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Faster Squaring in the Cyclotomic Subgroup of Sixth Degree Extensions [chapter]

Robert Granger, Michael Scott
<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;
This paper describes an extremely efficient squaring operation in the so-called 'cyclotomic subgroup' of F × q 6 , for q ≡ 1 mod 6.  ...  In particular we argue that such fields are ideally suited for the latter when the field characteristic satisfies p ≡ 1 (mod 6), and since torus-based techniques can be applied to the former, we present  ...  Acknowledgements The authors would like to thank Martijn Stam for answering some questions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-13013-7_13">doi:10.1007/978-3-642-13013-7_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oorlcvgkujbvre3znrsmbsjrby">fatcat:oorlcvgkujbvre3znrsmbsjrby</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20150429210300/http://www.iacr.org/archive/pkc2010/60560212/60560212.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/db/9f/db9f7068d7709f9b13e75300aab9080fdcdafb1c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-13013-7_13"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions [chapter]

Martijn Stam, Arjen K. Lenstra
<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;
This paper describes several speedups for computation in the order p + 1 subgroup of F * p 2 and the order p 2 − p + 1 subgroup of F * p 6 .  ...  These results are in a way complementary to LUC and XTR, where computations in these groups are sped up using trace maps. As a side result, we present an efficient method for XTR with p ≡ 3 mod 4.  ...  Special thanks go to Peter Beelen, whose remarks stimulated this research and improved the group arithmetic for quadratic extensions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36400-5_24">doi:10.1007/3-540-36400-5_24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/hvhn7zbw5nbzpe6yzam4hejtfe">fatcat:hvhn7zbw5nbzpe6yzam4hejtfe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170706154844/https://infoscience.epfl.ch/record/149725/files/EPFL-CONF-149725.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/ce/5a/ce5af00fbaf25ae6ab45327fc1805f9bb705d3d8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36400-5_24"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Discrete Logarithm Variants of VSH [chapter]

Arjen K. Lenstra, Daniel Page, Martijn Stam
<span title="">2006</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 article we explore a variant of VSH-DL that is based on cyclotomic subgroups of finite fields; we show that one can tradeoff performance against bandwidth by using known techniques in such groups  ...  Further, we investigate a variant of VSH-DL based on elliptic curves and extract a tighter reduction to the underlying DLP in comparison to the original VSH-DL proposal.  ...  To address this question, in this paper we investigate VSH-DL type schemes based on the cyclotomic subgroup of a sixth degree extension field and on elliptic curves.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11958239_15">doi:10.1007/11958239_15</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pjm6idipe5eahk5udv4pign3gm">fatcat:pjm6idipe5eahk5udv4pign3gm</a> </span>
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Squaring in cyclotomic subgroups

Koray Karabina
<span title="2012-06-27">2012</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5bz4zmidbngqxk6yv4msbkm54u" style="color: black;">Mathematics of Computation</a> </i> &nbsp;
We propose new squaring formulae for cyclotomic subgroups of certain finite fields.  ...  The squaring formulae lead to new exponentiation algorithms in cyclotomic subgroups which outperform the fastest previously-known exponentiation algorithms when the exponent has low Hamming weight.  ...  Acknowledgment The author would like to thank Diego F. Aranha and Alfred Menezes for fruitful discussion and for their useful comments on the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0025-5718-2012-02625-1">doi:10.1090/s0025-5718-2012-02625-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lgfu5wljpfbfjeu7twn2y4czju">fatcat:lgfu5wljpfbfjeu7twn2y4czju</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170705060713/http://cacr.uwaterloo.ca/techreports/2010/cacr2010-29.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/04/3c/043cd92baffd451c7cc8c54013a69e6258b7fb2a.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0025-5718-2012-02625-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

A Taxonomy of Pairing-Friendly Elliptic Curves

David Freeman, Michael Scott, Edlyn Teske
<span title="2009-06-18">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7jmi32x7o5hfnpkgh2irnuqpim" style="color: black;">Journal of Cryptology</a> </i> &nbsp;
In this paper we give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature.  ...  We also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees.  ...  We call the degree of this extension the "embedding degree" of E. Definition 2.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00145-009-9048-z">doi:10.1007/s00145-009-9048-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gwawq4j23refpe74ucw4oqjfsy">fatcat:gwawq4j23refpe74ucw4oqjfsy</a> </span>
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Parallelizing the Weil and Tate Pairings [chapter]

Diego F. Aranha, Edward Knapp, Alfred Menezes, Francisco Rodríguez-Henríquez
<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;
These pairings are faster than previous variants of the Weil pairing proposed in [47] and [27].  ...  Our experimental results suggest that the new Weil pairing is 1.25 times faster than the optimal ate pairing on 8-core extensions of the aforementioned machines.  ...  The advantage of this 'delayed squaring' trick is that α belongs to the order-(p 4 − p 2 + 1) cyclotomic subgroup of F * p 12 whence Karabina's squaring method [31] (see also [23] ) can be deployed  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-25516-8_17">doi:10.1007/978-3-642-25516-8_17</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3eijkeemarebpesps6hn4ycue4">fatcat:3eijkeemarebpesps6hn4ycue4</a> </span>
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Computing isomorphisms and embeddings of finite fields [article]

Ludovic Brieulle, Luca De Feo, Javad Doliskani, Jean-Pierre Flori and Éric Schost
<span title="2017-05-03">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We also implement most of the presented algorithms, compare them with the state of the art computer algebra software, and make the code available as open source.  ...  When deg f = deg g, this is also known as the isomorphism problem. This problem, a special instance of polynomial factorization, plays a central role in computer algebra software.  ...  In Figure 2c we group runs of the cyclotomic algorithm by the degree s of the auxiliary extension, and we plot median times against the degree r; only the graphs for s < 10 are shown in the figure.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1705.01221v1">arXiv:1705.01221v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ye2s2hibjvapbjcmhubrlggc5u">fatcat:ye2s2hibjvapbjcmhubrlggc5u</a> </span>
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Genus 2 Hyperelliptic Curve Families with Explicit Jacobian Order Evaluation and Pairing-Friendly Constructions [chapter]

Aurore Guillevic, Damien Vergnaud
<span title="">2013</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;
The use of elliptic and hyperelliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve.  ...  His approach is to obtain candidates for the zeta function of the Jacobian over F * q from its zeta function over an extension field where the Jacobian splits.  ...  Acknowledgments This work was supported in part by the French ANR-09-VERS-016 BEST Project and by the Commission of the European Communities through the ICT program under contract ICT-2007-216676 ECRYPT  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-36334-4_16">doi:10.1007/978-3-642-36334-4_16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cdp3jvavxfhr3h37fmz4c5pjvi">fatcat:cdp3jvavxfhr3h37fmz4c5pjvi</a> </span>
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Unbelievable Security Matching AES Security Using Public Key Systems [chapter]

Arjen K. Lenstra
<span title="">2001</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;
Despite the conclusions, this paper should not be interpreted as an endorsement of any particular public key system in favor of any other.  ...  Given a desired level of security for the AES, this paper discusses matching public key sizes for RSA and the ElGamal family of protocols.  ...  The author thanks Eric Verheul and Mike Wiener for their many insightful comments on earlier versions of this paper and Martijn Stam for his assistance with Figure 1 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-45682-1_5">doi:10.1007/3-540-45682-1_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2uc5yj5upncjbol6clxsdsygsm">fatcat:2uc5yj5upncjbol6clxsdsygsm</a> </span>
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The XTR Public Key System [chapter]

Arjen K. Lenstra, Eric R. Verheul
<span title="">2000</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;
This paper introduces the XTR public key system. XTR is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field.  ...  Application of XTR in cryptographic protocols leads to substantial savings both in communication and computational overhead without compromising security.  ...  We are greatly indebted to Mike Wiener for his permission to include his improvements of our earlier versions of Algorithms 2.37 and 2.48.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-44598-6_1">doi:10.1007/3-540-44598-6_1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fw6umabhqvazpj67gfon6htnt4">fatcat:fw6umabhqvazpj67gfon6htnt4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110912131103/http://infoscience.epfl.ch/record/149712/files/EPFL-CONF-149712.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/e5/33/e533be003f8b70d772932eead3bce3a9f927dbb0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-44598-6_1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

NEON Implementation of an Attribute-Based Encryption Scheme [chapter]

Ana Helena Sánchez, Francisco Rodríguez-Henríquez
<span title="">2013</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;
faster than the best timings previously reported on ARM platforms at this level of security.  ...  Some of the applications foreseen for this protocol lie in the context of mobile devices such a smartphones and tablets, which in a majority of instances are powered by an ARM processor supporting the  ...  for benchmarking our software in the Exynos 5 Cortex-A15 processor.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-38980-1_20">doi:10.1007/978-3-642-38980-1_20</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/a2mgs7s46jbj3a5whty22m65eq">fatcat:a2mgs7s46jbj3a5whty22m65eq</a> </span>
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Factor-4 and 6 compression of cyclotomic subgroups of and

Koray Karabina
<span title="">2010</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 pairing values lie in certain prime-order subgroups of the cyclotomic subgroups of orders 2 2m C 1 and 3 2m 3 m C 1, respectively, of the multiplicative groups F 2 4m and F 3 6m .  ...  Moreover, we present and compare several algorithms for performing exponentiation in the prime-order subgroups using the compressed representations.  ...  I also thank the two anonymous referees for their detailed and very helpful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/jmc.2010.001">doi:10.1515/jmc.2010.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dubwlxi66negxexlkp5zxscrse">fatcat:dubwlxi66negxexlkp5zxscrse</a> </span>
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The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms

Ueli M. Maurer, Stefan Wolf
<span title="">1999</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
logarithms in G to breaking the Di e-Hellman protocol in G and has complexity p maxf (p i )g (log jGj) O(1) , where (p) stands for the minimum of the set of largest prime factors of all the numbers d  ...  First, it is shown that in a cyclic group G of order jGj = Q p ei i , where all the multiple prime factors of jGj are polynomial in log jGj, there exists an algorithm that reduces the computation of discrete  ...  Acknowledgment The authors thank Dan Boneh, Dima Grigoriev, Hendrik Lenstra, Markus Metzger, Victor Shoup, and Igor Shparlinsky for interesting discussions on the subject of this paper, and two anonymous  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/s0097539796302749">doi:10.1137/s0097539796302749</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dpbheab4lnbxbdwjwirzbqhvge">fatcat:dpbheab4lnbxbdwjwirzbqhvge</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190218213112/http://pdfs.semanticscholar.org/10ce/d17f7db012e5f41d007c655d1e81ac3a55e1.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/10/ce/10ced17f7db012e5f41d007c655d1e81ac3a55e1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/s0097539796302749"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Exponentiating in Pairing Groups [chapter]

Joppe W. Bos, Craig Costello, Michael Naehrig
<span title="">2014</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;
curve representations for the non-pairing group operations in protocols.  ...  We study exponentiations in pairing groups for the most common security levels and show that, although the Weierstrass model is preferable for pairing computation, it can be worthwhile to map to alternative  ...  We thank the reviewer who pointed out that having 4 | #E(K) and #K ≡ 1 mod 4 is sufficient to write E/K in twisted Edwards form.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-43414-7_22">doi:10.1007/978-3-662-43414-7_22</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fohleebcbjfefitqex4mcugmly">fatcat:fohleebcbjfefitqex4mcugmly</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180729035801/https://link.springer.com/content/pdf/10.1007%2F978-3-662-43414-7_22.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/f5/78/f5788e225938ab14ab64d882313fd6e98754c060.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-662-43414-7_22"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Attractive Subfamilies of BLS Curves for Implementing High-Security Pairings [chapter]

Craig Costello, Kristin Lauter, Michael Naehrig
<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;
Curves from these particular subfamilies are defined over prime fields that support very efficient towering options for the full extension field.  ...  This paper shows that particular choices of the pairing-friendly search parameter give rise to four subfamilies of BLS curves, all of which offer highly efficient and implementationfriendly pairing instantiations  ...  Another alternative is to compute the coefficient by the algorithm described in [24] . Since r( x 0 ) | n(x 0 ), there is a subgroup of E(F p ) of prime order r(x 0 ).  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20150310141857/http://cryptosith.org/papers/blsfriendly-20111005.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/9d/cf/9dcf16e6d22f4055a1e5d438ddbe2eedfa546fe0.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-25578-6_23"> <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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