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Key Recovery on Hidden Monomial Multivariate Schemes [chapter]

Pierre-Alain Fouque, Gilles Macario-Rat, Jacques Stern
<i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5fby7eyrnbdfrnfd5ib4lkc2gu" style="color: black;">Advances in Cryptology – EUROCRYPT 2008</a> </i> &nbsp;
Here we show that if the hidden multivariate system is the projection of a quadratic monomial on a base finite field, as in C * , or a cubic (or higher) monomial as in the traitor tracing scheme, then  ...  In this paper, we study the key recovery problem for the C * scheme and generalisations where the quadratic monomial of C * (the product of two linearized monomials) is replaced by a product of three or  ...  Acknowledgements The first and last authors are very grateful to Adi Shamir and Vivien Dubois for interesting discussions on this subject.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-78967-3_2">doi:10.1007/978-3-540-78967-3_2</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/eurocrypt/FouqueMS08.html">dblp:conf/eurocrypt/FouqueMS08</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xfkbz7c2ejhvfju4ychsetk4hq">fatcat:xfkbz7c2ejhvfju4ychsetk4hq</a> </span>
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Small Public Keys and Fast Verification for $\mathcal{M}$ ultivariate $\mathcal{Q}$ uadratic Public Key Systems [chapter]

Albrecht Petzoldt, Enrico Thomae, Stanislav Bulygin, Christopher Wolf
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We do so by combining the previously unrelated concepts of reduced and equivalent keys. Our new scheme is based on the so-called Unbalanced Oil and Vinegar class of MQ-schemes.  ...  In this paper, we show how post-quantum signature systems based on Multivariate Quadratic (MQ) polynomials can be improved up by about 9/10, and 3/5, respectively, in terms of public key size and verification  ...  We thank Ishtiaq Shah for doing the implementation of our scheme. Furthermore we want to thank our financial supporters.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-23951-9_31">doi:10.1007/978-3-642-23951-9_31</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jrr5mzsnrvfg5kig7jbxdmx5tm">fatcat:jrr5mzsnrvfg5kig7jbxdmx5tm</a> </span>
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Algebraic Cryptanalysis of Hidden Field Equation (HFE) Cryptosystems Using Gröbner Bases [chapter]

Jean-Charles Faugère, Antoine Joux
<span title="">2003</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family.  ...  We also revisit the security estimates for existing schemes in the HFE family. D. Boneh (Ed.): CRYPTO  ...  General Description of Multivariate Cryptosystems The basic idea of multivariate cryptosystems from the HFE family is to build the secret key on a polynomial S in one unknown x over some finite field (  ... 
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On the Differential Security of the HFEv- Signature Primitive [chapter]

Ryann Cartor, Ryan Gipson, Daniel Smith-Tone, Jeremy Vates
<span title="">2016</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Multivariate Public Key Cryptography (MPKC) is one of the most attractive post-quantum options for digital signatures in a wide array of applications.  ...  The history of multivariate signature schemes is tumultuous, however, and solid security arguments are required to inspire faith in the schemes and to verify their security against yet undiscovered attacks  ...  key recovery attack.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-29360-8_11">doi:10.1007/978-3-319-29360-8_11</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tiguevch7ffs7c42qllqelub3m">fatcat:tiguevch7ffs7c42qllqelub3m</a> </span>
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Equivalent keys in ℳultivariate uadratic public key systems

Christopher Wolf, Bart Preneel
<span title="">2011</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/assei3huevd3bfn4hgcs7we3ji" style="color: black;">Journal of Mathematical Cryptology</a> </i> &nbsp;
Multivariate Quadratic public key schemes have been suggested as early as 1985 by Matsumoto and Imai as an alternative for the RSA scheme.  ...  Since then, several schemes have been proposed, for example Hidden Field Equations, Unbalanced Oil and Vinegar schemes, and Stepwise Triangular Schemes.  ...  Preneel there exists an additive sustainer in the case of Hidden Field Equations [26] but did not extend his result to other Multivariate Quadratic schemes.  ... 
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Improved Attacks for Characteristic-2 Parameters of the Cubic ABC Simple Matrix Encryption Scheme [chapter]

Dustin Moody, Ray Perlner, Daniel Smith-Tone
<span title="">2017</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
In the last few years multivariate public key cryptography has experienced an infusion of new ideas for encryption.  ...  hidden by an isomorphism of polynomials.  ...  Conclusion The ABC schemes offer an interesting new technique for the construction of multivariate pub lic key schemes.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-59879-6_15">doi:10.1007/978-3-319-59879-6_15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m2sdxehkuvbxfav2xae3ii4uzy">fatcat:m2sdxehkuvbxfav2xae3ii4uzy</a> </span>
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A family of weak keys in HFE and the corresponding practical key-recovery

Charles Bouillaguet, Pierre-Alain Fouque, Antoine Joux, Joana Treger
<span title="">2012</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/assei3huevd3bfn4hgcs7we3ji" style="color: black;">Journal of Mathematical Cryptology</a> </i> &nbsp;
The HFE (Hidden Field Equations) cryptosystem is one of the most interesting public-key multivariate scheme.  ...  Even though for schemes such as SFLASH or C * the hardness of key-recovery relies on the hardness of the IP problem, this is normally not the case for HFE, since the internal polynomial is kept secret.  ...  Hidden Field Equations The HFE scheme was designed in [29] by Patarin. Notice that specific variations of HFE do exist, but we will focus on the basic HFE scheme.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/jmc.2011.012">doi:10.1515/jmc.2011.012</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4qbffxamcrg6hgbcld6b3pehmi">fatcat:4qbffxamcrg6hgbcld6b3pehmi</a> </span>
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HFERP - A New Multivariate Encryption Scheme [chapter]

Yasuhiko Ikematsu, Ray Perlner, Daniel Smith-Tone, Tsuyoshi Takagi, Jeremy Vates
<span title="">2018</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The scheme suffered from the same MinRank weakness that has allowed effective attacks on several notable big field multivariate schemes: Hidden Field Equations (HFE), multi-HFE, HFE-, for example.  ...  In 2016, Yasuda et al. presented a new multivariate encryption technique based on the Square and Rainbow primitives and utilizing the plus modifier that they called Square Rainbow Plus (SRP).  ...  The purpose of this article is to confront this challenge, advancing a new multivariate encryption scheme Hidden Field Equations Rainbow Plus (HFERP), based on Square Rainbow Plus (SRP), see [3] , developed  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-79063-3_19">doi:10.1007/978-3-319-79063-3_19</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k4dnsr6vsbc7bofbifkttr3npm">fatcat:k4dnsr6vsbc7bofbifkttr3npm</a> </span>
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Cryptanalysis of the Hidden Matrix Cryptosystem [chapter]

Jean-Charles Faugère, Antoine Joux, Ludovic Perret, Joana Treger
<span title="">2010</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Until now, this scheme was exempt from cryptanalysis. We first present a distinguisher which uses a differential property of the public key.  ...  This distinguisher permits to break one perturbed version of HM. After that, we describe a practical message-recovery attack against HM using Gröbner bases.  ...  multivariate public-key schemes suffer from obvious to less obvious weaknesses ( [17, 13, 9, 7] for instance).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-14712-8_15">doi:10.1007/978-3-642-14712-8_15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gtxwk5rcgnbsfjkn6uoplmhrfu">fatcat:gtxwk5rcgnbsfjkn6uoplmhrfu</a> </span>
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Implementing 128-Bit Secure MPKC Signatures

Ming-Shing CHEN, Wen-Ding LI, Bo-Yuan PENG, Bo-Yin YANG, Chen-Mou CHENG
<span title="">2018</span> <i title="Institute of Electronics, Information and Communications Engineers (IEICE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/x2mklbxoqjfvjdjrdybgo53sdm" style="color: black;">IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences</a> </i> &nbsp;
Multivariate Public Key Cryptosystems (MPKCs) are often touted as future-proofing against Quantum Computers.  ...  We also present new techniques on evaluating multivariate polynomials, multiplications of large finite fields by additive Fast Fourier Transforms, and constant time linear solvers.  ...  Defeating a bipolar-form MPKC through solving an EIP is known as a "structural" or Key-Recovery attack.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1587/transfun.e101.a.553">doi:10.1587/transfun.e101.a.553</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/52czw23azzhb3foawn3znev444">fatcat:52czw23azzhb3foawn3znev444</a> </span>
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Roots of Square: Cryptanalysis of Double-Layer Square and Square+ [chapter]

Enrico Thomae, Christopher Wolf
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic elds and also X 2 as its central map.  ...  Both attacks recover the private key, given access to the public key. We also outline how possible variants such as Square or multi-Square can be attacked.  ...  Algorithm Attack Complexity over Double-Layer Square Key Recovery (n + )q +1 (2n + ) 3 Fq Square+ Key Recovery n+ +p 2 3 F q n+ As we have established a strong link between odd characteristic Hidden  ... 
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Algebraic Cryptanalysis of a Quantum Money Scheme The Noise-Free Case [chapter]

Marta Conde Pena, Jean-Charles Faugère, Ludovic Perret
<span title="">2015</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We emphasize that [1] proposes a non-noisy and a noisy version of the public-key quantum money scheme. The noisy version of the quantum money scheme remains secure.  ...  This problem underlies the security of the first public-key quantum money scheme that is proved to be cryptographically secure under a non quantum but classic hardness assumption.  ...  The private key of this scheme is a subspace A ⊂ F q n , and the polynomials p 1 , . . . , p m ∈ F q [x] (vanishing on A) and q 1 , . . . , q m ∈ F q [x] (vanishing on A ⊥ ) are the public key.  ... 
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Differential Properties of the HFE Cryptosystem [chapter]

Taylor Daniels, Daniel Smith-Tone
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Multivariate Public Key Cryptography (MPKC) has been put forth as a possible post-quantum family of cryptographic schemes.  ...  The effectiveness of differential attacks on various field-based systems has prompted the investigation of differential properties of multivariate schemes to determine the extent to which they are secure  ...  Along with rank attacks, differential attacks have made the greatest impact on MPKC among structural key recovery attacks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-11659-4_4">doi:10.1007/978-3-319-11659-4_4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ecasldrfgnexbe73xkdtj6fwma">fatcat:ecasldrfgnexbe73xkdtj6fwma</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190227155154/http://pdfs.semanticscholar.org/9263/cc7b3efc5a98d33f179710804b2201ec3969.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/92/63/9263cc7b3efc5a98d33f179710804b2201ec3969.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-11659-4_4"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Security Analysis and Key Modification for ZHFE [chapter]

Ray Perlner, Daniel Smith-Tone
<span title="">2016</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
., is one of the few promising candidates for a multivariate public-key encryption algorithm. In this article we extend and expound upon the existing security analysis on this scheme.  ...  We further suggest a modification, ZHF E − , a multivariate encryption scheme which retains the security and performance properties of ZHF E while optimizing key size in this theoretical framework.  ...  Differential Invariant As exemplified in [12] and [30] , invariant relations on the differential of a public key can be exploited in key recovery.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190303135237/http://pdfs.semanticscholar.org/b68a/ed4316522110a9de49173d9f6d3a52b3a422.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/b6/8a/b68aed4316522110a9de49173d9f6d3a52b3a422.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-29360-8_13"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Cryptanalysis of Multivariate and Odd-Characteristic HFE Variants [chapter]

Luk Bettale, Jean-Charles Faugère, Ludovic Perret
<span title="">2011</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
First, we propose an improved version of the basic Kipnis-Shamir key recovery attack against HFE. Second, we generalize the Kipnis-Shamir attack to Multi-HFE.  ...  We investigate the security of a generalization of HFE (multivariate and odd-characteristic variants).  ...  Introduction Multivariate Public-Key Cryptography (MPKC) is the set of public-key schemes using multivariate polynomials.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20151019140953/http://www-salsa.lip6.fr/%7Ejcf/Papers/pkc2011a.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c2/fd/c2fda059e2ad7c65018ce58fce39b7defaa02a17.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-19379-8_27"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>
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