BF BFA 2017 Wave velet tra ransf sforma rmation c 0 , c 1 , c 2 - PowerPoint PPT Presentation

Wave velet tra ransf sforma rmation and its s applica cation in informa rmation secu se curi rity Al Alla Levi vina BF BFA 2017

Wave velet tra ransf sforma rmation c 0 , c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 , . . . , c 2L − 1 c 1 , c 3 , c 5 , c 7 , . . . , c 2L − 1 Wavelet transformation: a j = (c 2j +c 2j+1 )/2, b j = (c 2j − c 2j+1 )/2, j = 0, 1, . . . , L − 1. c 2j =a j +b j , c 2j+1 =a j − b j , j=0,1,...,L − 1 Main stream: a 0 ,a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,...,a L − 1 Wavelet stream: b 0 ,b 1 ,b 2 ,b 3 ,b 4 ,b 5 ,b 6 ,...,b L − 1

Time meline First wavelet (Haar wavelet) by Alfréd Haar (1909) Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Ali Akansu, Victor Wickerhauser

Time meline Yves Meyer Abel Prize in 2017

Sp Spline-w -wave velet tra ransf sforma rmation In mathematics, a spline is a special function defined piecewise by polynomials . Let Z be the set of all integers. On finite or infinite interval ( α , β ) of the real axis R 1 consider the net: X , { x j } j 2 Z , { } X : . . . < x � 1 < x 0 < x 1 < . . . , for which α = j !�1 x j , β = lim j ! + 1 x j , 8 j 2 Z . lim

Cubic c Sp Spline

Time meline • The interpolatory spline wavelets introduced by C.K. Chui and J.Z. Wang 1991 «A cardinal spline approach to wavelet» • 1990 Demjanovish Y.K. « Локальная аппроксимация на многообразии »

Sp Spline-w -wave velet tra ransf sforma rmation a i = c i for i  k � 3 , defined a k � 2 = � ( x k � ξ )( ξ � x k � 2 ) � 1 c k � 3 + +( x k � x k � 2 )( ξ � x k � 2 ) � 1 c k � 2 , are a i = c i +1 for i � k � 1 , nodes for splines b j = 0 for j 6 = k � 1 , h b k � 1 = ( x k +1 � ξ )( x k � ξ ) c k � 3 � ( x k +1 � ξ )( x k � x k � 2 ) c k � 2 + sented to i +( x k +1 � x k � 1 )( ξ � x k � 2 ) c k � 1 � ( ξ � x k � 1 )( ξ � x k � 2 ) c k ⇥ ⇥ ( x k +1 � x k � 1 ) � 1 ( ξ � x k � 2 ) � 1 .

Sp Spline-w -wave velet tra ransf sforma rmation c j = a j + b j for j  k � 3 , c k � 2 = a k � 3 ( x k � ξ )( x k � x k � 2 ) � 1 + + a k � 2 ( ξ � x k � 2 )( x k � x k � 2 ) � 1 + b k � 2 , c k � 1 = a k � 2 ( x k +1 � ξ )( x k +1 � x k � 1 ) � 1 + + a k � 1 ( ξ � x k � 1 )( x k +1 � x k � 1 ) � 1 + b k � 1 , c j = a j � 1 + b j for j � k. (1 Theorem 3. For the second-order spline-wavelet decomp

Sp Spline-w -wave velets s (wave velets) tra ransf sforma rmation in informa rmation se secu curi rity Wavelet linear codes Spline-wavelets linear codes Spline-wavelets robust codes Wavelet robust codes/AMD codes Bent Functions build on spline-wavelet transformation

Wavelet codes

Wave velet tra ransf sforma rmation Wavelet transform can be represent in matrix form:  v 1  h 1 h 2 · · · h N   v 2 x 1     h N − 1 h N · · · h N − 2 . x 2   .   . · · · · · · · · · · · ·       x 3       v N/ 2 h 3 h 4 · · · h 2     .   = (1) , .       w 1 g 1 g 2 · · · g N .           . w 2 g N − 1 g N · · · g N − 2      .    .     . · · · · · · · · · · · ·   .   . x N   g 3 g 4 · · · g 2 w N/ 2 where { x 1 , x 2 , · · · , x N } is the original sequence, { v 1 , v 2 , · · · , v N/ 2 } is the main sequence, { w 1 , w 2 , · · · , w N/ 2 } is wavelet sequence, { h 1 , h 2 , · · · , h N } and { g 1 , g 2 , · · · , g N } are coefficients of scaling function.

Linear r wave velets s co codes

Robust st co codes s (n (nonlinear r co codes) s) Mark Karpovsky, Boston University

Robust st co codes cyclic codes from the previous section. Robust codes are nonlinear systematic error-detecting codes that provide uni- form protection against all errors without any (or that minimize) assumptions about the error and fault distributions, capabilities and methods of an attacker. One of the main criteria for evaluating the e ff ectiveness of a robust code is the error masking probability . The error masking probability Q ( e ) can be defined as: Q ( e ) = |{ x | ∈ C, x + e ∈ C }| , M where C is the robust code, x is a codeword that belongs to the code C , e is an error, and M is the number of codewords in the code C .

Robust st co codes

Me Method of co const stru ruct cting Syst Systema matic c Robust st co code fro rom m Linear r Codes Let be a binary linear code with length and amount of n C L r redundant elements . Code can be made into a nonlinear L systematic robust code: r 1. by taking multiplicative inverse in of r redundant bits: GF ( 2 ) 1 ∈ ( ) ( ) ( ) ( ) − k r C = x, v | x GF 2 , v = Px GF 2 ∈ L 2. by calculation the cube in of r redundant bits: r GF ( 2 ) 3 ∈ ( ) ( ) ( ) ( ) k r C = x, v | x GF 2 , v = Px GF 2 ∈ L

Pro Propose sed Robust st Code Sch Scheme me r GF ( 2 ) by taking multiplicative inverse in of r redundant bits: r GF ( 2 ) by calculation the cube in of r redundant bits:

Be Benefits s of wave velet co codes s are shown in Table 1. Q(e) Undetectabl Code errors 2 k Hamming linear code 1 2 k − r Partially robust Hamming code 1 2 − r Robust quadratic systematic code [2] 0 2 − k Robust duplication code [2] 0 2 k Wavelet linear code 1 Wavelet robust code 2 − k 0 with encoding function 1 /x Wavelet robust code 2 − k 0 with encoding function x 3

Spline-wavelet codes

Imp mpleme mentation in AD ADV6 V612 Table 1. Comparison of the maximum error masking probability Q ( e ) and number of undetected errors for the ADV612 computer model. Wavelet code Number of the undetected parameters max Q ( e ) errors System without codes 1 All errors 2 16 (32 , 16)-linear 1 wavelet code 2 − 15 (32 , 16)-robust 0 wavelet code

Imp mpleme mentation in AD ADV6 V612 Compared Encoding rate in Encoding rate in constructions system without wavelets ADV612 computer model System without codes 3 , 14 c 2 , 93 c Linear 3 , 32 c 3 , 18 c wavelet code Robust wavelet code 3 , 51 c 3 , 36 c with w − 1 nonlinear part

AMD AMD co codes 2008 Ronald Cramer AMD codes

Wave velet AMD AMD co codes

BF BF on wave velet tra ransf sforma rmation

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