• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.9 no.4
• /
• pp.3-10
• /
• 1999

More concerns are concentrated in finite fields arithmetic as finite fields being applied for Elliptic curve cryptosystem coding theory and etc. Finite fields arithmetic is affected in represen -tation of those. Optimal normal basis is effective in hardware implementation and polynomial field which is effective in the basis conversion with optimal normal basis and show that the arithmetic of finite field with the basis is effective in software implementation.

• The Journal of the Korea institute of electronic communication sciences
• /
• v.6 no.3
• /
• pp.389-395
• /
• 2011

Since Elliptic Curve Cryptosystems(ECCs) support the same security as RSA cryptosystem and ElGamal cryptosystem with 1/6 size key, ECCs are the most efficient to smart cards, cellular phone and small-size computers restricted by high memory capacity and power of process. In this paper, we explicitly explain methods for finite fields operations used in ECC, and then construct some composite fields over finite fields which are secure under Weil's decent attack and maximize the speed of operations. Lastly, we propose a fast multiplier over our composite fields.

• Proceedings of the Korea Information Processing Society Conference
• /
• 2011.11a
• /
• pp.925-927
• /
• 2011

본 논문에서는 확장 이진 최대공약수 알고리듬 (Extended Binary GCD algorithm)을 기본으로 GF($2^m$) 상에서 유한체 나눗셈 연산을 위한 고속 알고리듬을 제안하고, 제안한 알고리듬을 기본으로 한 나눗셈 연산기의 FPGA 설계 구현에 관하여 기술한다. 제안한 알고리듬은 Verilog HDL 로 기술하였고, Xilinx FPGA virtex4-xc4vlx15 디바이스를 타겟으로 하였다.

• The Journal of the Korea institute of electronic communication sciences
• /
• v.9 no.4
• /
• pp.409-416
• /
• 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 Institute of Electronics Engineers of Korea SD
• /
• v.42 no.9 s.339
• /
• pp.51-60
• /
• 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
• /
• 2008.05a
• /
• pp.651-654
• /
• 2008

In finite field operations based on $GF(2^m)$, additions and subtractions are easily implemented. On the other hand, multiplications and divisions require mathematical elaboration of complex equations. There are two dominant way of approaching the solutions of finite filed operations, normal basis approach and polynomial basis approach, each of which has both benefits and weakness respectively. In this study, we adopted the mathematically feasible polynomial basis approach and suggest the optimization techniques of finite field operations based of mathematical principles.

• Journal of the Korea Institute of Information and Communication Engineering
• /
• v.10 no.2
• /
• pp.328-332
• /
• 2006

Efficient finite field operation in the elliptic curve (EC) public key cryptography algorithm, which attracts much of latest issues in the applications in information security, is very important. Traditional serial finite multipliers root from Mastrovito's serial multiplication architecture. In this paper, we adopt the polynomial basis and propose a new finite field multiplier, inducing numerical expressions which can be applied to exhibit 3 times as much performance as the Mastrovito's. We described the proposed multiplier with HDL to verify and evaluate as a proper hardware IP. HDL-implemented serial GF (Galois field) multiplier showed 3 times as fast speed as the traditional serial multiplier's adding only partial-sum block in the hardware. So far, there have been grossly 3 types of studies on GF($2^m$) multiplier architecture, such as serial multiplication, array multiplication, and hybrid multiplication. 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.

• Proceedings of the Korean Information Science Society Conference
• /
• 2005.07a
• /
• pp.187-189
• /
• 2005

최적확장체(Optimal Extension Field:OEF)는 유한체의 일종으로서, 타원곡선 암호시스템의 소프트웨어 구현에 있어 매우 유용하다. Bailey및 Paar는 $p^i$거듭제곱 연산을 비롯하여 다수의 효율적인 OEF연산 알고리즘을 제안하였으며, 또한 암호 응용에 적합한 OEF를 생성하기 위한 효과적인 알고리즘을 제안하였다. 본 논문에서는 Bailey-Paar의 $p^i$거듭제곱 알고리즘이 적용되지 않는 반례를 제시하며, 또한 그들의 OEF생성 알고리즘은 실제로 OEF가 아닌 유한체를 OEF로 출력하는 오류가 있음을 보인다. 본 논문에서는 이러한 문제들을 해결한 개선된 알고리즘들을 제시하고, OEF의 개수에 관한 수정된 통계치를 제시한다.

• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.29 no.3
• /
• pp.511-514
• /
• 2019

Elliptic Curves Cryptosystem(ECC) provides the same level of security with relatively small key sizes, as compared to the traditional cryptosystems. The performance of ECC over GF(2m) and GF(p) depends on the efficiency of finite field arithmetic, especially the modular multiplication which is based on the reduction algorithm. In this paper, we propose a new modular reduction algorithm which provides high-speed ECC over NIST prime P-256. Detailed experimental results show that the proposed algorithm is about 25% faster than the previous methods.

• Proceedings of the IEEK Conference
• /
• 2001.06b
• /
• pp.341-344
• /
• 2001

본 논문에서는 타원곡선 알고리즘에 기반한 공개키암호시스템 구현을 다룬다. 공개키의 길이는 193비트를 갖고 기약다항식은 p(x)=x/sup 193＋x/sup 15＋1을 사용하였다. 타원곡선은 polynomial basis 로 표현하였으며 SEC 2 파라메터를 기준으로 하였다 암호시스템은 polynomial basis 유한체 연산기로 구성되며 특히, digit-serial 구조로 스마트카드와 같이 제한된 면적에서 구현이 가능하도록 하였다. 시스템의 회로는 VHDL, SYNOPSYS 시뮬레이션 및 회로합성을 이용하여 XILINX FPGA로 회로를 구현하였다. 본 시스템 은 Diffie-Hellman 키교환에 적용하여 동작을 검증하였다.