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In this presentation slide i have Explained how to reducing Computational time complexity of Discrete Fourier transform(DFT) from O(n^2 ) to nlogn through Radix-2 .FFT Algorithm in this work i have also introduced how we can use Radix-2 FFT in encrypted signal processing application by considering homomarphic properties(RSA) of Paillier cryptosystem.

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- 1. Implementation of Decimation in Time-Radix-2 FFT Algorithms in Signal Processing of Encrypted Domain(SPED) AJAY KUMAR.M.ANNAIAH Ph.D Research scholar Dept of IT NITK-Surathkal ajaymodaliger@gmail.com
- 2. 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
- 3. 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
- 4. 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-InFrequency (DIF) frameworks for separate parts of the computation to achieve a reduction in the computational complexity.
- 5. Discrete Fourier Transform(DFT) Allows us to compute an approximation of the Fourier Transform on a discrete set of frequencies from a discrete set of time samples. N 1 Xk xn e j2 k n N for k 0, 1,, N 1 n 0 Where k are the index of the discrete frequencies and n the index of the time samples N complex multiplies N-1 complex addition
- 6. Symmetries properties of FFT FFT algorithms exploits two symmetric properties Complex conjugate symmetry K W N N-n WN Kn * N Kn W Periodicity n W K N K N n N W k N n N W Finally WN e j2 k n N Kn or WN e j2 k n N
- 7. Fast Fourier Transform Cooley-Tukey algorithm: Kn WN e j2 k n N Based on decimation, leads to a factorization of computations. Let us first look at the classical radix 2 decimation in time. FFT uses the Divide and conquer rule split the Big DFT computation between odd and even part N 1 Xk xn e j2 k n N for k 0, 1,, N 1 n 0 N 1 N 1 kn x n WN X k n 0 kn x n WN n 0
- 8. Fast Fourier Transform Consider and replace even and odd indices part Even part of n2r Odd part of n2r+1 for all r=0,1…N/2-1 N /2 1 N /2 1 k2r x 2r WN X k r 0 N/2 1 Xk k x 2r 1 WN 2r n 0 2 kr N N/2 1 2kr x 2r 1 WN x 2r W r 0 2 kr N x 2r W r 0 K WN n 0 N /2 1 X k 1 N /2 1 W K N x 2r 1 W n 0 2 Kr N
- 9. Fast Fourier Transform N /2 1 X k 2 kr N x 2r W N /2 1 W K N x 2r 1 W r 0 n 0 Simplify the term N/2 1 r 0 Xk X k e WN WN 2 N/2 1 kr x 2r WN 2 Xk 2 Kr N K WN x 2r 1 WNkr2 n 0 K WN X 0 k Now the sum of two N/2 point DFT’s we can use to get a N point DFT
- 10. 2 point Butterfly Example if N=8 the even number[0,2,4,8] odd number[0.3.5.7] X(0) X(1) x(0) x(2) TFD N/2 •N/2(N/2-1) complex ‘+’ for each N/2 DFT. •(N/2)2 complex ‘ ’ for each DFT. x(N-2) X(N/2-1) W0 W1 x(1) x(3) We need: - X(N/2) X(N/2+1) •N/2 complex ‘ ’ at the input of the butterflies. •N complex ‘+’ for the butterflies. •Grand total: N2/2 complex ‘+’ TFD N/2 N/2(N/2+1) complex ‘ ’ WN/2-1 x(N-1) N 2 2 - X(N-1) .2.......... .......... ...... N
- 11. Fast Fourier Transform 2.point FFT splitting in to multiple pass i.e. N N 2 2 2 .2.......... .......... ...... N ..... * N 4 ......... * N 8 ...... 8 * N 16 2 4 till simplify the given form by applying mathematical rule … Finally computational time complexity of Radix-2 FFT algorithm is N log 2 N
- 12. Algorithm Parameters 2/2 The parameters are shown below: 1st stage Node Spacing Butterflies per group Number of groups Twiddle factor 2nd stage 3rd stage … Last stage 1 2 3 … N/2 1 2 3 … N/2 N/2 N/4 N/8 … 1 …
- 13. Algorithm Parameters The FFT can be computed according to the following pseudo-code: For each stage For each group of butterfly For each butterfly compute butterfly end end end
- 14. Number of Operations If N=2r, we have r=log2(N) stages. For each one we have: N/2 complex ‘ ’ (some of them are by ‘1’). N complex ‘+’. Thus the grand total of operations is: N/2 log2(N) complex ‘ ’. N log2(N) complex ‘+’. N 128 1024 4096 + 896 10240 49152 x 448 5120 24576 These counts can be compared with the ones for the DFT
- 15. 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.
- 16. 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
- 17. 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 radix2, 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
- 18. 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
- 19. 7.Public key cryptography for signal encryption
- 20. 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)
- 21. Homomorphism signal encryption Signal encryption Homomorphism encryption
- 22. 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.
- 23. Encrypted domain DFT (e-DFT) Consider the DFT sequence x(n) is defined as : N 1 X k xn e j2 k n N for k 0, 1, , N 1 n 0 N 1 X k xnW nk for k 0, 1,, N 1 n 0 w and x(n) is a finite duration sequences with length M Consider the scenario the where electronic processor fed the input data signal as encrypted data format as in digital form such as 0’s and 1’s
- 24. Encrypted domain DFT (e-DFT) Encrypted input data signal in the form of digital 0’s and 1’s in the form of equation E(X)=(E[x(0)],E[x(1)],…..E[x(N-1)] in order to make possible linear computation for encrypted input signal use homographic technique of Additive that is represented by E(x + y) = E(x) . E(y)
- 25. 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
- 26. . THANK YOU

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