Multiplication using the triangular basis representation over GF(2/sup m/)

Furness, Robert W.; Fenn, S.T.J.; Benaïssa, Mohammed

doi:10.1109/glocom.1996.585975

Cited by 2 publications

(3 citation statements)

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“…Recently the authors have suggested forcing the Hasan-Bhargava multiplier (HBM) to operate over the TB [SI and so only one basis conversion is required rather than two. In [8] it was also proposed carrying out this conversion with an XOR tree rather than a sequential circuit, thereby reducing both the hardware requirements and overall delays of the multiplier. In addition, this approach has been extended to allow for the design of bit-parallel multipliers [SI.…”

Section: Introductionmentioning

confidence: 99%

“…The multiplier shown in Figure 2 therefore operates over the PB. In [8] the multiplier structure shown in Figure 3 was proposed. This circuit again comprises a central multiplier implementing Equation (2) but only one basis converter is required, a TB to PB conversion circuit on the input.…”

Section: Introductionmentioning

confidence: 99%

“…Further let x = (xo, xl, ..., xm-T d x = (Xo,Xl, ..., X m -1) be the GTB and PB representations of x E G F ( 2 ) respectively. Then x = (Xk-pXk-2, ...'X1,XO+Xk-l'Xm-l +Xk'Xm-2'Xm-3'... , X k + l ' X k )(8) andx = W k-1 + X 0 , -5 -2 J -3 , ...,X,,&J,Xm-l,Xm-2'. .…”

mentioning

confidence: 99%

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Generalised triangular basis multipliers for the design of Reed-Solomon codecs

Furness

Benaïssa²,

Fenn³

1997 IEEE Workshop on Signal Processing Systems. SiPS 97 Design and Implementation Formerly VLSI Signal Processing

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In this paper a generalised definition of a triangular basis for CF(F) ispresented. This definition is more flexible than the traditional definition and allows a number of triangular bases to be determined to any given basis. The triangular basis to the polynomial basis can then be used which has the simplest basis transformation in modified Hasan-Bhargava multipliers. It is shown that when the defining irreducible polynomial for the field is a trinomial, the triangular basis is a permutation of the polynomial basis elements. Furthermore, if the defining irreducible polynomial is a pentanomial of a certain form, the triangular basis can be obtained from the polynomial basis with a reordering of the coefficients and two GF(2) additions. This new definition of triangularity allows for reduced complexity modified Hasan-Bhargava multipliers to be designed which in turn reduces the hardware complexity of Reed-Solomon codecs using these multipliers. Finally, a simple method of making large hardware savings in Reed-Solomon decoders utilising frequency domain decoding is presented.0-7803-3806-5/97/$10.00 0 1 997 IEEE 202 expense of a hardware increase in the encoder. Similarly in the case of an RS (255,223) code, the saving of 223 constant multipliers equates to a hardware saving of around 3,500 XOR gates when using bit-parallel multipliers at the expense of an increase of 48 XOR gates in the encoder. J J

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%