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

• The Journal of the Korea institute of electronic communication sciences
• /
• v.8 no.8
• /
• pp.1207-1212
• /
• 2013

Arithmetic operations over finite fields are widely used in coding theory and cryptography. In both of these applications, there is a need to design low complexity finite field arithmetic units. The complexity of such a unit largely depends on how the field elements are represented. Among them, representation of elements using a optimal normal basis is quite attractive. Using an algorithm minimizing the number of 1's of multiplication matrix, in this paper, we propose a multiplier which is time and area efficient over finite fields with optimal normal basis.

• ETRI Journal
• /
• v.35 no.3
• /
• pp.523-529
• /
• 2013

Subquadratic space complexity multipliers for optimal normal bases (ONBs) have been proposed for practical applications. However, for the Gaussian normal basis (GNB) of type t > 2 as well as the normal basis (NB), there is no known subquadratic space complexity multiplier. In this paper, we propose the first subquadratic space complexity multipliers for the type 4 GNB. The idea is based on the fact that the finite field GF($2^n$) with the type 4 GNB can be embedded into fields with an ONB.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.6 no.2
• /
• pp.75-84
• /
• 2002

Using good properties of an optimal normal basis of type I in a finite field ${\mathbb{F}}_{2^m}$, we present a design of a bit serial multiplier of Berlekamp type, which is very effective in computing $xy^2$. It is shown that our multiplier does not need a basis conversion process and a squaring operation is a simple permutation in our basis. Therefore our multiplier provides a fast and an efficient hardware architecture for a bit serial multiplication of two elements in ${\mathbb{F}}_{2^m}$.

• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.14 no.3
• /
• pp.41-48
• /
• 2004

Finite field multiplication and division are important arithmetic operation in error-correcting codes and cryptosystems. The elements of the finite field GF($2^m$) are represented by bases with a primitive polynomial of degree m over GF(2). We can be easily realized for multiplication or computing multiplicative inverse in GF($2^m$) based on a normal basis representation. The number of product terms of logic function determines a complexity of the Messay-Omura multiplier. A normal basis exists for every finite field. It is not easy to find the optimal normal element for a given primitive polynomial. In this paper, the generating method of normal basis is investigated. The normal bases whose product terms are less than other bases for multiplication in GF($2^m$) are found. For each primitive polynomial, a list of normal elements and number of product terms are presented.

• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.25 no.5
• /
• pp.979-984
• /
• 2015

In this paper, we proposed a Semi-Systolic multiplier of $GF(2^n)$ with Type II optimal Normal Basis. Comparing the complexity of the proposed multiplier with Chiou's multiplier proposed in 2012, it is saved $2n^2+44n+26$ in total transistor numbers and decrease 4 clocks in time delay. This means that, for $GF(2^{333})$ of the field recommended by NIST for ECDSA, the space complexity is 6.4% less and the time complexity of the 2% decrease. In addition, this structure has an advantage as applied to Chiou's method of concurrent error detection and correction in multiplication of $GF(2^n)$.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.33 no.1C
• /
• pp.140-148
• /
• 2008

Using the self duality of an optimal normal basis(ONB) of type II, we present a bit parallel and bit serial systolic arrays over GF($2^m$) which has a low hardware complexity and a low latency. We show that our multiplier has a latency m+1 and the basic cell of our circuit design needs 5 latches(flip-flops). Comparing with other arrays of the same kinds, we find that our array has significantly reduced latency and hardware complexity.

• 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.43 no.2 s.344
• /
• pp.84-95
• /
• 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
• /
• v.43 no.1 s.343
• /
• pp.45-58
• /
• 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