Implementation of Decimation in Time-Radix-2 Implementation of Decimation in Time-Radix-2 FFT Algorithms in Signal Processing of FFT Algorithms in Signal Processing of

Encrypted Domain(SPED)Encrypted Domain(SPED)

AJAY KUMAR.M.ANNAIAHAJAY KUMAR.M.ANNAIAHPh.D Research scholar Ph.D Research scholar

Dept of ITDept of IT

NITK-SurathkalNITK-Surathkal

ajaymodaliger@gmail.comajaymodaliger@gmail.com

CONTENT INTRODUCTION

REVIEW OF DFT –FFT ALGORITHM

TIME COMPLEXITY ANALYSIS OF RADIX-2

S.P.E.D

HOMOMORPHISM V/S SIGNAL ENCRYPTION

IMPLEMENTATION OF e-DFT

1.Introduction

• Discrete Fourier Transform(DFT) invented around 1805 by Carls Friedrich Gauss.

Limitation –computation time.

• In the mid-1965 DFT is reinvented as The Fast Fourier Transform (FFT), By Cooley-Tukey

• FFT reduced the complexity of a Discrete Fourier Transform from O(N²), to O(N·logN),

purpose of reducing time complexity

large number of Application developed

• FFT algorithms became known as the Radix- 2 algorithm and was shortly followed by the Radix-3, Radix-4, and Mixed Radix algorithms

• Evaluation of research Fast Hartley Transform (FHT)and the Split Radix (SRFFT), QFT. DITF

2.Review of FFT Algorithms• The basic principle behind most Radix-based FFT algorithms is to exploit the

symmetry properties of a complex exponential that is the cornerstone of the Discrete Fourier Transform (DFT). These algorithms

Complex conjugate symmetry

Periodicity

• Divide the problem into similar sub-problems (butterfly computations) and achieve a reduction in computational complexity.

• All Radix algorithms are similar in structure differing only in the core computation of the butterflies. The FHT differs from the other algorithms in that it uses a real kernel

• The DITF algorithm uses both the Decimation-In-Time (DIT) and Decimation-In-Frequency (DIF) frameworks for separate parts of thecomputation to achieve a reduction in the computational complexity.

Discrete Fourier Transform(DFT)Discrete Fourier Transform(DFT) Allows us to compute an approximation of Allows us to compute an approximation of

the Fourier Transform on a discrete set of the Fourier Transform on a discrete set of frequencies from a discrete set of time frequencies from a discrete set of time samples.samples.

Where Where k k are the index of the discrete are the index of the discrete frequencies and frequencies and n n the index of the time samplesthe index of the time samples

N N complex multiplies complex multiplies N-1 N-1 complex addition complex addition

1N10kenxkXn

N

k2j1N

0n

,, ,for

1N10kenxkXn

N

k2j1N

0n

,, ,for

Symmetries properties of FFTSymmetries properties of FFT FFT algorithms exploits two symmetric properties FFT algorithms exploits two symmetric properties

Complex conjugate symmetryComplex conjugate symmetry

Periodicity Periodicity

FinallyFinally

or or

*N

K

N Wn -N

W KnN

Kn W

*N

K

N Wn -N

W KnN

Kn W

nNkN

nNK W N K

N Wn W nNkN

nNK W N K

N Wn W

nN

k2j

N eW

nN

k2j

N eW

n

N

k2j

KnN eW

n

N

k2j

KnN eW

Fast Fourier Transform Fast Fourier Transform Cooley-Tukey algorithm:Cooley-Tukey algorithm:

Based on decimation, leads to a factorization of Based on decimation, leads to a factorization of computations.computations.

Let us first look at the classical Let us first look at the classical radix 2 radix 2 decimation in time.decimation in time.

FFT uses the Divide and conquer rule split the FFT uses the Divide and conquer rule split the Big DFT computation between Big DFT computation between odd and even odd and even partpart

knN

1

0

knN

1

0

W W

N

n

N

n

nxnxkX knN

1

0

knN

1

0

W W

N

n

N

n

nxnxkX

1N10kenxkXn

N

k2j1N

0n

,, ,for

1N10kenxkXn

N

k2j1N

0n

,, ,for

nN

k2j

KnN eW

n

N

k2j

KnN eW

Fast Fourier Transform Fast Fourier Transform Consider and replace even and odd indices partConsider and replace even and odd indices part Even part of nEven part of n2r2r Odd part of nOdd part of n2r+1 for all r=0,1…N/2-12r+1 for all r=0,1…N/2-1

12rkN

12/

0

k2rN

12/

0

W12 W2

N

n

N

r

rxrxkX 12rkN

12/

0

k2rN

12/

0

W12 W2

N

n

N

r

rxrxkX

KN

12N

0n

kr12N

0r

W1r2xr2xkX

2krN

/2N

/

WW KN

12N

0n

kr12N

0r

W1r2xr2xkX

2krN

/2N

/

WW

Kr212/

0

2N

12/

0

12W 2 N

N

n

KN

krN

r

WrxWrxkX

Kr212/

0

2N

12/

0

12W 2 N

N

n

KN

krN

r

WrxWrxkX

Fast Fourier Transform Fast Fourier Transform

Simplify the termSimplify the term

Now the sum of two N/2 point DFT’s we Now the sum of two N/2 point DFT’s we can use to get a N point DFTcan use to get a N point DFT

NWNW 2

NW2

NW

Kr212/

0

2N

12/

0

12W 2 N

N

n

KN

krN

r

WrxWrxkX

Kr212/

0

2N

12/

0

12W 2 N

N

n

KN

krN

r

WrxWrxkX

kr12N

0n

KN

12N

0r2

1r2xWr2xkX N2

N W W/

kr/

kr12N

0n

KN

12N

0r2

1r2xWr2xkX N2

N W W/

kr/

kXWke

XkX 0KN kXWk

eXkX 0

KN

2 point Butterfly2 point Butterfly Example if N=8 the even number[0,2,4,8] Example if N=8 the even number[0,2,4,8]

odd number[0.3.5.7] odd number[0.3.5.7]

TFD N/2TFD N/2

TFD N/2TFD N/2

x(0)x(0)x(2)x(2)

x(N-2)x(N-2)

x(1)x(1)x(3)x(3)

x(N-1)x(N-1)

X(0)X(0)X(1)X(1)

X(N/2-1)X(N/2-1)

X(N/2)X(N/2)X(N/2+1)X(N/2+1)

X(N-1)X(N-1)

WW00

WW11

WWN/2-1N/2-1

----

--

We need:We need:

•N/2(N/2-1) N/2(N/2-1) complex ‘complex ‘+’ for +’ for each each N/2 N/2 DFT.DFT.

•((N/2N/2))2 2 complex ‘complex ‘×’ for each ×’ for each DFT.DFT.

•N/2 N/2 complex ‘complex ‘×’ at the input of ×’ at the input of the butterflies.the butterflies.

•N N complex ‘complex ‘+’ for the butter-+’ for the butter-flies.flies.

•Grand total:Grand total:

NN22/2 /2 complex ‘complex ‘+’+’

N/2(N/2+1) N/2(N/2+1) complex ‘complex ‘×’×’

N222

N ........................... N222

N ...........................

Fast Fourier Transform Fast Fourier Transform 2.point FFT splitting in to multiple pass i.e.2.point FFT splitting in to multiple pass i.e.

simplify the given form by applying simplify the given form by applying mathematical rule …mathematical rule …

Finally computational time complexity of Finally computational time complexity of Radix-2 FFT algorithm is Radix-2 FFT algorithm is

N222

N ........................... N222

N ...........................

16842 *8......*4.........*2..... NNNN till16842 *8......*4.........*2..... NNNN till

NN 2log NN 2log

Algorithm Parameters 2/2Algorithm Parameters 2/2

The parameters are shown below:The parameters are shown below:

1st stage 2nd stage 3rd stage … Last stage

Node Spacing

1 2 3 … N/2

Butterflies per group

1 2 3 … N/2

Number of groups

N/2 N/4 N/8 … 1

Twiddle factor

… / 2

0

N kNW

k

/ 4

0,1

N kNW

k

/8

0, ,3

N kNW

k

0, , / 2 1

kNW

k N

Algorithm Parameters Algorithm Parameters

The FFT can be computed according to the The FFT can be computed according to the following pseudo-code:following pseudo-code: For each For each stagestage

For each For each group of butterflygroup of butterfly For each For each butterflybutterfly

compute butterflycompute butterfly endend

endend endend

Number of OperationsNumber of Operations

If If NN==22rr, , we have we have r=r=loglog22((NN) stages. For each ) stages. For each one we have:one we have: N/2N/2 complex ‘ complex ‘×’ (some of them are by ‘1’).×’ (some of them are by ‘1’). NN complex ‘+’. complex ‘+’.

Thus the grand total of operations is:Thus the grand total of operations is: N/2N/2 log log22((NN)) complex ‘ complex ‘×’.×’. NN log log22((NN)) complex ‘ complex ‘+’.+’.

N + x

128 896 4481024 10240 51204096 49152 24576

These counts can be compared with the ones for the DFTThese counts can be compared with the ones for the DFT

3.Signal processing in encrypted domain

• Signal processing is an area of systems engineering electrical engineering and  applied mathematics that deals with operations on or analysis of analog  as well as digitized  signals representing time-varying or spatially varying physical quantities.

• Signals of interest can include sound. Electromagnetic radiation images and sensor  readings telecommunication transmission signals, and many others

• Signal transmission using electronic signal processing. Transducers  convert signals from other physical waveforms  to electrical current  or voltage waveforms, which then are processed, transmitted as electromagnetic waves, received and converted by another transducer to final form.

4. Encrypted Signal processing

• Statistical signal processing – analyzing and extracting information from signals and noise based on their stochastic properties

• Spectral estimation – for determining the spectral content (i.e., the distribution of power over frequency) of a time series

• Audio signal processing – for electrical signals representing sound, such as speech or music

• Speech signal processing – for processing and interpreting spoken words

• Image processing – in digital cameras, computers and various imaging systems

• Video processing – for interpreting moving pictures

• Filtering – used in many fields to process signals

• Time-frequency analysis – for processing non-stationary signals

5.Signal processing module v/s cryptosystem • Signal processing modules working directly on encrypted Signal data

provide better solution to application scenarios

• valuable signals must be protected from a malicious processing device.

• investigate the implementation of the discrete Fourier transform (DFT) in the encrypted domain, by using the homomorphic properties of the underlying cryptosystem.

• Several important issues are considered for the direct DFT, the radix-2, and the radix-4 fast Fourier algorithms, including the error analysis and the maximum size of the sequence that can be transformed.

• The results show that the radix-4 FFT is best suited for an encrypted domain implementation. With computational complexity and error analysis

6.Traditional approach of signal Encryption • Most of technological solutions proposed issues of multimedia security

rely on the use of cryptography.

• Early works in this direction by applying cryptographic primitives, is to build a secure layer on top of signal application.

• secure layer is able to protect them from leakage of critical information Signal processing modules.

• Examples of such an approach include the encryption of content before its transmission or storage (like encrypted digital TV channels), or wrapping multimedia objects into an encrypted system with an application (the reader)

• encryption layer is used only to protect the data against third parties and authorized to access the data.

• signal processing tools capable of operating directly on encrypted data highlighting the benefits offered by the availability

7.Public key cryptography for signal encryption

Homomorphism for encrypted domain • Homomorphic encryption is a concept where specific computations can be

performed on the cipher text of a message. The result of these computations is the same as if the operations were performed on the plaintext first and encrypted afterwards. So homomorphic encryption allows parties who do not have an decryption key and thus don't know the plaintext value, still perform computation on this value

• The two group homomorphism operations are the arithmetic addition and multiplication.

• A homomorphic encryption is additive is

E(x + y) = E(x) . E(y)

1) where E denotes an encryption function, 2) . denotes an operation depending on cipher

3) x and y are plaintext messages.

• A homomorphic encryption is multiplicative if:

E(x y) = E(x) . E(y)

Homomorphism signal encryption Signal encryption

Homomorphism  encryption       

simple example of how a homomorphic encryption scheme might work in cloud computing:

• Company X  has a very important data set (VIDS) that consists of the numbers 5 and 10.  To encrypt the data set, Company X multiplies each element in the set by 2, creating a new set whose members are 10 and 20.

• Company X  sends the encrypted VIDS set to the cloud for safe storage.  A few months later, the government contacts Company X and requests the sum of VIDS elements.   

• Company X  is very busy, so it asks the cloud provider to perform the operation.  The cloud provider, who only has access to the encrypted data set,  finds the sum of 10 + 20 and returns the answer 30.

• Company X decrypts the cloud provider’s reply and provides the government with the decrypted answer, 15.

Encrypted domain DFT (e-DFT)Encrypted domain DFT (e-DFT) Consider the DFT sequence x(n) is defined as :Consider the DFT sequence x(n) is defined as :

w and x(n) is a finite duration sequences with w and x(n) is a finite duration sequences with length M length M

Consider the scenario the where electronic Consider the scenario the where electronic processor fed the input data signal as encrypted processor fed the input data signal as encrypted data format as in digital form such as 0’s and 1’s data format as in digital form such as 0’s and 1’s

1 ,,1 ,0for 21

0

NkenxkX

nN

kjN

n

1 ,,1 ,0for 21

0

NkenxkX

nN

kjN

n

1 ,,1 ,0for 1

0

NkWnxkX nkN

n

1 ,,1 ,0for 1

0

NkWnxkX nkN

n

Encrypted domain DFT (e-DFT)Encrypted domain DFT (e-DFT)

Encrypted input data signal in the form Encrypted input data signal in the form of digital 0’s and 1’s in the form of of digital 0’s and 1’s in the form of equation equation

E(X)=E(X)=((E[x(0)],E[x(1)],…..E[x(N-1)]E[x(0)],E[x(1)],…..E[x(N-1)]

in order to make possible linear in order to make possible linear computation for encrypted input signal computation for encrypted input signal use homographic technique of Additive use homographic technique of Additive that is represented by that is represented by

E(x + y) = E(x)E(x + y) = E(x) . . E(y)E(y)

Encrypted domain DFT (e-DFT)

• Issues of DFT in SPED is

• both input sample of encrypted signal and DFT coefficients need to represented as integer values

• Paillier homographic cryptosystem uses modular operation

• Uses of FFT-Radix 4 reduces the time complexity in SPED and best suited for encryption

.

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