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 se secu curi rity Al Alla Levi vina BF BFA 2017

Wave velet tra ransf sforma rmation

c0, c1, c2, c3, c4, c5, c6, c7, . . . , c2L−1 c1, c3, c5, c7, . . . , c2L−1 Wavelet transformation: aj = (c2j +c2j+1)/2, bj = (c2j −c2j+1)/2, j = 0, 1, . . . , L−1. c2j =aj +bj, c2j+1 =aj −bj, j=0,1,...,L−1 Main stream: a0,a1,a2,a3,a4,a5,a6,...,aL−1 Wavelet stream: b0,b1,b2,b3,b4,b5,b6,...,bL−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

Let Z be the set of all integers. On finite or infinite interval (α,β) of the real axis R1 consider the net: X , {xj}j2Z,

{ } X : . . . < x1 < x0 < x1 < . . . , for which α = lim

j!1 xj, β =

lim

j!+1 xj, 8j 2 Z.

In mathematics, a spline is a special function defined piecewise by polynomials.

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

defined are nodes for splines sented to ai = ci for i  k 3, ak2 = (xk ξ)(ξ xk2)1ck3+ +(xk xk2)(ξ xk2)1ck2, ai = ci+1 for i k 1, bj = 0 for j 6= k 1, bk1 = h (xk+1ξ)(xkξ)ck3(xk+1ξ)(xkxk2)ck2+ +(xk+1 xk1)(ξ xk2)ck1 (ξ xk1)(ξ xk2)ck i ⇥ ⇥(xk+1 xk1)1(ξ xk2)1.

Sp Spline-w

• wave

velet tra ransf sforma rmation

cj = aj + bj for j  k 3, ck2 = ak3(xk ξ)(xk xk2)1+ +ak2(ξ xk2)(xk xk2)1 + bk2, ck1 = ak2(xk+1 ξ)(xk+1 xk1)1+ +ak1(ξ xk1)(xk+1 xk1)1 + bk1, cj = aj1 + bj 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:              v1 v2 . . . vN/2 w1 w2 . . . wN/2              =           h1 h2 · · · hN hN−1 hN · · · hN−2 · · · · · · · · · · · · h3 h4 · · · h2 g1 g2 · · · gN gN−1 gN · · · gN−2 · · · · · · · · · · · · g3 g4 · · · g2                    x1 x2 x3 . . . . . . xN          (1), where {x1, x2, · · · , xN} is the

• riginal

sequence, {v1, v2, · · · , vN/2} is the main sequence, {w1, w2, · · · , wN/2} is wavelet sequence, {h1, h2, · · · , hN} and {g1, g2, · · · , gN} 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 effectiveness 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 redundant elements . Code can be made into a nonlinear systematic robust code:

• 1. by taking multiplicative inverse in of r redundant bits:
• 2. by calculation the cube in of r redundant bits:

L

n

r

L

C

) 2 (

r

GF

( )

( )

( )

( )

r k L

GF Px = v , GF x v x, = C 2 2 |

1 ∈

∈

−

) 2 (

r

GF

( )

( )

( )

( )

r k L

GF Px = v , GF x v x, = C 2 2 |

3 ∈

∈

Pro Propose sed Robust st Code Sch Scheme me

by taking multiplicative inverse in of r redundant bits: by calculation the cube in of r redundant bits:

) 2 (

r

GF ) 2 (

r

GF

Be Benefits s of wave velet co codes

s are shown in Table 1. Code Q(e) Undetectabl errors Hamming linear code 1 2k Partially robust Hamming code 1 2k−r Robust quadratic systematic code [2] 2−r Robust duplication code [2] 2−k Wavelet linear code 1 2k Wavelet robust code with encoding function 1/x 2−k Wavelet robust code with encoding function x3 2−k

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 (32, 16)-linear 1 216 wavelet code (32, 16)-robust 2−15 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

Thank you for your attention!