition theme-admonition-note alert alert--secondary admonition_LlT9"><div class="admonitionHeading_tbUL"><span class="admonitionIcon_kALy"><svg viewBox="0 0 14 16"><path fill-rule="evenodd" d="M6.3 5.69a.942.942 0 0 1-.28-.7c0-.28.09-.52.28-.7.19-.18.42-.28.7-.28.28 0 .52.09.7.28.18.19.28.42.28.7 0 .28-.09.52-.28.7a1 1 0 0 1-.7.3c-.28 0-.52-.11-.7-.3zM8 7.99c-.02-.25-.11-.48-.31-.69-.2-.19-.42-.3-.69-.31H6c-.27.02-.48.13-.69.31-.2.2-.3.44-.31.69h1v3c.02.27.11.5.31.69.2.2.42.31.69.31h1c.27 0 .48-.11.69-.31.2-.19.3-.42.31-.69H8V7.98v.01zM7 2.3c-3.14 0-5.7 2.54-5.7 5.68 0 3.14 2.56 5.7 5.7 5.7s5.7-2.55 5.7-5.7c0-3.15-2.56-5.69-5.7-5.69v.01zM7 .98c3.86 0 7 3.14 7 7s-3.14 7-7 7-7-3.12-7-7 3.14-7 7-7z"></path></svg></span>See an error in this documentation? </div><div class="admonitionContent_S0QG"><p>Submit a pull request on the development branch of <a href="https://github.com/apache/incubator-milagro" target="_blank" rel="noopener noreferrer">Milagro Website Repo</a>.</p></div></div><p>Supported admonition types are: caution, note, important, tip, warning.</p><div class="footnotes"><hr><ol><li id="fn-first"><a href="https://en.wikipedia.org/wiki/Post-quantum_cryptography" target="_blank" rel="noopener noreferrer">Wikipedia article</a><a href="#fnref-first" class="footnote-backref">↩</a></li><li id="fn-second"><a href="https://en.wikipedia.org/wiki/Zero-knowledge_proof" target="_blank" rel="noopener noreferrer">Wikipedia article</a><a href="#fnref-second" class="footnote-backref">↩</a></li></ol></div></div></article><nav class="pagination-nav docusaurus-mt-lg" aria-label="Docs pages navigation"><a class="pagination-nav__link pagination-nav__link--prev" href="/docs/milagro-intro"><div class="pagination-nav__sublabel">Previous</div><div class="pagination-nav__label">Milagro Introduction</div></a><a class="pagination-nav__link pagination-nav__link--next" href="/docs/milagro-protocols"><div class="pagination-nav__sublabel">Next</div><div class="pagination-nav__label">Milagro Protocols</div></a></nav></div></div><div class="col col--3"><div class="tableOfContents_bqdL thin-scrollbar theme-doc-toc-desktop"><ul class="table-of-contents table-of-contents__left-border"><li><a href="#elliptic-curve-cryptography" class="table-of-contents__link toc-highlight">Elliptic Curve Cryptography</a></li><li><a href="#pairing-based-cryptography" class="table-of-contents__link toc-highlight">Pairing Based Cryptography</a></li><li><a href="#identity-based-encryption" class="table-of-contents__link toc-highlight">Identity Based Encryption</a></li><li><a href="#post-quantum-cryptography" class="table-of-contents__link toc-highlight">Post-Quantum Cryptography</a></li><li><a href="#zero-knowledge-proof" class="table-of-contents__link toc-highlight">Zero Knowledge Proof</a></li><li><a href="#summary" class="table-of-contents__link toc-highlight">Summary</a></li></ul></div></div></div></div></main></div></div><footer class="footer"><div class="container container-fluid"><div class="footer__bottom text--center"><div class="margin-bottom--sm"><img src="/img/milagro.svg" class="themedImage_ToTc themedImage--light_HNdA footer__logo"><img src="/img/milagro.svg" class="themedImage_ToTc themedImage--dark_i4oU footer__logo"></div><div class="footer__copyright">Copyright © 2022 The Apache Software Foundation. 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