Pub Date : 2016-03-01DOI: 10.1112/S1461157016000036
P. Rowley, Ben Wright
The point-line collinearity graph ${mathcal{G}}$ of the maximal 2-local geometry for the largest simple Fischer group, $Fi_{24}^{prime }$ , is extensively analysed. For an arbitrary vertex $a$ of ${mathcal{G}}$ , the $itext{th}$ -disc of $a$ is described in detail. As a consequence, it follows that ${mathcal{G}}$ has diameter $5$ . The collapsed adjacency matrix of ${mathcal{G}}$ is given as well as accompanying computer files which contain a wealth of data about ${mathcal{G}}$ . Supplementary materials are available with this article.
{"title":"Structure of the maximal -local geometry point-line collinearity graph","authors":"P. Rowley, Ben Wright","doi":"10.1112/S1461157016000036","DOIUrl":"https://doi.org/10.1112/S1461157016000036","url":null,"abstract":"The point-line collinearity graph ${mathcal{G}}$\u0000 of the maximal 2-local geometry for the largest simple Fischer group, $Fi_{24}^{prime }$\u0000 , is extensively analysed. For an arbitrary vertex $a$\u0000 of ${mathcal{G}}$\u0000 , the $itext{th}$\u0000 -disc of $a$\u0000 is described in detail. As a consequence, it follows that ${mathcal{G}}$\u0000 has diameter $5$\u0000 . The collapsed adjacency matrix of ${mathcal{G}}$\u0000 is given as well as accompanying computer files which contain a wealth of data about ${mathcal{G}}$\u0000 . Supplementary materials are available with this article.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"105-154"},"PeriodicalIF":0.0,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-02-16DOI: 10.1112/S146115701600022X
Tony Quertier
We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over the reals. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists, and if so, to compute one.
{"title":"Effective Hasse principle for the intersection of two quadrics","authors":"Tony Quertier","doi":"10.1112/S146115701600022X","DOIUrl":"https://doi.org/10.1112/S146115701600022X","url":null,"abstract":"We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over the reals. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists, and if so, to compute one.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"73-82"},"PeriodicalIF":0.0,"publicationDate":"2016-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701600022X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-02-11DOI: 10.1112/S146115701600019X
A. Booker, J. Sijsling, Andrew V. Sutherland, J. Voight, D. Yasaki
We describe the construction of a database of genus-$2$curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated$L$-function. This data has been incorporated into the$L$-Functions and Modular Forms Database (LMFDB).
{"title":"A database of genus-2 curves over the rational numbers","authors":"A. Booker, J. Sijsling, Andrew V. Sutherland, J. Voight, D. Yasaki","doi":"10.1112/S146115701600019X","DOIUrl":"https://doi.org/10.1112/S146115701600019X","url":null,"abstract":"<jats:p>We describe the construction of a database of genus-<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701600019X_inline1\" /><jats:tex-math>$2$</jats:tex-math></jats:alternatives></jats:inline-formula>curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701600019X_inline2\" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>-function. This data has been incorporated into the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S146115701600019X_inline3\" /><jats:tex-math>$L$</jats:tex-math></jats:alternatives></jats:inline-formula>-Functions and Modular Forms Database (LMFDB).</jats:p>","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"11 1","pages":"235-254"},"PeriodicalIF":0.0,"publicationDate":"2016-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701600019X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-02-05DOI: 10.1112/S1461157016000188
Abhinav Kumar, R. E. Mukamel
We describe a method to compute equations for real multiplica- tion on the divisors of genus two curves via algebraic correspondences. We implement our method for various examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies{Kumar. We also compute a correspondence over the universal family over the Hilbert modular surface of discriminant 5 and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.
{"title":"Real multiplication through explicit correspondences","authors":"Abhinav Kumar, R. E. Mukamel","doi":"10.1112/S1461157016000188","DOIUrl":"https://doi.org/10.1112/S1461157016000188","url":null,"abstract":"We describe a method to compute equations for real multiplica- tion on the divisors of genus two curves via algebraic correspondences. We implement our method for various examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies{Kumar. We also compute a correspondence over the universal family over the Hilbert modular surface of discriminant 5 and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"29-42"},"PeriodicalIF":0.0,"publicationDate":"2016-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000188","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157015000339
N. M. Khan
In recent years, Com–Poisson has emerged as one of the most popular discrete models in the analysis of count data owing to its flexibility in handling different types of dispersion. However, in a stationary longitudinal Com–Poisson count data set-up where the covariates are time independent, estimation of regression and dispersion parameters based on a generalized quasi-likelihood (GQL) approach involves some major computational difficulties particularly in the inversion of the joint covariance matrix. On the other hand, in practical real-life longitudinal studies, time-dependent covariates leading to non-stationary responses are more frequently encountered. This implies that further computational problems will now arise when estimating parameters under non-stationary set-ups. This paper overcomes this problem by approximating the inverse of the ill-conditioned covariance matrix in the GQL approach through a multidimensional conjugate gradient method. The performance of this novel version of the GQL approach is then assessed on simulations of AR(1) stationary and AR(1) non-stationary longitudinal Com–Poisson counts and on real-life epileptic seizure counts. However, there is not yet an algorithm to generate non-stationary longitudinal Com–Poisson counts nor a GQL algorithm to estimate the parameters under non-stationary set-ups. Thus, the paper also provides a framework to generate non-stationary AR(1) Com–Poisson counts along with the construction of a GQL equation under non-stationary set-ups.
近年来，康-泊松模型由于其处理不同类型色散的灵活性而成为计数数据分析中最流行的离散模型之一。然而，在协变量与时间无关的平稳纵向com -泊松计数数据设置中，基于广义拟似然(GQL)方法的回归和分散参数估计涉及一些主要的计算困难，特别是在联合协方差矩阵的反演中。另一方面，在实际的纵向研究中，导致非平稳响应的时间相关协变量更常见。这意味着在非平稳设置下估计参数时将出现进一步的计算问题。本文通过一种多维共轭梯度法逼近GQL方法中病态协方差矩阵的逆，克服了这一问题。然后通过模拟AR(1)平稳和AR(1)非平稳纵向como - poisson计数以及实际癫痫发作计数来评估这种新型GQL方法的性能。然而，目前还没有一种算法来产生非平稳的纵向com -泊松计数，也没有一种GQL算法来估计非平稳设置下的参数。因此，本文还提供了一个框架来生成非平稳AR(1) com -泊松计数以及非平稳设置下GQL方程的构造。
{"title":"A robust algorithm for estimating regression and dispersion parameters in non-stationary longitudinally correlated Com–Poisson data","authors":"N. M. Khan","doi":"10.1112/S1461157015000339","DOIUrl":"https://doi.org/10.1112/S1461157015000339","url":null,"abstract":"In recent years, Com–Poisson has emerged as one of the most popular discrete models in the analysis of count data owing to its flexibility in handling different types of dispersion. However, in a stationary longitudinal Com–Poisson count data set-up where the covariates are time independent, estimation of regression and dispersion parameters based on a generalized quasi-likelihood (GQL) approach involves some major computational difficulties particularly in the inversion of the joint covariance matrix. On the other hand, in practical real-life longitudinal studies, time-dependent covariates leading to non-stationary responses are more frequently encountered. This implies that further computational problems will now arise when estimating parameters under non-stationary set-ups. This paper overcomes this problem by approximating the inverse of the ill-conditioned covariance matrix in the GQL approach through a multidimensional conjugate gradient method. The performance of this novel version of the GQL approach is then assessed on simulations of AR(1) stationary and AR(1) non-stationary longitudinal Com–Poisson counts and on real-life epileptic seizure counts. However, there is not yet an algorithm to generate non-stationary longitudinal Com–Poisson counts nor a GQL algorithm to estimate the parameters under non-stationary set-ups. Thus, the paper also provides a framework to generate non-stationary AR(1) Com–Poisson counts along with the construction of a GQL equation under non-stationary set-ups.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"25-36"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000339","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157016000243
Tom Fisher
We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.
{"title":"Visualizing elements of order in the Tate–Shafarevich group of an elliptic curve","authors":"Tom Fisher","doi":"10.1112/S1461157016000243","DOIUrl":"https://doi.org/10.1112/S1461157016000243","url":null,"abstract":"We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"100-114"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000243","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157016000280
W. Castryck, Ilia Iliashenko, F. Vercauteren
Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus q and degree n number field K, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod q of a certain fractional ideal O-K(V) subset of K called the codifferent or 'dual', rather than from the ring of integers O-K itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by vertical bar Delta(K)vertical bar(1/2n) with Delta(K) the discriminant of K. As a main result, we provide, for any epsilon > 0, a family of number fields K for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by vertical bar Delta(K)vertical bar((1-epsilon)/n).
自2010年由Lyubashevsky, Peikert和Regev引入以来，带错误的环学习问题(ring- lwe)由于其强大的通用性和由理想晶格问题组成的(量子)约简的硬度证明，已经成为加密原语的流行构建块。但是，对于给定的模数q和n度数域K，生成环lwe样本可能会被认为是麻烦的，因为密钥必须从K的某个分数理想O-K(V)子集的约简模q中提取，称为协差或“对偶”，而不是从整数O-K本身的环中提取。这导致了环lwe的各种非对偶变体，其中通过放大误差来补偿非对偶性。我们给这些版本的比较,重新审视一些不幸的选择已经在最近的文献中,其中之一是扩大由竖线δ(K)竖线(1/2n)δ(K) K .作为一个主要的判别结果,我们提供,对于任何ε> 0,一个家庭的字段数K的变体ring-LWE就可以被很容易的错误是由竖线δ(K)扩大竖线((1 -ε)/ n)。
{"title":"On error distributions in ring-based LWE","authors":"W. Castryck, Ilia Iliashenko, F. Vercauteren","doi":"10.1112/S1461157016000280","DOIUrl":"https://doi.org/10.1112/S1461157016000280","url":null,"abstract":"Since its introduction in 2010 by Lyubashevsky, Peikert and Regev, the ring learning with errors problem (ring-LWE) has become a popular building block for cryptographic primitives, due to its great versatility and its hardness proof consisting of a (quantum) reduction from ideal lattice problems. But, for a given modulus q and degree n number field K, generating ring-LWE samples can be perceived as cumbersome, because the secret keys have to be taken from the reduction mod q of a certain fractional ideal O-K(V) subset of K called the codifferent or 'dual', rather than from the ring of integers O-K itself. This has led to various non-dual variants of ring-LWE, in which one compensates for the non-duality by scaling up the errors. We give a comparison of these versions, and revisit some unfortunate choices that have been made in the recent literature, one of which is scaling up by vertical bar Delta(K)vertical bar(1/2n) with Delta(K) the discriminant of K. As a main result, we provide, for any epsilon > 0, a family of number fields K for which this variant of ring-LWE can be broken easily as soon as the errors are scaled up by vertical bar Delta(K)vertical bar((1-epsilon)/n).","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"130-145"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000280","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157016000115
A. Bucur, A. Ernvall-Hytönen, Almasa Odžak, L. Smajlovic
The Li coefficients $unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $text{Re}(z)>unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $unicode[STIX]{x1D70F}$ aspects.
{"title":"On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients","authors":"A. Bucur, A. Ernvall-Hytönen, Almasa Odžak, L. Smajlovic","doi":"10.1112/S1461157016000115","DOIUrl":"https://doi.org/10.1112/S1461157016000115","url":null,"abstract":"The Li coefficients $unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $text{Re}(z)>unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $unicode[STIX]{x1D70F}$ aspects.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"259-280"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157016000371
J. Cheon, Jinhyuck Jeong, Changmin Lee
Let f and g be polynomials of a bounded Euclidean norm in the ring Z[X]/⟨X+1⟩. Given the polynomial [f/g]q ∈ Zq[X]/⟨X+1⟩, the NTRU problem is to find a, b ∈ Z[X]/⟨X + 1⟩ with a small Euclidean norm such that [a/b]q = [f/g]q. We propose an algorithm to solve the NTRU problem, which runs in 2 2 λ) time when ∥g∥, ∥f∥, and ∥g−1∥ are within some range. The main technique of our algorithm is the reduction of a problem on a field to one in a subfield. Recently, the GGH scheme, the first candidate of a (approximate) multilinear map, was found to be insecure by the Hu–Jia attack using low-level encodings of zero, but no polynomial-time attack was known without them. In the GGH scheme without low-level encodings of zero, our algorithm can be directly applied to attack this scheme if we have some top-level encodings of zero and a known pair of plaintext and ciphertext. Using our algorithm, we can construct a level-0 encoding of zero and utilize it to attack a security ground of this scheme in the quasi-polynomial time of its security parameter using the parameters suggested by [GGH13].
{"title":"An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero","authors":"J. Cheon, Jinhyuck Jeong, Changmin Lee","doi":"10.1112/S1461157016000371","DOIUrl":"https://doi.org/10.1112/S1461157016000371","url":null,"abstract":"Let f and g be polynomials of a bounded Euclidean norm in the ring Z[X]/⟨X+1⟩. Given the polynomial [f/g]q ∈ Zq[X]/⟨X+1⟩, the NTRU problem is to find a, b ∈ Z[X]/⟨X + 1⟩ with a small Euclidean norm such that [a/b]q = [f/g]q. We propose an algorithm to solve the NTRU problem, which runs in 2 2 λ) time when ∥g∥, ∥f∥, and ∥g−1∥ are within some range. The main technique of our algorithm is the reduction of a problem on a field to one in a subfield. Recently, the GGH scheme, the first candidate of a (approximate) multilinear map, was found to be insecure by the Hu–Jia attack using low-level encodings of zero, but no polynomial-time attack was known without them. In the GGH scheme without low-level encodings of zero, our algorithm can be directly applied to attack this scheme if we have some top-level encodings of zero and a known pair of plaintext and ciphertext. Using our algorithm, we can construct a level-0 encoding of zero and utilize it to attack a security ground of this scheme in the quasi-polynomial time of its security parameter using the parameters suggested by [GGH13].","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"255-266"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000371","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.1112/S1461157016000231
H. Heer, G. McGuire, Oisín Robinson
We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion $mathbb{Z}/6mathbb{Z}oplus mathbb{Z}/6mathbb{Z}$ and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over $mathbb{Q}$ and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion $mathbb{Z}/2mathbb{Z}oplus mathbb{Z}/8mathbb{Z}$ of Bernstein et al. are completely recovered by the $mathbb{Z}/4mathbb{Z}oplus mathbb{Z}/8mathbb{Z}$ family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.
{"title":"JKL-ECM: an implementation of ECM using Hessian curves","authors":"H. Heer, G. McGuire, Oisín Robinson","doi":"10.1112/S1461157016000231","DOIUrl":"https://doi.org/10.1112/S1461157016000231","url":null,"abstract":"We present JKL-ECM, an implementation of the elliptic curve method of integer factorization which uses certain twisted Hessian curves in a family studied by Jeon, Kim and Lee. This implementation takes advantage of torsion subgroup injection for families of elliptic curves over a quartic number field, in addition to the ‘small parameter’ speedup. We produced thousands of curves with torsion $mathbb{Z}/6mathbb{Z}oplus mathbb{Z}/6mathbb{Z}$ and small parameters in twisted Hessian form, which admit curve arithmetic that is ‘almost’ as fast as that of twisted Edwards form. This allows JKL-ECM to compete with GMP-ECM for finding large prime factors. Also, JKL-ECM, based on GMP, accepts integers of arbitrary size. We classify the torsion subgroups of Hessian curves over $mathbb{Q}$ and further examine torsion properties of the curves described by Jeon, Kim and Lee. In addition, the high-performance curves with torsion $mathbb{Z}/2mathbb{Z}oplus mathbb{Z}/8mathbb{Z}$ of Bernstein et al. are completely recovered by the $mathbb{Z}/4mathbb{Z}oplus mathbb{Z}/8mathbb{Z}$ family of Jeon, Kim and Lee, and hundreds more curves are produced besides, all with small parameters and base points.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"83-99"},"PeriodicalIF":0.0,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000231","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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