• Journal of KIISE:Computer Systems and Theory
• /
• v.35 no.7
• /
• pp.327-331
• /
• 2008

Optimal Extension Fields (OEFs) are finite fields of a special form which are very useful for software implementation of elliptic curve cryptosystems. Bailey and Paar introduced efficient OEF arithmetic algorithms including the $p^ith$ powering operation, and an efficient algorithm to construct OEFs for cryptographic use. In this paper, we give a counterexample where their $p^ith$ powering algorithm does not work, and show that their OEF construction algorithm is faulty, i.e., it may produce some non-OEFs as output. We present improved algorithms which correct these problems, and give improved statistics for the number of OEFs.

• Journal of the Korea Society of Computer and Information
• /
• v.28 no.2
• /
• pp.69-75
• /
• 2023

Finite field arithmetic operations play an important role in a variety of applications, including modern cryptography and error correction codes. In this paper, we propose an efficient multiplication algorithm over finite fields using the Montgomery multiplication algorithm. Existing multipliers can be implemented using AND and XOR gates, but in order to reduce time and space complexity, we propose an algorithm using NAND and NOR gates. Also, based on the proposed algorithm, an efficient semi-systolic finite field multiplier with low space and low latency is proposed. The proposed multiplier has a lower area-time complexity than the existing multipliers. Compared to existing structures, the proposed multiplier over finite fields reduces space-time complexity by about 71%, 66%, and 33% compared to the multipliers of Chiou et al., Huang et al., and Kim-Jeon. As a result, our multiplier is proper for VLSI and can be successfully implemented as an essential module for various applications.

• Journal of KIISE:Computer Systems and Theory
• /
• v.31 no.1_2
• /
• pp.51-58
• /
• 2004

Most of pubic-key cryptosystems are built on the basis of arithmetic operations defined over the finite field GF$GF(2^m)$ .The other operations of finite fields except addition can be computed by repeated multiplications. Therefore, it is very important to implement the multiplication operation efficiently in public-key cryptosystems. We propose an efficient bit-parallel normal basis multiplier for$GF(2^m)$ fields defined by All-One Polynomials. The gate count and time complexities of our proposed multiplier are lower than or equal to those of the previously proposed multipliers of the same class. Also, since the architecture of our multiplier is regular, it is suitable for VLSI implementation.

• Journal of the Korea Institute of Information Security & Cryptology
• /
• v.18 no.3
• /
• pp.33-44
• /
• 2008

The efficiency of pairing based cryptosystems depends on the computation of pairings. pairings is defined over finite fileds GF$(3^m)$ by trinomials due to efficiency. The hardware architectures for pairings have been widely studied. This paper proposes new adder and multiplier for GF(3) which are more efficient than previous results. Furthermore, this paper proposes a new unified adder-subtractor for GF$(3^m)$ based on the proposed adder and multiplier. Finally, this paper proposes new multiplier for GF$(3^m)$. The proposed MSB-first bit-serial multiplier for GF$(p^m)$ reduces the time delay by approximately 30 % and the size of register by half than previous LSB-first multipliers. The proposed multiplier can be applied to all finite fields defined by trinomials.

• 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 KIISE:Computer Systems and Theory
• /
• v.33 no.1_2
• /
• pp.62-68
• /
• 2006

The important arithmetic operations over finite fields include multiplication and exponentiation. An exponentiation operation can be implemented using a series of squaring and multiplication operations over GF($2^m$) using the binary method. Hence, it is important to develop a fast algorithm and efficient hardware for multiplication. This paper presents an efficient bit-serial systolic array for MSB-first multiplication in GF($2^m$) based on the polynomial representation. As compared to the related multipliers, the proposed systolic multiplier gains advantages in terms of input-pin and area-time complexity. Furthermore, it has regularity, modularity, and unidirectional data flow, and thus is well suited to VLSI implementation.

• 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) .

• The Journal of Society for e-Business Studies
• /
• v.11 no.2
• /
• pp.1-11
• /
• 2006

This paper proposes an efficient inversion algorithm for Galois field GF(2n) by using a modified multi-bit shifting method based on the Montgomery algorithm. It is well known that the efficiency of arithmetic algorithms depends on the basis and many foregoing papers use either polynomial or optimal normal basis. An inversion algorithm, which modifies a multi-bit shifting based on the Montgomery algorithm, is studied. Trinomials and AOPs (all-one polynomials) are tested to calculate the inverse. It is shown that the suggested inversion algorithm reduces the computation time up to 26 % of the forgoing multi-bit shifting algorithm. The modified algorithm can be applied in various applications and is easy to implement.

• The KIPS Transactions:PartA
• /
• v.9A no.3
• /
• pp.281-286
• /
• 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.

• Journal of the Institute of Electronics Engineers of Korea SD
• /
• v.45 no.2
• /
• pp.49-54
• /
• 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.