Complexity bounds on Semaev’s naive index calculus method for ECDLP

Since Semaev introduced summation polynomials in 2004, a number of studies have been devoted to improving the index calculus method for solving the elliptic curve discrete logarithm problem (ECDLP) with better complexity than generic methods such as Pollard’s rho method and the baby-step and giant-step method (BSGS). In this paper, we provide a deep analysis of Gröbner basis computation for solving polynomial systems appearing in the point decomposition problem (PDP) in Semaev’s naive index calculus method. Our analysis relies on linear algebra under simple statistical assumptions on summation polynomials. We show that the ideal derived from PDP has a special structure and Gröbner basis computation for the ideal is regarded as an extension of the extended Euclidean algorithm. This enables us to obtain a lower bound on the cost of Gröbner basis computation. With the lower bound, we prove that the naive index calculus method cannot be more efficient than generic methods.

Quasi-subfield Polynomials and the Elliptic Curve Discrete Logarithm Problem

We initiate the study of a new class of polynomials which we call quasi-subfield polynomials. First, we show that this class of polynomials could lead to more efficient attacks for the elliptic curve discrete logarithm problem via the index calculus approach. Specifically, we use these polynomials to construct factor bases for the index calculus approach and we provide explicit complexity bounds. Next, we investigate the existence of quasi-subfield polynomials.

IDENTIFIKASI MISKONSEPSI MAHASISWA MENGGUNAKAN CRI PADA MATA KULIAH KALKULUS II

AbstrakMiskonsepsi merupakan masalah yang selalu muncul dalam kegiatan belajar mengajar. Untuk itu, penelitian ini bertujuan untuk mengidentifikasi miskonsepsi mahasiswa yang sedang mengambil matakuliah kalkulus II. CRI (Certainty of Response Index) digunakan untuk mengidentifikasi mahasiswa yang mengalami miskonsepsi, tidak paham konsep, dan sama sekali tidak tahu konsep. Penelitian ini melibatkan 20 responden yang sedang mengambil matakuliah kalkulus II. Instrument yang digunakan adalah tes diasnogtik dalam bentuk essay. Selain menjawab soal, mahasiswa juga diminta untuk menuliskan tingkat keyakinan kebenaran dari jawaban mereka. Selain itu, wawancara terstruktur juga digunakan untuk mengvalidasi tingkat keyakinan mahasiswa dan penyebab terjadinya miskonsepsi. Hasil penelitian menunjukkan bahwa CRI dapat dengan mudah membedakan siswa yang memahami konsep dengan baik dengan mahasiswa yang mengalami kesalahan. Berdasarkan analisis kesalahan mahasiswa diperoleh 46% mahasiswa mengalami miskonsepsi, 53,4% mahasiswa tidak mengetahui konsep dan 0,6% mahasiswa yang sama sekali tidak tahu konsep (lucky guess).Kata Kunci: Miskonsepsi, CRI (Certainty of Response Index), Kalkulus II.AbstractMisconception is an issue that always arises in teaching and learning activities. Therefore, this study aims to identify the misconception of students who are taking calculus II courses. CRI (Certainty of Response Index) is used to identify students with misconceptions, not understanding concepts, and not knowing concepts at all. This study involved 20 respondents who are taking calculus II courses. Instrument used is diagnogtic test in essay form. In addition to answering questions, students are also asked to write down the level of confidence in their answers. In addition, structured interviews are also used to validate student confidence and the causes of misconceptions. The results show that CRI can easily distinguish students who understand the concept well with students who experience errors. Based on the student error analysis, 46% of students have misconception, 53,4% of students do not know concept and 0,6% of students do not know concept (lucky guess).Keyword: Misconceptions, CRI (Certainty of Response Index), Calculus II.