• The Journal of the Korea institute of electronic communication sciences
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• v.9 no.4
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• pp.409-416
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• 2014

In H/W implementation for the finite field, the use of normal basis has several advantages, especially the optimal normal basis is the most efficient to H/W implementation in $GF(2^m)$. The finite field $GF(2^m)$ with type I optimal normal basis(ONB) has the disadvantage not applicable to some cryptography since m is even. The finite field $GF(2^m)$ with type II ONB, however, such as $GF(2^{233})$ are applicable to ECDSA recommended by NIST. In this paper, we propose a bit-parallel multiplier over $GF(2^m)$ having a type II ONB, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{2m})$. The time and area complexity of the proposed multiplier is the same as or partially better than the best known type II ONB bit-parallel multiplier.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.16 no.4
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• pp.83-89
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• 2006

In H/W implementation for the finite field, the use of normal basis has several advantages, especially, the optimal normal basis is the most efficient to H/W implementation in GF($2^m$). In this paper, we propose a new, simpler, parallel multiplier over GF($2^m$) having a type II optimal normal basis, which performs multiplication over GF($2^m$) in the extension field GF($2^{2m}$). The time and area complexity of the proposed multiplier is same as the best of known type II optimal normal basis parallel multiplier.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.8 no.2
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• pp.27-36
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• 1998

최근들어 타원곡선 암호법(ECC)이 RSA암호법을 대체할 것으로 기대되면서ECC의 연산속도를 결정하는 중요한 요소인 유한체의 연산 속도에 관심이 고조되고 있다. 본 논문에서는 Modified 최적 정규 기저의 성질 규명과 GF(q)(q=2$^{k}$ , k=8또는 16)위에서 GF(q$^{m}$ )(m: 홀수)의 Mofdified trinomial 기가 존재하는 m들을 제시하고, GF(r$^{n}$ )위에서 GF(r$^{nm}$ )dml Modified 최적 정규기저와 Modified trinomial 기저를 이용한 연산의 회수와 각 기저를 이용한 연산의 회수와 각 기저를 이용한 유한체 GF(q$^{m}$ )의 연산을 S/W화한 결과를 비교 하였다.

• Journal of the Korea Institute of Information and Communication Engineering
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• v.8 no.6
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• pp.1188-1193
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• 2004

So far, there have been grossly 3 types of studies on GF(2m) multiplier architecture, such as serial multiplication, array multiplication, and hybrid multiplication. Serial multiplication method was first suggested by Mastrovito (1), to be known as the basic CF(2m) multiplication architecture, and this method was adopted in the array multiplier (2), consuming m times as much resource in parallel to extract m times of speed. In 1999, Paar studied further to get the benefit of both architecture, presenting the hybrid multiplication architecture (3). However, the hybrid architecture has defect that only complex ordo. of finite field should be used. In this paper, we propose a novel approach on developing serial multiplier architecture based on Mastrovito's, by modifying the numerical formula of the polynomial-basis serial multiplication. The proposed multiplier architecture was described and implemented in HDL so that the novel architecture was simulated and verified in the level of hardware as well as software. The implemented GF(2m) multiplier shows t times as fast as the traditional one, if we modularized the numerical expression by t number of parts.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.43 no.2 s.344
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• pp.84-95
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• 2006

In H/W implementation for the finite field the use of normal basis has several advantages, especially, the optimal normal basis is the most efficient to H/W implementation in $GF(2^m)$. In this paper, we propose a new, simpler, parallel multiplier over $GF(2^m)$ having a Gaussian normal basis of type k, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{mk})$ containing a type-I optimal normal basis. For k=2,4,6 the time and area complexity of the proposed multiplier is the same as tha of the best known Reyhani-Masoleh and Hasan multiplier.

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.43 no.1 s.343
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• pp.45-58
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• 2006

In H/W implementation for the finite field, the use of normal basis has several advantages, especially, the optimal normal basis is the most efficient to H/W implementation in $GF(2^m)$. In this paper, we propose a new, simpler, parallel multiplier over $GF(2^m)$ having a Gaussian normal basis of type k, which performs multiplication over $GF(2^m)$ in the extension field $GF(2^{mk})$ containing a type-I optimal normal basis. For k=2,4,6 the time and area complexity of the proposed multiplier is the same as tha of the best known Reyhani-Masoleh and Hasan multiplier

• Proceedings of the Korean Information Science Society Conference
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• 2003.04a
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• pp.272-274
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• 2003

유한체 GF(2$^{m}$ ) 상의 산술 연산 중 곱셈 연산의 효율적인 구현은 암호이론 분야의 어플리케이션에서 매우 중요하다. 본 논문에서는 All-One 다항식에 의해 정의된 GF(2$^{m}$ ) 상의 효율적인 Bit-Parallel 정규기저 곱셈기를 제안한다. 게이트 및 시간 면에서 본 논문의 곱셈기의 complexity는 이전에 제안된 같은 종류의 곱셈기 보다 낮거나 동일하다. 그리고 본 논문의 곱셈기는 이전 곱셈기 보다 더 모듈적이어서 VLSI 구현에 적합하다.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.12 no.6
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• pp.17-28
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• 2002

XTR is a new method to represent elements of a subgroup of a multiplicative group of a finite field GF( $p^{6m}$) and it can be generalized to the field GF( $p^{6m}$)$^{[6,9]}$ This paper progress optimal extention fields for XTR among Galois fields GF ( $p^{6m}$) which can be aplied to XTR. In order to select such fields, we introduce a new notion of Generalized Opitimal Extention Fields(GOEFs) and suggest a condition of prime p, a defining polynomial of GF( $p^{2m}$) and a fast method of multiplication in GF( $p^{2m}$) to achieve fast finite field arithmetic in GF( $p^{2m}$). From our implementation results, GF( $p^{36}$ )longrightarrowGF( $p^{12}$ ) is the most efficient extension fields for XTR and computing Tr( $g^{n}$ ) given Tr(g) in GF( $p^{12}$ ) is on average more than twice faster than that of the XTR system on Pentium III/700MHz which has 32-bit architecture.$^{［6,10］/ ［6,10］/6,10］}$

• Journal of the Institute of Electronics Engineers of Korea SD
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• v.42 no.9 s.339
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• pp.51-60
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• 2005

The design of efficient cryptosystems is mainly appointed by the efficiency of the underlying finite field arithmetic. Especially, among the basic arithmetic over finite field, the rnultiplicative inversion is the most time consuming operation. In this paper, a fast inversion algerian in finite field $GF(2^m)$ with the standard basis representation is proposed. It is based on the Extended binary gcd algorithm (EBGA). The proposed algorithm executes about $18.8\%\;or\;45.9\%$ less iterations than EBGA or Montgomery inverse algorithm (MIA), respectively. In practical applications where the dimension of the field is large or may vary, systolic array sDucture becomes area-complexity and time-complexity costly or even impractical in previous algorithms. It is not suitable for low-weight and low-power systems, i.e., smartcard, the mobile phone. In this paper, we propose a new hardware architecture to apply an area-efficient and a synchronized inverter on low-complexity systems. It requires the number of addition and reduction operation less than previous architectures for computing the inverses in $GF(2^m)$ furthermore, the proposed inversion is applied over either prime or binary extension fields, more specially $GF(2^m)$ and GF(P) .

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• 2019.05a
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• pp.254-256
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• 2019

소수체 GF(p)와 이진체 $GF(2^m)$ 상의 다중 타원곡선을 지원하는 듀얼 필드 ECC (DF-ECC) 프로세서를 설계하였다. DF-ECC 프로세서의 저면적 설와 다양한 타원곡선의 지원이 가능하도록 워드 기반 몽고메리 곱셈 알고리듬을 적용한 유한체 곱셈기를 저면적으로 설계하였으며, 페르마의 소정리(Fermat's little theorem)를 유한체 곱셈기에 적용하여 유한체 나눗셈을 구현하였다. 설계된 DF-ECC 프로세서는 스칼라 곱셈과 점 연산, 그리고 모듈러 연산 기능을 가져 다양한 공개키 암호 프로토콜에 응용이 가능하며, 유한체 및 모듈러 연산에 적용되는 파라미터를 내부 연산으로 생성하여 다양한 표준의 타원곡선을 지원하도록 하였다. 설계된 DF-ECC는 FPGA 구현을 하드웨어 동작을 검증하였으며, 0.18-um CMOS 셀 라이브러리로 합성한 결과 22,262 GEs (gate equivalences)와 11 kbit RAM으로 구현되었으며, 최대 100 MHz의 동작 주파수를 갖는다. 설계된 DF-ECC 프로세서의 연산성능은 B-163 Koblitz 타원곡선의 경우 스칼라 곱셈 연산에 885,044 클록 사이클이 소요되며, B-571 슈도랜덤 타원곡선의 스칼라 곱셈에는 25,040,625 사이클이 소요된다.