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| In [[abstract algebra]], the '''Chien search''', named after R. T. Chien, is a fast algorithm for determining [[Root of a function|root]]s of [[polynomial]]s defined over a [[finite field]]. The most typical use of the Chien search is in finding the roots of error-locator polynomials encountered in decoding [[Reed-Solomon code]]s and [[BCH code]]s.
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| ==Algorithm==
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| We denote the polynomial (over the finite field GF(<math>q</math>)) whose roots we wish to determine as:
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| :<math>\ \Lambda(x) = \lambda_0 + \lambda_1 x + \lambda_2 x^2 + \cdots + \lambda_t x^t </math>
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| Conceptually, we may evaluate <math>\Lambda(\beta)</math> for each non-zero <math>\beta</math> in GF(<math>q</math>). Those resulting in 0 are roots of the polynomial.
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| The Chien search is based on two observations:
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| * Each non-zero <math>\beta</math> may be expressed as <math>\alpha^{i_\beta}</math> for some <math>i_\beta</math>, where <math>\alpha</math> is a [[primitive element (finite field)|primitive element]] of <math>\mathrm{GF}(q)</math>, <math>i_\beta</math> is the power number of primitive element <math>\alpha</math>. Thus the powers <math>\alpha^i</math> for <math>0 \leq i < (q-1)</math> cover the entire field (excluding the zero element).
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| * The following relationship exists:
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| :<math>
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| \begin{array}{lllllllllll}
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| \Lambda(\alpha^i) &=& \lambda_0 &+& \lambda_1 (\alpha^i) &+& \lambda_2 (\alpha^i)^2 &+& \cdots &+& \lambda_t (\alpha^i)^t \\ | |
| &\triangleq& \gamma_{0,i} &+& \gamma_{1,i} &+& \gamma_{2,i} &+& \cdots &+& \gamma_{t,i}
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| \end{array}
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| </math>
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| :<math>
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| \begin{array}{lllllllllll}
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| \Lambda(\alpha^{i+1}) &=& \lambda_0 &+& \lambda_1 (\alpha^{i+1}) &+& \lambda_2 (\alpha^{i+1})^2 &+& \cdots &+& \lambda_t (\alpha^{i+1})^t \\
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| &=& \lambda_0 &+& \lambda_1 (\alpha^i)\,\alpha &+& \lambda_2 (\alpha^i)^2\,\alpha^2 &+& \cdots &+& \lambda_t (\alpha^i)^t\,\alpha^t \\
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| &=& \gamma_{0,i} &+& \gamma_{1,i}\,\alpha &+& \gamma_{2,i}\,\alpha^2 &+& \cdots &+& \gamma_{t,i}\,\alpha^t \\
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| &\triangleq& \gamma_{0,i+1} &+& \gamma_{1,i+1} &+& \gamma_{2,i+1} &+& \cdots &+& \gamma_{t,i+1}
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| \end{array}
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| </math>
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| In other words, we may define each <math>\Lambda(\alpha^i)</math> as the sum of a set of terms <math>\{\gamma_{j,i} | 0\leq j \leq t\}</math>, from which the next set of coefficients may be derived thus:
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| :<math>\ \gamma_{j,i+1} = \gamma_{j,i}\,\alpha^j</math> | |
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| In this way, we may start at <math>i = 0</math> with <math>\gamma_{j,0} = \lambda_j</math>, and iterate through each value of <math>i</math> up to <math>(q-1)</math>. If at any stage the resultant summation is zero, i.e.
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| :<math>\ \sum_{j=0}^t \gamma_{j,i} = 0,</math>
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| then <math>\Lambda(\alpha^i) = 0</math> also, so <math>\alpha_i</math> is a root. In this way, we check every element in the field.
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| When implemented in hardware, this approach significantly reduces the complexity, as all multiplications consist of one variable and one constant, rather than two variables as in the brute-force approach.
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| ==References==
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| *{{Citation
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| |first= R. T.
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| |last= Chien
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| |title= Cyclic Decoding Procedures for the Bose-Chaudhuri-Hocquenghem Codes
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| |journal= IEEE Transactions on Information Theory
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| |volume= IT-10
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| |issue= 4
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| |pages=357–363
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| |date= October 1964
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| |url=
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| |issn= 0018-9448
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| |doi=}}
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| *{{Citation
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| |first= S.
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| |last= Lin
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| |first2= D.
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| |last2= Costello
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| |title= Error Control Coding: Fundamentals and Applications
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| |publisher= Prentice-Hall
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| |location= Englewood Cliffs, NJ
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| |year= 2004
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| |isbn=
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| |doi= }}
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| *{{Citation
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| |last=Gill
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| |first= John
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| |title= EE387 Notes #7, Handout #28
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| |year= unknown
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| |accessdate= April 21, 2010
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| |pages=42–45
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| |publisher= Stanford University
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| |url= http://www.stanford.edu/class/ee387/handouts/notes7.pdf
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| |doi=}}
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| [[Category:Error detection and correction]]
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| [[Category:Finite fields]]
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