• Title/Summary/Keyword: Finite field arithmetic

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Implementation of a LSB-First Digit-Serial Multiplier for Finite Fields GF(2m) (유한 필드 GF(2m)상에서의 LSB 우선 디지트 시리얼 곱셈기 구현)

  • Kim, Chang-Hun;Hong, Chun-Pyo;U, Jong-Jeong
    • The KIPS Transactions:PartA
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    • v.9A no.3
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    • pp.281-286
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    • 2002
  • In this paper we, implement LSB-first digit-serial systolic multiplier for computing modular multiplication $A({\times})B$mod G ({\times})in finite fields GF $(2^m)$. If input data come in continuously, the implemented multiplier can produce multiplication results at a rate of one every [m/L] clock cycles, where L is the selected digit size. The analysis results show that the proposed architecture leads to a reduction of computational delay time and it has more simple structure than existing digit-serial systolic multiplier. Furthermore, since the propose architecture has the features of regularity, modularity, and unidirectional data flow, it shows good extension characteristics with respect to m and L.

Fast Bit-Serial Finite Field Multipliers (고속 비트-직렬 유한체 곱셈기)

  • Chang, Nam-Su;Kim, Tae-Hyun;Lee, Ok-Suk;Kim, Chang-Han
    • Journal of the Institute of Electronics Engineers of Korea SD
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    • v.45 no.2
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    • pp.49-54
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    • 2008
  • In cryptosystems based on finite fields, a modular multiplication operation is the most crucial part of finite field arithmetic. Also, in multipliers with resource constrained environments, bit-serial output structures are used in general. This paper proposes two efficient bit-serial output multipliers with the polynomial basis representation for irreducible trinomials. The proposed multipliers have lower time complexity compared to previous bit-serial output multipliers. One of two proposed multipliers requires the time delay of $(m+1){\cdot}MUL+(m+1){\cdot}ADD$ which is more efficient than so-called Interleaved Multiplier with the time delay of $m{\cdot}MUL+2m{\cdot}ADD$. Therefore, in elliptic curve cryptosystems and pairing based cryptosystems with small characteristics, the proposed multipliers can result in faster overall computation. For example, if the characteristic of the finite fields used in cryprosystems is small then the proposed multipliers are approximately two times faster than previous ones.

A small-area implementation of cryptographic processor for 233-bit elliptic curves over binary field (233-비트 이진체 타원곡선을 지원하는 암호 프로세서의 저면적 구현)

  • Park, Byung-Gwan;Shin, Kyung-Wook
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.21 no.7
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    • pp.1267-1275
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    • 2017
  • This paper describes a design of cryptographic processor supporting 233-bit elliptic curves over binary field defined by NIST. Scalar point multiplication that is core arithmetic in elliptic curve cryptography(ECC) was implemented by adopting modified Montgomery ladder algorithm, making it robust against simple power analysis attack. Point addition and point doubling operations on elliptic curve were implemented by finite field multiplication, squaring, and division operations over $GF(2^{233})$, which is based on affine coordinates. Finite field multiplier and divider were implemented by applying shift-and-add algorithm and extended Euclidean algorithm, respectively, resulting in reduced gate counts. The ECC processor was verified by FPGA implementation using Virtex5 device. The ECC processor synthesized using a 0.18 um CMOS cell library occupies 49,271 gate equivalents (GEs), and the estimated maximum clock frequency is 345 MHz. One scalar point multiplication takes 490,699 clock cycles, and the computation time is 1.4 msec at the maximum clock frequency.

CARTIER OPERATORS ON COMPACT DISCRETE VALUATION RINGS AND APPLICATIONS

  • Jeong, Sangtae
    • Journal of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.101-129
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    • 2018
  • From an analytical perspective, we introduce a sequence of Cartier operators that act on the field of formal Laurent series in one variable with coefficients in a field of positive characteristic p. In this work, we discover the binomial inversion formula between Hasse derivatives and Cartier operators, implying that Cartier operators can play a prominent role in various objects of study in function field arithmetic, as a suitable substitute for higher derivatives. For an applicable object, the Wronskian criteria associated with Cartier operators are introduced. These results stem from a careful study of two types of Cartier operators on the power series ring ${\mathbf{F}}_q$[[T]] in one variable T over a finite field ${\mathbf{F}}_q$ of q elements. Accordingly, we show that two sequences of Cartier operators are an orthonormal basis of the space of continuous ${\mathbf{F}}_q$-linear functions on ${\mathbf{F}}_q$[[T]]. According to the digit principle, every continuous function on ${\mathbf{F}}_q$[[T]] is uniquely written in terms of a q-adic extension of Cartier operators, with a closed-form of expansion coefficients for each of the two cases. Moreover, the p-adic analogues of Cartier operators are discussed as orthonormal bases for the space of continuous functions on ${\mathbf{Z}}_p$.

Design of an Efficient User Authentication Protocol Using subgroup of Galois Field (유한체의 부분군을 이용한 효율적인 사용자 인증 프로로콜 설계)

  • 정경숙
    • Journal of the Korea Society of Computer and Information
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    • v.9 no.2
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    • pp.105-113
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    • 2004
  • If the protocol has fast operations and short key length, it can be efficient user authentication protocol Lenstra and Verheul proposed XTR. XTR have short key length and fast computing speed. Therefore, this can be used usefully in complex arithmetic. In this paper, to design efficient user authentication protocol we used a subgroup of Galois Field to problem domain. Proposed protocol does not use GF($p^6$) that is existent finite field, and uses GF($p^2$) that is subgroup and solves problem. XTR-ElGamal based user authentication protocol reduced bit number that is required when exchange key by doing with upside. Also, Proposed protocol provided easy calculation and execution by reducing required overhead when calculate. In this paper, we designed authentication protocol that is required to do user authentication.

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Practical Implementation and Performance Evaluation of Random Linear Network Coding (랜덤 선형 네트워크 코딩의 실용적 설계 및 성능 분석)

  • Lee, Gyujin;Shin, Yeonchul;Koo, Jonghoe;Choi, Sunghyun
    • The Journal of Korean Institute of Communications and Information Sciences
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    • v.40 no.9
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    • pp.1786-1792
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    • 2015
  • Random linear network coding (RLNC) is widely employed to enhance the reliability of wireless multicast. In RLNC encoding/decoding, Galois Filed (GF) arithmetic is typically used since all the operations can be performed with symbols of finite bits. Considering the architecture of commercial computers, the complexity of arithmetic operations is constant regardless of the dimension of GF m, if m is smaller than 32 and pre-calculated tables are used for multiplication/division. Based on this, we show that the complexity of RLNC inversely proportional to m. Considering additional overheads, i.e., the increase of header length and memory usage, we determine the practical value of m. We implement RLNC in a commercial computer and evaluate the codec throughput with respect to the type of the tables for multiplication/division and the number of original packets to encode with each other.

On Implementations of Algorithms for Fast Generation of Normal Bases and Low Cost Arithmetics over Finite Fields (유한체위에서 정규기저의 고속생성과 저비용 연산 알고리즘의 구현에 관한 연구)

  • Kim, Yong-Tae
    • The Journal of the Korea institute of electronic communication sciences
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    • v.12 no.4
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    • pp.621-628
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    • 2017
  • The efficiency of implementation of the arithmetic operations in finite fields depends on the choice representation of elements of the field. It seems that from this point of view normal bases are the most appropriate, since raising to the power 2 in $GF(2^n)$ of characteristic 2 is reduced in these bases to a cyclic shift of the coordinates. We, in this paper, introduce our algorithm to transform fastly the conventional bases to normal bases and present the result of H/W implementation using the algorithm. We also propose our algorithm to calculate the multiplication and inverse of elements with respect to normal bases in $GF(2^n)$ and present the programs and the results of H/W implementations using the algorithm.

MoTE-ECC Based Encryption on MSP430

  • Seo, Hwajeong;Kim, Howon
    • Journal of information and communication convergence engineering
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    • v.15 no.3
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    • pp.160-164
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    • 2017
  • Public key cryptography (PKC) is the basic building block for the cryptography applications such as encryption, key distribution, and digital signature scheme. Among many PKC, elliptic curve cryptography (ECC) is the most widely used in IT systems. Recently, very efficient Montgomery-Twisted-Edward (MoTE)-ECC was suggested, which supports low complexity for the finite field arithmetic, group operation, and scalar multiplication. However, we cannot directly adopt the MoTE-ECC to new PKC systems since the cryptography is not fully evaluated in terms of performance on the Internet of Things (IoT) platforms, which only supports very limited computation power, energy, and storage. In this paper, we fully evaluate the MoTE-ECC implementations on the representative IoT devices (16-bit MSP processors). The implementation is highly optimized for the target platform and compared in three different factors (ROM, RAM, and execution time). The work provides good reference results for a gradual transition from legacy ECC to MoTE-ECC on emerging IoT platforms.

On spanning column rank of matrices over semirings

  • Song, Seok-Zun
    • Bulletin of the Korean Mathematical Society
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    • v.32 no.2
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    • pp.337-342
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    • 1995
  • A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.

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MEAN VALUES OF DERIVATIVES OF L-FUNCTIONS IN FUNCTION FIELDS: IV

  • Andrade, Julio;Jung, Hwanyup
    • Journal of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1529-1547
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    • 2021
  • In this series, we investigate the calculation of mean values of derivatives of Dirichlet L-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields. The present paper generalizes the results obtained in the first paper. For µ ≥ 1 an integer, we compute the mean value of the µ-th derivative of quadratic Dirichlet L-functions over the rational function field. We obtain the full polynomial in the asymptotic formulae for these mean values where we can see the arithmetic dependence of the lower order terms that appears in the asymptotic expansion.