A Variant of the XL Algorithm Using the Arithmetic over Polynomial Matrices (Computer Algebra : Foundations and Applications)

The title of this paper has been changed from the title of talk “Polynomial XL: A Variant of the XL Algorithm Using Macaulay Matrices over Polynomial Rings” at “Computer Algebra -Foundations and Applications”.Solving a system of multivariate polynomials is a classical but very important problem in many areas of mathematics and its applications, and in particular quadratic systems over finite fields play a major role in the multivariate public key cryptography. The XL algorithm is known to be one of the main approaches for solving a multivariate system, as well as Groebner basis approaches, and so far many variants of XL have been proposed. In this talk, we present a new variant of XL, which we name “Polynomial XL”, by using Macaulay matrices over polynomial rings

Synthesis and Resist Properties of Calixarene Polymers with Pendant Haloalkyl Groups

We examined the condensation polymerization of t-butylcalix[8]arene (BCA[8]), p-t-butylcalix[4]arene (BCA[4]), and C-(4-t-butylbenz)calix[4]resorcinarene (BCRA[4]) with 1,3-adamantane dibromoacetate (ADB), yielding soluble polymers poly(BCA[8]-co-ADB), poly(BCA[4]-co-ADB), and poly(BCRA[4]-co-ADB), respectively. These polymers had good solubility, good film forming ability, and good thermal stability. It was anticipated that these polymers were applicable to positive-type resist materials. However, by the examination on the resist sensitivity using EUV exposure system, these polymers were applicable to negative type resist materials using THF as a developer. Furthermore, a negative clear resist pattern with 100nm resolution could be obtained by EB exposure system. These results indicated that poly(BCA[8]-co-ADB) and poly(BCRA[4]-co-ADB) have high potential to offer higher resolution negative pattern using EUV lithography system

Polynomial XL: A Variant of the XL Algorithm Using Macaulay Matrices over Polynomial Rings

Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for solving the MQ problem with linearization over a coefficient field. Furthermore, the hybrid approach with XL (h-XL) is a variant of XL guessing some variables beforehand. In this paper, we present a variant of h-XL, which we call the \textit{polynomial XL (PXL)}. In PXL, the whole $n$ variables are divided into $k$ variables to be fixed and the remaining $n-k$ variables as ``main variables\u27\u27, and we generate a Macaulay matrix with respect to the $n-k$ main variables over a polynomial ring of the $k$ (sub-)variables. By eliminating some columns of the Macaulay matrix over the polynomial ring before guessing $k$ variables, the amount of manipulations required for each guessed value can be reduced. Our complexity analysis of PXL gives a new theoretical bound, and it indicates that PXL is efficient in theory on the random system with $n=m$, which is the case of general multivariate signatures. For example, on systems over ${\mathbb F}_{2^8}$ with $n=m=80$, the numbers of manipulations deduced from the theoretical bounds of the hybrid approaches with XL and Wiedemann XL and PXL with optimal $k$ are estimated as $2^{252}$, $2^{234}$, and $2^{220}$, respectively

Study on resist performance of chemically amplified molecular resists based on cyclic oligomers

Novel resist materials are required for lithographic processing with ionization radiation such as extreme ultraviolet (EUV) and electron beam (EB) exposure tools. In this study, we synthesized positive-tone chemically amplified molecular resist materials with pendant adamantyl ester (AD) and cyclohexyl 2-propyl ether moieties based on cyclic oligomers such as noria, calixarene dimer, cyclodextrin, and pillar arene, and we examined the lithographic performances of sensitivity, etching durability, and patterning under EUV and EB exposure. We clarified that the sensitivity of the resist materials was consistent with the structure of the cyclic oligomers, i.e., the hole size of the molecular structure might be an important factor relevant to high-resolution resist materials. We found that chemically amplified molecular resists based on cyclic oligomers such as noria, calixarene dimer, cyclodextrin, and pillar arene are promising candidates for higher-resolution resist materials

Synthesis and Property of Tellurium-Containing Polymer for Extreme Ultraviolet Resist Material

Synthesis, physical properties, and resist properties of tellurium containing polymer with pendant adamantyl ester groups poly(Re-co-Te)-AD were examined, relevant to the application of resist material for extreme ultraviolet laser photolithography (EUVL) system. A tellurium containing polymer with pendant hydroxyl groups poly(Re-co-Te) was synthesized by the condensation reaction of resorcinol (Re) and tellurium tetrachloride (TeCl4), followed by the condensation reaction with adamantyl bromo acetate to give a corresponding polymer poly(Re-co-Te)-AD. Their physical properties (solubility, film-forming ability, thermal stability) and resist properties (thickness loss property after soaking in 2.38 wt% TMAH aq. solution, out-gassing on EUV exposure tool, and resist sensitivity under EUV exposure tool) were also examined

Synthesis and Resist Properties of Hyperbranched Polyacetals

The synthesized noria-AD offered 40 nm resolution resist pattern with LWR = 9.5 nm in the case of EB exposure tool and a clear 26 nm resolution pattern with LWR = 8.3 nm by means of EUV exposure tool. These results indicate that the present poly(THPE-co-BVOC) would have higher potential to offer higher resolution pattern using EUV lithography system

Synthesis and Property of Tannic Acid Derivatives and Their Application for Extreme Ultraviolet Laser Lithography System

We synthesized tannic acid derivatives with pendant cyclohexyl acetal moieties (TA-CVEn), butyl acetal moieties (TA-BVEn), and adamantyl ester moieties (TA-ADn) by the reaction of tannnic acid (TA) with cyclohexyl vinyl ether (CVE), butyl vinyl ether (BVE), and adamantyl bromo acetate (AD) in various feeds ratios. The synthesized TA-CVEn, TA-BVEn, and TA-ADn had good solubility, good film-forming ability, and high thermal stability relevant to application of photolithography materials. However, only TA-BVE97 and TA-AD74 can be used as positive-type photo-resist materials using 2.38 wt% TMAH aq. as developer due to the result of thickness loss property. Furthermore, their resist-sensitivity upon EUV exposure tool and etching durability were adequate and they have high potential as next-generation resist material for EUV lithography