assets/js/e6851893.36b6ad27.js

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This section focuses specifically on a few protocols that are used extensively as key building blocks within the Milagro Zero-Knowledge Proof Multi-Factor Authentication (ZKP-MFA) server and clients, and Milagro Decentralized Trust Authority (D-TA) applications. Both applications implement these protocols using the Milagro crypto libraries.",source:"@site/../docs/milagro-protocols.md",sourceDirName:".",slug:"/milagro-protocols",permalink:"/docs/milagro-protocols",draft:!1,tags:[],version:"current",frontMatter:{id:"milagro-protocols",title:"Milagro Protocols",sidebar_label:"Milagro Protocols"},sidebar:"docs",previous:{title:"Milagro Crypto",permalink:"/docs/milagro-crypto"},next:{title:"Milagro Design",permalink:"/docs/milagro-design"}},i={},k=[{value:"M-Pin Protocol - Introduction",id:"m-pin-protocol---introduction",level:2},{value:"Chow-Choo Protocol - Introduction",id:"chow-choo-protocol---introduction",level:2},{value:"BLS Signatures - Introduction",id:"bls-signatures---introduction",level:2},{value:"Supersingular Isogeny Key Encapsulation - Introduction",id:"supersingular-isogeny-key-encapsulation---introduction",level:2},{value:"Protocols In Depth",id:"protocols-in-depth",level:2},{value:"M-Pin 1-Pass",id:"m-pin-1-pass",level:3},{value:"M-Pin 2-Pass",id:"m-pin-2-pass",level:3},{value:"M-Pin FULL",id:"m-pin-full",level:3},{value:"Chow-Choo Protocol",id:"chow-choo-protocol",level:3},{value:"Secret Revocation",id:"secret-revocation",level:3},{value:"BLS Subgroup Multi-Signatures",id:"bls-subgroup-multi-signatures",level:3},{value:"Supersingular Isogeny Key Encapsulation (SIKE)",id:"supersingular-isogeny-key-encapsulation-sike",level:3}],c={toc:k};function o(a){var e=a.components,r=(0,n.Z)(a,p);return(0,m.kt)("wrapper",(0,s.Z)({},c,r,{components:e,mdxType:"MDXLayout"}),(0,m.kt)("p",null,"The Apache Milagro crypto libraries contain an (almost) overwhelming choice of algorithms and cryptographic primitives for robust and rapid application development. This section focuses specifically on a few protocols that are used extensively as key building blocks within the Milagro Zero-Knowledge Proof Multi-Factor Authentication (ZKP-MFA) server and clients, and Milagro Decentralized Trust Authority (D-TA) applications. Both applications implement these protocols using the Milagro crypto libraries."),(0,m.kt)("h2",{id:"m-pin-protocol---introduction"},"M-Pin Protocol - Introduction"),(0,m.kt)("p",null,"The genesis of the M-Pin Protocol was first put forward in a research paper by ",(0,m.kt)("a",{parentName:"p",href:"https://scholar.google.com/citations?user=GsM-aeEAAAAJ&hl=en"},"Dr. Michael Scott")," in 2002",(0,m.kt)("sup",{parentName:"p",id:"fnref-first"},(0,m.kt)("a",{parentName:"sup",href:"#fn-first",className:"footnote-ref"},"first")),"."),(0,m.kt)("p",null,"The M-Pin Protocol has been iterated on several times over the years since, to develop three distinct modes, which will be explored in the following sections."),(0,m.kt)("p",null,"Because of the characteristics that M-Pin inherits from the four techniques above, the M-Pin Protocol and its variants are able to deliver:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"Multi-factor authentication (MFA) using Zero Knowledge Proof"),(0,m.kt)("li",{parentName:"ul"},"Authenticated Key Agreement"),(0,m.kt)("li",{parentName:"ul"},"Distribution, or splitting, of Trust Authorities"),(0,m.kt)("li",{parentName:"ul"},(0,m.kt)("a",{parentName:"li",href:"https://en.wikipedia.org/wiki/Subliminal_channel"},"Subliminal Channel Communication"),(0,m.kt)("sup",{parentName:"li",id:"fnref-second"},(0,m.kt)("a",{parentName:"sup",href:"#fn-second",className:"footnote-ref"},"second")))),(0,m.kt)("p",null,"The three modes of operation of the M-Pin Protocol are as follows:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},(0,m.kt)("strong",{parentName:"li"},"M-Pin 1-pass"),": Client to server authentication via digital signature, this mode implements a ",(0,m.kt)("em",{parentName:"li"},"non-interactive")," zero knowledge proof and is resistant to ",(0,m.kt)("a",{parentName:"li",href:"https://en.wikipedia.org/wiki/Man-in-the-middle_attack"},"MITM (man in the middle)")," attacks."),(0,m.kt)("li",{parentName:"ul"},(0,m.kt)("strong",{parentName:"li"},"M-Pin 2-pass"),": Client to server authentication via a ",(0,m.kt)("em",{parentName:"li"},"interactive")," zero knowledge proof, resistant to ",(0,m.kt)("a",{parentName:"li",href:"https://en.wikipedia.org/wiki/Man-in-the-middle_attack"},"MITM")," and ",(0,m.kt)("a",{parentName:"li",href:"https://kcitls.org"},"KCI (Key Compromise Impersonation)")," attacks."),(0,m.kt)("li",{parentName:"ul"},(0,m.kt)("strong",{parentName:"li"},"M-Pin FULL"),": Mutual client to server authentication via a ",(0,m.kt)("em",{parentName:"li"},"interactive")," zero knowledge proof, resistant to MITM and KCI attacks and able to drive an Authenticated ",(0,m.kt)("a",{parentName:"li",href:"https://en.wikipedia.org/wiki/Key-agreement_protocol"},"Key Agreement")," between client and server, resulting in 128 bit shared secret key.")),(0,m.kt)("p",null,"Note that the M-Pin Full Authenticated Key Agreement possesses the quality of ",(0,m.kt)("a",{parentName:"p",href:"https://en.wikipedia.org/wiki/Forward_secrecy"},"perfect forward secrecy (PFS)"),", meaning, even if the client and server long term keys are compromised, the past session keys (used to encrypt TLS traffic, for example) are not compromised."),(0,m.kt)("h2",{id:"chow-choo-protocol---introduction"},"Chow-Choo Protocol - Introduction"),(0,m.kt)("p",null,"The Chow-Choo Protocol was developed by Sherman S.M. Chow and Kim-Kwang Raymond Choo and published in 2007 via a research paper titled Strongly-Secure Identity-based Key Agreement",(0,m.kt)("sup",{parentName:"p",id:"fnref-third"},(0,m.kt)("a",{parentName:"sup",href:"#fn-third",className:"footnote-ref"},"third")),".  The Chow-Choo Protocol can be technically described as an identity-based key agreement protocol."),(0,m.kt)("p",null,"The Chow-Choo Protocol is of these classifications and exploits the features of:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"Elliptic Curve Cryptography"),(0,m.kt)("li",{parentName:"ul"},"Pairing Based Cryptography"),(0,m.kt)("li",{parentName:"ul"},"Identity Based Encryption")),(0,m.kt)("p",null,"Because of the characteristics that Chow-Choo inherits from the three techniques above, the Chow-Choo Protocol can deliver:"),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"Authenticated Key Agreement"),(0,m.kt)("li",{parentName:"ul"},"Distribution, or splitting, of Trust Authorities")),(0,m.kt)("p",null,"Note that the Chow-Choo Protocol is not a Zero Knowledge Proof protocol."),(0,m.kt)("h2",{id:"bls-signatures---introduction"},"BLS Signatures - Introduction"),(0,m.kt)("p",null,"The BLS signature (Boneh-Lynn-Schacham) scheme",(0,m.kt)("sup",{parentName:"p",id:"fnref-fourth"},(0,m.kt)("a",{parentName:"sup",href:"#fn-fourth",className:"footnote-ref"},"fourth"))," uses a bilinear pairing for verification, and signatures are elements of an elliptic curve group. Working in an elliptic curve group provides some defense against index calculus attacks (with the caveat that such attacks are still possible in the target group ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"G"),(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"_"),(0,m.kt)("mi",{parentName:"mrow"},"T")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"G\\_{T}")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.9933em",verticalAlign:"-0.31em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"G"),(0,m.kt)("span",{parentName:"span",className:"mord",style:{marginRight:"0.02778em"}},"_"),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.13889em"}},"T"))))))," of the pairing), allowing shorter signatures than other systems for similar levels of security. "),(0,m.kt)("p",null,"BLS signatures have become the subject of much work as they are seen as being a possible way forward to solve privacy issues within cryptocurrencies through a process of signature aggregation. Apache Milagro uses signature aggregation and key splitting, and further aggregation of signatures from those split keys, as a key component of the Milagro Custody Node digital asset safekeeping protocol."),(0,m.kt)("h2",{id:"supersingular-isogeny-key-encapsulation---introduction"},"Supersingular Isogeny Key Encapsulation - Introduction"),(0,m.kt)("p",null,"Supersingular isogeny Diffie\u2013Hellman key exchange (SIDH) and the key encapsulation protocol, SIKE",(0,m.kt)("sup",{parentName:"p",id:"fnref-fifth"},(0,m.kt)("a",{parentName:"sup",href:"#fn-fifth",className:"footnote-ref"},"fifth"))," ","(","derived from SIDH), are post-quantum cryptographic algorithms used to establish a secret key between two parties over an otherwise insecure communications channel. SIKE boasts one of the smallest key sizes of all post-quantum key encapsulations; with compression, SIKE uses 2688-bit public keys at a 128-bit quantum security level. "),(0,m.kt)("p",null,"SIKE also distinguishes itself from similar systems such as NTRU and Ring-LWE by supporting perfect forward secrecy, a property that prevents compromised long-term keys from compromising the confidentiality of old communication sessions. These properties make SIKE a natural candidate to replace Diffie\u2013Hellman (DHE and elliptic curve Diffie\u2013Hellman (ECDHE), which are widely used in Internet communication."),(0,m.kt)("p",null,"Since it is the only post-quantum cryptography protocol which is constructed on elliptic curves, hybrid cryptography protocols can be derived from SIKE and classical elliptic curve cryptography (ECC) to make the transition towards post-quantum cryptography more convenient and practical."),(0,m.kt)("p",null,"Milagro Custody Node implements SIKE for key encapsulation within Milagro's encrypted envelope format."),(0,m.kt)("hr",null),(0,m.kt)("table",null,(0,m.kt)("thead",{parentName:"table"},(0,m.kt)("tr",{parentName:"thead"},(0,m.kt)("th",{parentName:"tr",align:null},"Protocols"),(0,m.kt)("th",{parentName:"tr",align:null},"Use Cases"))),(0,m.kt)("tbody",{parentName:"table"},(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"M-Pin 1-Pass"),(0,m.kt)("td",{parentName:"tr",align:null},"Digital signature authentication in battery or bandwidth constrained environments such as IoT devices, embedded applications and mobile apps. ",(0,m.kt)("br",null),"This should be considered the default implementation for client to server authentication suitable for almost all use cases.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"M-Pin 1-Pass + ",(0,m.kt)("br",null),"M-Pin 2-Pass"),(0,m.kt)("td",{parentName:"tr",align:null},"Digital signature and client to server authentication in smartphones apps, desktop browsers and software applications.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"M-Pin 2-Pass"),(0,m.kt)("td",{parentName:"tr",align:null},"Client to server authentication in smartphone apps, desktop browsers and software applications.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"M-Pin FULL"),(0,m.kt)("td",{parentName:"tr",align:null},"Mutual client and server authentication with authenticated key agreement for use in smartphone apps, hardware and software applications. ",(0,m.kt)("br",null),"Authenticated Key Agreement with PFS can be used as the basis for TLS sessions between clients and servers.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"Chow-Choo"),(0,m.kt)("td",{parentName:"tr",align:null},"Mutual peer to peer authentication with authenticated key agreement for use in smartphone apps, hardware and software applications. ",(0,m.kt)("br",null),"Authenticated Key Agreement with PFS can be used as the basis for TLS sessions between clients and servers and peer to peer.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"BLS Signing + ",(0,m.kt)("br",null),"Key Splitting + ",(0,m.kt)("br",null),"Signature Aggregation"),(0,m.kt)("td",{parentName:"tr",align:null},"A simple approach for splitting keys and aggregating many BLS signatures (from the shares of keys acting as signing keys) on a common message. Public keys are not needed for verifying the multi-signature. ",(0,m.kt)("br",null),"An important property of the construction is that the scheme is secure against a rogue public-key attack without requiring users to prove knowledge of their secret keys.")),(0,m.kt)("tr",{parentName:"tbody"},(0,m.kt)("td",{parentName:"tr",align:null},"SIKE (Key Encapsulation)"),(0,m.kt)("td",{parentName:"tr",align:null},"SIKE is a family of post-quantum key encapsulation mechanisms based on the Supersingular Isogeny Diffie-Hellman (SIDH) key exchange protocol. ",(0,m.kt)("br",null),"The algorithms use arithmetic operations on elliptic curves defined over finite fields and compute maps, so-called isogenies, between such curves. ",(0,m.kt)("br",null),"SIKE can also delivery perfect forward secrecy.")))),(0,m.kt)("hr",null),(0,m.kt)("h2",{id:"protocols-in-depth"},"Protocols In Depth"),(0,m.kt)("h3",{id:"m-pin-1-pass"},"M-Pin 1-Pass"),(0,m.kt)("p",null,"As opposed to Chow-Choo, which can be used in a client to server as well as a peer to peer setting, M-Pin is strictly a client-server protocol."),(0,m.kt)("p",null,"To embellish the security of the client-server protocol, it is important that client and server secrets should be kept distinct."),(0,m.kt)("p",null,"A simple way to do this is to exploit the structure of a Type-3 pairing and put client secrets in ",(0,m.kt)("span",{parentName:"p",className:"math 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math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("msub",{parentName:"mrow"},(0,m.kt)("mi",{parentName:"msub"},"G"),(0,m.kt)("mn",{parentName:"msub"},"2"))),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"G_2")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.8333em",verticalAlign:"-0.15em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"G"),(0,m.kt)("span",{parentName:"span",className:"msupsub"},(0,m.kt)("span",{parentName:"span",className:"vlist-t 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a challenge by the server as part of the second step within the protocol, after the first step whereby the client announced her identity to the server."),(0,m.kt)("p",null,"In a later proposal, it was realised that an M-Pin 1-Pass Protocol could be obtained if the client itself derived the challenge as ",(0,m.kt)("span",{parentName:"p",className:"math 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exploit -- in any communication context there is an initiator and a responder, or a ",(0,m.kt)("em",{parentName:"p"},"sender")," and ",(0,m.kt)("em",{parentName:"p"},"receiver"),", if you will."),(0,m.kt)("p",null,"In the above example, Alice and Bob both were issued ",(0,m.kt)("em",{parentName:"p"},"sender")," and ",(0,m.kt)("em",{parentName:"p"},"receiver")," keys respectively, as this describes where they can appear in the pairing."),(0,m.kt)("p",null,"An obvious advantage is to issue each Thing with two keys, one in ",(0,m.kt)("span",{parentName:"p",className:"math 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"),(0,m.kt)("p",null,"In June of 2018 Dan Boneh, Manu Drijvers and Gregory Neven released research that constructs the first practical, short accountable-subgroup multi-signature (ASM) scheme based on BLS signatures",(0,m.kt)("sup",{parentName:"p",id:"fnref-sixth"},(0,m.kt)("a",{parentName:"sup",href:"#fn-sixth",className:"footnote-ref"},"sixth")),"."),(0,m.kt)("p",null,"An ASM scheme enables any subset ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"S")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"S")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"S")))))," of a set of ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"n")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"n")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"n")))))," parties to sign a message ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"m")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"m")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"m")))))," so that a valid signature discloses which subset generated the signature (hence the subset ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"S")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"S")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"S")))))," is accountable for signing ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"m")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"m")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.4306em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"m"))))),")."),(0,m.kt)("p",null,"In addition to the ASM scheme, Milagro exploits a unique property of BLS signatures: Signing Keys can be split using Shamir's Secret Sharing",(0,m.kt)("sup",{parentName:"p",id:"fnref-seventh"},(0,m.kt)("a",{parentName:"sup",href:"#fn-seventh",className:"footnote-ref"},"seventh"))," in which the 'shares' of the keys, when distributed to signers, themselves become signing keys. By using a 'key splitter' and 'signature aggregator' role who also performs the Shamir Secret Sharing (SSS) dealer function, several benefits emerge."),(0,m.kt)("p",null,"Thresholds can be set on the distribution of key shares just as in a normal SSS routine. So once the shares of the original signing key are distributed, the original signing key can be securely deleted, and is never re-created again. Instead, signatures derived from the shares of the original signing key are created. Again, the properties of BLS enable these shares of signature keys to be themselves used a signature keys. The object of the exercise is for the signature aggregator, who is collecting the signatures created from these shares, to complete the threshold and achieve the signature that the original signature would have created, which can be validated by its original public key component."),(0,m.kt)("p",null,"In other words, if the aggregator takes m-of-n of these signature shares, and perform the same polynomial interpolation as one would usually do with the secret shares, you\u2019ll recover a complete signature which is identical to what would have been created if the original complete private key would have been used."),(0,m.kt)("p",null,"In this context, signing is a single round protocol and is non-interactive."),(0,m.kt)("hr",null),(0,m.kt)("h3",{id:"supersingular-isogeny-key-encapsulation-sike"},"Supersingular Isogeny Key Encapsulation (SIKE)"),(0,m.kt)("p",null,"A key encapsulation mechanism (KEM) is a set of three algorithms."),(0,m.kt)("ul",null,(0,m.kt)("li",{parentName:"ul"},"key 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An isogeny with degree ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\ell")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113")))))," map is called ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\ell")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113"))))),"-isogeny. Let ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"G")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"G")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"G")))))," be a subgroup of points on ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"E")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"E")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"E")))))," which contains ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\ell")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113")))))," + 1 cyclic subgroups of order ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\ell")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113"))))),". This subgroup is the torsion group ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"E"),(0,m.kt)("mo",{parentName:"mrow",stretchy:"false"},"["),(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113"),(0,m.kt)("mo",{parentName:"mrow",stretchy:"false"},"]")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"E[\\ell]")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal",style:{marginRight:"0.05764em"}},"E"),(0,m.kt)("span",{parentName:"span",className:"mopen"},"["),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113"),(0,m.kt)("span",{parentName:"span",className:"mclose"},"]")))))," and each element of this group is corresponding to an isogeny of degree ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow",mathvariant:"normal"},"\u2113")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"\\ell")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6944em"}}),(0,m.kt)("span",{parentName:"span",className:"mord"},"\u2113"))))),"; accordingly, an isogeny also can be identified by ",(0,m.kt)("span",{parentName:"p",className:"math math-inline"},(0,m.kt)("span",{parentName:"span",className:"katex"},(0,m.kt)("span",{parentName:"span",className:"katex-mathml"},(0,m.kt)("math",{parentName:"span",xmlns:"http://www.w3.org/1998/Math/MathML"},(0,m.kt)("semantics",{parentName:"math"},(0,m.kt)("mrow",{parentName:"semantics"},(0,m.kt)("mi",{parentName:"mrow"},"G")),(0,m.kt)("annotation",{parentName:"semantics",encoding:"application/x-tex"},"G")))),(0,m.kt)("span",{parentName:"span",className:"katex-html","aria-hidden":"true"},(0,m.kt)("span",{parentName:"span",className:"base"},(0,m.kt)("span",{parentName:"span",className:"strut",style:{height:"0.6833em"}}),(0,m.kt)("span",{parentName:"span",className:"mord mathnormal"},"G"))))),", i.e., the kernel of isogeny."),(0,m.kt)("p",null,"This section provides a brief presentation of the SIKE protocol. We refer readers to ",(0,m.kt)("sup",{parentName:"p",id:"fnref-eighth"},(0,m.kt)("a",{parentName:"sup",href:"#fn-eighth",className:"footnote-ref"},"eighth"))," and ",(0,m.kt)("sup",{parentName:"p",id:"fnref-fifth"},(0,m.kt)("a",{parentName:"sup",href:"#fn-fifth",className:"footnote-ref"},"fifth"))," for more detailed explanation of the supersignular isogeny problem and the base key-exchange protocol which the SIKE is constructed upon."),(0,m.kt)("admonition",{title:"See an error in this documentation? ",type:"tip"},(0,m.kt)("p",{parentName:"admonition"},"Submit a pull request on the development branch of ",(0,m.kt)("a",{parentName:"p",href:"https://github.com/apache/incubator-milagro"},"Milagro Website Repo"),".")),(0,m.kt)("div",{className:"footnotes"},(0,m.kt)("hr",{parentName:"div"}),(0,m.kt)("ol",{parentName:"div"},(0,m.kt)("li",{parentName:"ol",id:"fn-first"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://eprint.iacr.org/2002/164"},"M-Pin protocol"),". As noted in ",(0,m.kt)("a",{parentName:"p",href:"milagro-crypto"},"Milagro Crypto"),", the M-Pin Protocol is of these classifications and exploits the features of:"),(0,m.kt)("ul",{parentName:"li"},(0,m.kt)("li",{parentName:"ul"},"Elliptic Curve Cryptography"),(0,m.kt)("li",{parentName:"ul"},"Pairing Based Cryptography"),(0,m.kt)("li",{parentName:"ul"},"Identity Based Encryption"),(0,m.kt)("li",{parentName:"ul"},"Zero Knowledge Proof")),(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"#fnref-first",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-second"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://eprint.iacr.org/2015/576"},"The Carnac protocol -- or how to read the contents of a sealed envelope"),(0,m.kt)("a",{parentName:"p",href:"#fnref-second",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-third"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://eprint.iacr.org/2007/018.pdf"},"Strongly-Secure Identity-based Key Agreement and Anonymous Extension"),(0,m.kt)("a",{parentName:"p",href:"#fnref-third",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-fourth"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://www.iacr.org/archive/asiacrypt2001/22480516.pdf"},"Short Signatures from the Weil Pairing"),(0,m.kt)("a",{parentName:"p",href:"#fnref-fourth",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-fifth"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://sike.org/"},"Supersingular isogeny key encapsulation - NIST Submission"),(0,m.kt)("a",{parentName:"p",href:"#fnref-fifth",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-ninth"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://reader.elsevier.com/reader/sd/pii/S2213020915000592?token=5C2E08C17803B9549FB79F3A91A1ED2382360D3E840087C8C065BD06EFCABB55C4A5A300566388A2920786DCC63E631E"},"A brief review of revocable ID-based public key cryptosystem"),(0,m.kt)("a",{parentName:"p",href:"#fnref-ninth",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-sixth"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://eprint.iacr.org/2018/483"},"Compact Multi-Signatures for Smaller Blockchains"),(0,m.kt)("a",{parentName:"p",href:"#fnref-sixth",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-seventh"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://cs.jhu.edu/~sdoshi/crypto/papers/shamirturing.pdf"},"How to Share a Secret"),(0,m.kt)("a",{parentName:"p",href:"#fnref-seventh",className:"footnote-backref"},"\u21a9"))),(0,m.kt)("li",{parentName:"ol",id:"fn-eighth"},(0,m.kt)("p",{parentName:"li"},(0,m.kt)("a",{parentName:"p",href:"https://eprint.iacr.org/2011/506.pdf"},"Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies"),(0,m.kt)("a",{parentName:"p",href:"#fnref-eighth",className:"footnote-backref"},"\u21a9"))))))}o.isMDXComponent=!0},6349:(a,e,t)=>{t.d(e,{Z:()=>s});const s=t.p+"assets/images/Chow-ChooProtocol-1aa57e8d61146f9a9a0741dd614493be.png"}}]);