BCH codes
Howon Kim
2017. 6.5
BCH code1– Designing BCH codes
참고[A]:
- minimal polynomial?
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BCH code
참고[B]:
- BCH 코드의 generator polynomial g(X)를 계산할 때, 실제 a^1, a^2,
a^3, …, a^(2t)까지 할 필요 없음. a^1, a^3, …, a^(2t-1)만 하면 됨
참고[C]: LCM(최소공배수):
- LCM( polynomial, polynomial2, …, ). 각 polynomial이 m 차원이고 총
개수가 t개 이므로, LCM 결과 차수는 m*t보다는 작거나 같음
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BCH code
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BCH code
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BCH code
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BCH code
Parity Check Matrix of BCH Codes
“0”이 되도록 만들어짐 오류 없는 경우
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BCH code
“0”이 되도록 만들어짐 오류 없는 경우
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Decoding BCH Codes
General Approach
r(X) = c(X) + e(X)임. 에러가 없으면, r(X)=c(X)임
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…
Refer to BOOK …
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Thank you !
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참고 A – Basic Properties of GF(2m)
참고[A]: What is minimal polynomial ? (!= irreducible
polynomial)
Basic Properties of a GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 A – Basic Properties of GF(2m)
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참고 B
참고[B]
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참고 C
참고[C] : LCM(Least Common Multiples) of polynomials
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참고 – Basic Properties of GF(2m)
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참고 – Basic Properties of GF(2m)
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참고 – Basic Properties of GF(2m)
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참고 – Basic Properties of GF(2m)
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Fuzzy Extractor 구현 사례 1
1. Noise가 있는 PUF 출력값 w’과 수신된 s값을 XOR하면, r’값을 얻음.
2. 이 값 255비트 값을 BCH decoding하며, K 비트(131비트) 값을 얻음.
3. 다시 이 값( r 값)을 수신된 S와 XOR하면, w 값을 얻을 수 있음.
이 w와 x값을 XOR해서, SHA를 통과하면 R 값을 얻게됨.
이 R값의 역할은?
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Fuzzy Extractor 구현 사례 2
• Hash 함수 대신 BCH codes의 syndrome을 사용함
• The proposed method is more simply constructed by replacing the hash function
output with the syndrome from the BCH code
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Fuzzy Extractor 구현 사례 3
• Generation process:
• Secure Sketch (SS) is applied to PUF output w, as
• shown in Fig. 8. The second input to SS is the key
K, generated using a True Random Number
Generator, RNG1. The output of SS, denoted by s,
is stored as helper data in the database.
• Reproduction process:
• During the reproduction process, the helper data is used to regenerate the key K from a noisy PUF response w’. BCH
decoder is used to regenerate the 131 bit key as shown in the figure. We propose to use BCH with the following
parameters: (n=255, k=131, t=18) code.
The meaning of these parameters is as follows: n=255 is the output block size, k=131 is the input block size (in our
case, the size of the key to be encoded), and t=18 is the number of errors that can be corrected by this code. We
chose these parameters because the code with these parameters can easily correct the worse case errors
Please note that in our scheme, the key K is not a function of the PUF response during the Generation
process, but becomes a function of the PUF response during the Reproduction process in the field.
참고) Efficient SR-Latch PUF by Bilal Habib, Jens-Peter Kaps, Kris Gaj
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