Skip to main content
SpringerLink
Log in

Finite field trigonometric transforms

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

In this paper, the finite field trigonometric transforms (FFTT) are introduced. The FFTT family includes eight types of cosine transforms and eight types of sine transforms, which are developed from concepts related to the finite field trigonometry. New lemmas are presented and some properties of the FFTT are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baktir, S., Sunar, B.: Finite field polynomial multiplication in the frequency domain with application to elliptic curve cryptography. In: Proceedings of the 21st International Symposium on Computer and Information Sciences (ISCIS 2006), Lecture Notes in Computer Science (LNCS), vol. 4263, pp. 991–1001. Springer, Heidelberg (2006)

  2. Blahut R.E.: Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA (1983)

    MATH  Google Scholar 

  3. Blahut R.E.: Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading (1985)

    MATH  Google Scholar 

  4. Bracewell R.: The Fourier Transform and its Applications. 3rd edn. McGraw-Hill, New York (1999)

    Google Scholar 

  5. Campello de Souza, R.M., Freire, E.S.V., de Oliveira, H.M.: Fourier codes. In: Proceeding of the 10th International Symposium on Communication Theory and Application. Ambleside (2009)

  6. Campello de Souza R.M., de Oliveira H.M.: The complex Hartley transform over a finite field. In: Farnell, P.G., Darnell, M., Honary, B. (eds) Coding, Communications and Broadcasting, pp. 267–276. Research Studies Press, Jonh Wiley, Hertfordshire (2000)

    Google Scholar 

  7. Campello de Souza, R.M., de Oliveira, H.M., Kauffman, A.N., Pachoal, A.J.A.: Trigonometry in finite fields and a new Hartley transform. In: Proceeding of IEEE International Symposium on Information Theory, p. 293. Boston (1998)

  8. Chan K.S., Fekri F.: A block cipher cryptosystem using wavelet transforms over finite fields. IEEE Trans. Signal Process. 52(10), 2975–2991 (2004)

    Article  MathSciNet  Google Scholar 

  9. Cintra R., Dimitrov V., Campello de Souza R., de Oliveira H.: Fragile watermarking using finite field trigonometric transforms. Signal Process. Image Commun. 24(7), 587–597 (2009)

    Article  Google Scholar 

  10. Creutzburg R., Tasche M.: Number-theoretic transforms of prescribed length. Math. Comput. 47, 693–701 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. de Oliveira H.M., Campello de Souza R.M.: Orthogonal multilevel spreading sequence design. In: Farrell, P.G., Darnell, M., Honary, B. (eds) Coding, Communications and Broadcasting, pp. 291–303. Research Studies Press, Jonh Wiley, Hertfordshire (2000)

    Google Scholar 

  12. de Oliveira, H.M., Campello de Souza, R.M., Kauffman, A.N.: Efficient multiplex for band-limited channels: galois-field multiple access. In: Proceeding of the Workshop on Coding and Criptography, pp. 235–241. Cambridge, (1999)

  13. de Souza M.M.C., de Oliveira H.M., Campello de Souza R.M., Vasconcelos M.M.: The discrete cosine transform over prime finite fields. In: de Souza, J.N., Dini, P., Lorenz, P. (eds) International Conference on Telecommunications, Lecture Notes in Computer Science, vol. 3124, pp. 482–487. Springer, Berlin (2004)

    Google Scholar 

  14. Dubois E., Venetsanopoulos A.N.: The generalized discrete fourier transform in rings of algebraic integers. IEEE Trans. Acoust. Speech Signal Process. 28(1), 169–175 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fekri F., McLaughlin S.W., Mersereau R.M., Schafer R.W.: Block error correcting codes using finite-field wavelet transforms. IEEE Trans. Signal Process. 54(3), 991–1004 (2006)

    Article  Google Scholar 

  16. Fekri F., Sartipi M.W., Mersereau R.M., Schafer R.W.: Convolutional codes using finite-field wavelets: time-varying codes and more. IEEE Trans. Signal Process. 53(5), 1881–1896 (2005)

    Article  MathSciNet  Google Scholar 

  17. Guralnick, R.M., Lorenz, M.: Orders of Finite Groups of Matrices. http://www.citebase.org/abstract?id=oai:arXiv.org:math/0511191 (2005)

  18. Lima J.B., Campellode Souza R.M.: The fractional fourier transform over finite fields. Signal Processing 92(2), 465–476 (2012)

    Article  Google Scholar 

  19. Lima J.B., Campello de Souza R.M., Panario D.: The eigenstructure of finite field trigonometric transforms. Linear Algebra Appl. 435, 1956–1971 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Martucci S.A.: Symmetric convolution and the discrete sine and cosine transforms. IEEE Trans. Signal Process. 42(5), 1038–1051 (1994)

    Article  Google Scholar 

  21. Merched R.: On OFDM and single-carrier frequency-domain systems based on trigonometric transforms. IEEE Signal Process. Lett. 13(8), 473–476 (2006)

    Article  Google Scholar 

  22. Oppenheim A.V., Schafer R.W., Buck J.R.: Discrete-Time Signal Processing. 2nd edn. Prentice Hall, Englewood Cliffs, NJ (1999)

    Google Scholar 

  23. Paar C.: A new architecture for a parallel finite field multiplier with low complexity based on composite fields. IEEE Trans. Comput. 45(7), 856–861 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  24. Pollard J.M.: The fast Fourier transform in a finite field. Math. Comput. 25(114), 365–374 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rao K.R., Yip P.: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic, San Diego (1990)

    MATH  Google Scholar 

  26. Reed I.S., Truong T.K.: The use of finite fields to compute convolutions. IEEE Trans. Inf. Theory 21(2), 208–213 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rubanov N.S., Bovbel E.I., Kukharchik P.D., Bodrov V.J.: The modified number theoretic transform over the direct sum of finite fields to compute the linear convolution. IEEE Trans. Signal Process. 46(3), 813–817 (1998)

    Article  MathSciNet  Google Scholar 

  28. Steidl, G., Creutzburg, R.: Number-theoretic transforms in rings of cyclotomic integers. Elektronische Informationsverarbeitung und Kybernetik, pp. 573–584 (1988)

  29. Toivonen T., Heikkilä J.: Video filtering with Fermat number theoretic transforms using residue number system. IEEE Trans. Circuits Syst. Video Tech. 16(1), 92–101 (2006)

    Article  Google Scholar 

  30. Ye G.: Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit. Lett. 31(5), 347–354 (2010)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Polytechnic School of Pernambuco, University of Pernambuco, Rua Benfica, 455, Madalena, Recife, PE, 50750-470, Brazil

    Juliano B. Lima

  2. Department of Electronics and Systems, Federal University of Pernambuco, Av. Acad. Hélio Ramos, S/N, 4° andar—Cidade Universitária, Recife, PE, 50740-530, Brazil

    Ricardo M. Campello de Souza

Authors
  1. Juliano B. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo M. Campello de Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juliano B. Lima.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lima, J.B., Campello de Souza, R.M. Finite field trigonometric transforms. AAECC 22, 393–411 (2011). https://doi.org/10.1007/s00200-011-0158-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-011-0158-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation