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Comparison of Arithmetic Architectures for ReedSolomon Decoders in Reconfigurable Hardware
 IEEE Transactions on Computers
, 1997
"... ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. T ..."
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Cited by 13 (2 self)
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ReedSolomon (RS) error correction codes are being widely used in modern communication systems such as compact disk players or satellite communication links. RS codes rely on arithmetic in finite, or Galois fields. The specific field GF (2 8 ) is of central importance for many practical systems. The most costly, and thus most critical, elementary operations in RS decoders are multiplication and inversion in Galois fields. Although there have been considerable efforts in the area of Galois field arithmetic architectures, there appears to be very little reported work for Galois field arithmetic for reconfigurable hardware. This contribution provides a systematic comparison of two promising arithmetic architecture classes. The first one is based on a standard base representation, and the second one is based on composite fields. For both classes a multiplier and an inverter for GF (2 8 ) are described and theoretical gate counts are provided. Using a design entry based on a VHDL descr...
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Cited by 9 (0 self)
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Vlsi Architecture For Datapath Integration Of Arithmetic Over GF(2^m) On Digital Signal Processors
 in Proc. IEEE ICASSP'97
, 1997
"... This paper examines the implementation of Finite Field arithmetic, i.e. multiplication, division, and exponentiation, for any standard basis GF(2 ) with m8 on a DSP datapath. We introduce an opportunity to exploit cells and the interconnection structure of a typical binary multiplier unit for the ..."
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Cited by 8 (4 self)
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This paper examines the implementation of Finite Field arithmetic, i.e. multiplication, division, and exponentiation, for any standard basis GF(2 ) with m8 on a DSP datapath. We introduce an opportunity to exploit cells and the interconnection structure of a typical binary multiplier unit for the Finite Field operations by adding just a small overhead of logic. We develop division and exponentiation based on multiplication on the algorithm level and present a simple scheme for implementation of all operations on a processor datapath.
Scalable and unified hardware to compute montgomery inverse
 in GF(p) and GF(2 n ),” Cryptographic Hardware and Embedded Systems  CHES 2002, 4th International Workshop
, 2003
"... Abstract. Computing the inverse of a number in finite fields GF(p) or GF(2 n) is equally important for cryptographic applications. This paper proposes a novel scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2 n) fields. We adjust and modify a GF ..."
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Cited by 7 (1 self)
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Abstract. Computing the inverse of a number in finite fields GF(p) or GF(2 n) is equally important for cryptographic applications. This paper proposes a novel scalable and unified architecture for a Montgomery inverse hardware that operates in both GF(p) and GF(2 n) fields. We adjust and modify a GF(2 n) Montgomery inverse algorithm to accommodate multibit shifting hardware, making it very similar to a previously proposed GF(p) algorithm. The architecture is intended to be scalable, which allows the hardware to compute the inverse of long precision numbers in a repetitive way. After implementing this unified design it was compared with other designs. The unified hardware was found to be eight times smaller than another reconfigurable design, with comparable performance. Even though the unified design consumes slightly more area and it is slightly slower than the scalable inverter implementations for GF(p) only, it is a practical solution whenever arithmetic in the two finite fields is needed. 1
VLSI Architecture For NonSequential Inversion Over GF(2^m) Using The Euclidean Algorithm
 Proc. of ICSPAT 97
, 1997
"... This paper examines a VLSI implementation scheme of inversion over Finite Fields GF(2 based on the Euclidean algorithm. The multiplicative inverse can be used to perform division over finite fields. In contrast to previously published solutions, we developed a new logic circuit without memory ele ..."
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Cited by 3 (1 self)
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This paper examines a VLSI implementation scheme of inversion over Finite Fields GF(2 based on the Euclidean algorithm. The multiplicative inverse can be used to perform division over finite fields. In contrast to previously published solutions, we developed a new logic circuit without memory elements calculating the inverse element over a finite field in a combinational manner. The application target for this circuit is datapath of Domain Specific DSPs (DSDSP) which recently attain attention in research and development. The proposed architecture is highly regular and well suited for VLSI implementation.
Parallel ItohTsujii Multiplicative Inversion Algorithm for a Special Class of Trinomials
, 2006
"... In this contribution, we derive a novel parallel formulation of the standard ItohTsujii algorithm for multiplicative inverse computation over GF(2 ). ..."
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Cited by 2 (2 self)
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In this contribution, we derive a novel parallel formulation of the standard ItohTsujii algorithm for multiplicative inverse computation over GF(2 ).
i Preface
"... This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k). ..."
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This thesis describes various efficient architectures for computation in Galois fields of the type GF(2^k).