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Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $
The differential spectrum of a class of power functions over finite fields
1. | School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China |
2. | State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China |
3. | College of Mathematical sciences, Dezhou University, Dezhou, 253023, China |
Functions with good differential-uniformity properties have important applications in coding theory and sequence design in addition to the applications in cryptography. The differential spectrum of a cryptographic function is useful for estimating its resistance to some variants of differential cryptanalysis. The objective of this paper is to determine the differential spectrum of the power function $ x^{p^{2k}-p^k+1} $ over $ \mathbb F_{p^n} $, where $ p $ is an odd prime, $ n, k, e $ are integers with $ \gcd(n,k) = e $ and $ \frac{n}{e} $ being odd. In particular, when $ n $ is odd and $ e = 1 $, our result includes a recent one (IEEE Trans. Inform. Theory 65(10): 6819-6826) as a special case.
References:
[1] |
T. P. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy,
On almost perfect nonlinear functions over $F_{2^n}$, IEEE Trans. Inform. Theory, 52 (2006), 4160-4170.
doi: 10.1109/TIT.2006.880036. |
[2] |
E. Biham and A. Shamir,
Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.
doi: 10.1007/BF00630563. |
[3] |
C. Blondeau, A. Canteaut and P. Charpin,
Differential properties of power functions, Int. J. Inf. Coding Theory, 1 (2010), 149-170.
doi: 10.1504/IJICOT.2010.032132. |
[4] |
C. Blondeau, A. Canteaut and P. Charpin,
Differential properties of $x \mapsto x^{2^t-1}$, IEEE Trans. Inform. Theory, 57 (2011), 8127-8137.
doi: 10.1109/TIT.2011.2169129. |
[5] |
C. Blondeau and L. Perrin,
More differentially $6$-uniform power functions, Des. Codes Cryptogr., 73 (2014), 487-505.
doi: 10.1007/s10623-014-9948-2. |
[6] |
A. Canteaut and M. Videau,
Degree of composition of highly nonlinear functions and applications to higher order differential cryptanalysis, Advances in Cryptology - EUROCRYPT, Lecture Notes in Comput. Sci., Springer, Berlin, 2332 (2002), 518-533.
doi: 10.1007/3-540-46035-7_34. |
[7] |
C. Carlet, P. Charpin and V. Zinoviev,
Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.
doi: 10.1023/A:1008344232130. |
[8] |
C. Carlet and C. S. Ding,
Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.
doi: 10.1016/j.jco.2003.08.008. |
[9] |
C. Carlet, C. S. Ding and J. Yuan,
Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.
doi: 10.1109/TIT.2005.847722. |
[10] |
S.-T. Choi, S. Hong, J.-S. No and H. Chung,
Differential spectrum of some power functions in odd prime characteristic, Finite Fields Appl., 21 (2013), 11-29.
doi: 10.1016/j.ffa.2013.01.002. |
[11] |
R. S. Coulter and R. W. Matthews,
Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10 (1997), 167-184.
doi: 10.1023/A:1008292303803. |
[12] |
N. T. Courtois and J. Pieprzyk,
Cryptanalysis of block ciphers with overdefined systems of equations, Advances in Cryptology - ASIACRYPT, Lecture Notes in Comput. Sci., Springer, Berlin, 2501 (2002), 267-287.
doi: 10.1007/3-540-36178-2_17. |
[13] |
C. S. Ding, M. J. Moisio and J. Yuan,
Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610.
doi: 10.1109/TIT.2007.899545. |
[14] |
C. S. Ding and J. Yuan,
A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113 (2006), 1526-1535.
doi: 10.1016/j.jcta.2005.10.006. |
[15] |
H. Dobbertin, D. Mills, E. N. Muller and A. P. Willems,
APN functions in odd characteristic, Discrete Math., 267 (2003), 95-112.
doi: 10.1016/S0012-365X(02)00606-4. |
[16] |
H. Dobbertin,
Almost perfect nonlinear power functions on $GF(2^n)$: The Welch case, IEEE Trans. Inform. Theory, 45 (1999), 1271-1275.
doi: 10.1109/18.761283. |
[17] |
Y. Edel, G. Kyureghyan and A. Pott,
A new APN functions which is not equivalent to a power mapping, IEEE Trans. Inform. Theory, 52 (2006), 744-747.
doi: 10.1109/TIT.2005.862128. |
[18] |
T. Helleseth, C. M. Rong and D. Sandberg,
New families of almost perfect nonlinear power mapping, IEEE Trans. Inform. Theory, 45 (1999), 475-485.
doi: 10.1109/18.748997. |
[19] |
T. Helleseth and D. Sandberg,
Some power mappings with low differential uniformity, Appl. Algebra Engrg. Comm. Comput., 8 (1997), 363-370.
doi: 10.1007/s002000050073. |
[20] |
T. Helleseth,
Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.
doi: 10.1016/0012-365X(76)90100-X. |
[21] |
T. Jakobsen and L. R. Knudsen,
The interpolation attack on block ciphers, Fast Software Encryption - FSE, Lecture Notes in Comput. Sci., Springer, Berlin, 1267 (1997), 28-40.
doi: 10.1007/BFb0052332. |
[22] |
P. V. Kumar and O. Moreno,
Prime-phase sequences with periodic correlation properties better than binary sequences, IEEE Trans. Inform. Theory, 37 (1991), 603-616.
doi: 10.1109/18.79916. |
[23] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[24] |
S. X. Ma, H. L. Zhang, W. D. Jin and X. H. Niu,
A new family of optimal ternary cyclic codes, IEICE Trans. Fund., E97 (2014), 690-693.
doi: 10.1587/transfun.E97.A.690. |
[25] |
G. J. Ness and T. Helleseth,
A new family of ternary almost perfect nonlinear mappings, IEEE Trans. Inform. Theory, 53 (2007), 2581-2586.
doi: 10.1109/TIT.2007.899508. |
[26] |
A. Pott,
Almost perfect and planar functions, Des. Codes Cryptogr., 78 (2016), 141-195.
doi: 10.1007/s10623-015-0151-x. |
[27] |
H. Trachtenberg, On the Cross-Correlation Functions of Maximal Linear Sequences, Ph. D. thesis, University of Southern California, 1970. Google Scholar |
[28] |
M. S. Xiong and H. D. Yan,
A note on the differential spectrum of a differentially 4-uniform power function, Finite Fields Appl., 48 (2017), 117-125.
doi: 10.1016/j.ffa.2017.07.008. |
[29] |
M. S. Xiong, H. D. Yan and P. Z. Yuan,
On a conjecture of differentially 8-uniform power functions, Des. Codes Cryptogr., 86 (2018), 1601-1621.
doi: 10.1007/s10623-017-0416-7. |
[30] |
G. K. Xu, X. W. Cao and S. D. Xu,
Several classes of polynomials with low differential uniformity over finite fields of odd characteristic, Appl. Algebra Engrg. Comm. Comput., 27 (2016), 91-103.
doi: 10.1007/s00200-015-0272-5. |
[31] |
H. D. Yan and D. C. Han, A class of 3-uniform ternary power function and related codes, IEICE Trans. Fund., E102-A (2019), 849-853. Google Scholar |
[32] |
H. D. Yan, Z. C. Zhou, J. Weng, J. M. Wen, T. Helleseth and Q. Wang,
Differencial spectrum of Kasami power permutation over odd characteristic finite fields, IEEE Trans. Inform. Theory, 65 (2019), 6819-6826.
doi: 10.1109/TIT.2019.2910070. |
[33] |
X. Y. Zeng, L. Hu, W. F. Jiang, Q. Yue and X. W. Cao,
The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.
doi: 10.1016/j.ffa.2009.12.001. |
[34] |
Z. B. Zha and X. L. Wang,
Almost perfect nonlinear power functions in odd characteristic, IEEE Trans. Inform. Theory, 57 (2011), 4826-4832.
doi: 10.1109/TIT.2011.2145130. |
[35] |
Z. B. Zha and X. L. Wang,
Power functions with low uniformity on odd characteristic finite fields, Sci. China Math., 53 (2010), 1931-1940.
doi: 10.1007/s11425-010-3149-x. |
[36] |
Z. C. Zhou and C. S. Ding,
A class of three-weight codes, Finite Fields Appl., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
show all references
References:
[1] |
T. P. Berger, A. Canteaut, P. Charpin and Y. Laigle-Chapuy,
On almost perfect nonlinear functions over $F_{2^n}$, IEEE Trans. Inform. Theory, 52 (2006), 4160-4170.
doi: 10.1109/TIT.2006.880036. |
[2] |
E. Biham and A. Shamir,
Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.
doi: 10.1007/BF00630563. |
[3] |
C. Blondeau, A. Canteaut and P. Charpin,
Differential properties of power functions, Int. J. Inf. Coding Theory, 1 (2010), 149-170.
doi: 10.1504/IJICOT.2010.032132. |
[4] |
C. Blondeau, A. Canteaut and P. Charpin,
Differential properties of $x \mapsto x^{2^t-1}$, IEEE Trans. Inform. Theory, 57 (2011), 8127-8137.
doi: 10.1109/TIT.2011.2169129. |
[5] |
C. Blondeau and L. Perrin,
More differentially $6$-uniform power functions, Des. Codes Cryptogr., 73 (2014), 487-505.
doi: 10.1007/s10623-014-9948-2. |
[6] |
A. Canteaut and M. Videau,
Degree of composition of highly nonlinear functions and applications to higher order differential cryptanalysis, Advances in Cryptology - EUROCRYPT, Lecture Notes in Comput. Sci., Springer, Berlin, 2332 (2002), 518-533.
doi: 10.1007/3-540-46035-7_34. |
[7] |
C. Carlet, P. Charpin and V. Zinoviev,
Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.
doi: 10.1023/A:1008344232130. |
[8] |
C. Carlet and C. S. Ding,
Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.
doi: 10.1016/j.jco.2003.08.008. |
[9] |
C. Carlet, C. S. Ding and J. Yuan,
Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.
doi: 10.1109/TIT.2005.847722. |
[10] |
S.-T. Choi, S. Hong, J.-S. No and H. Chung,
Differential spectrum of some power functions in odd prime characteristic, Finite Fields Appl., 21 (2013), 11-29.
doi: 10.1016/j.ffa.2013.01.002. |
[11] |
R. S. Coulter and R. W. Matthews,
Planar functions and planes of Lenz-Barlotti class II, Des. Codes Cryptogr., 10 (1997), 167-184.
doi: 10.1023/A:1008292303803. |
[12] |
N. T. Courtois and J. Pieprzyk,
Cryptanalysis of block ciphers with overdefined systems of equations, Advances in Cryptology - ASIACRYPT, Lecture Notes in Comput. Sci., Springer, Berlin, 2501 (2002), 267-287.
doi: 10.1007/3-540-36178-2_17. |
[13] |
C. S. Ding, M. J. Moisio and J. Yuan,
Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610.
doi: 10.1109/TIT.2007.899545. |
[14] |
C. S. Ding and J. Yuan,
A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113 (2006), 1526-1535.
doi: 10.1016/j.jcta.2005.10.006. |
[15] |
H. Dobbertin, D. Mills, E. N. Muller and A. P. Willems,
APN functions in odd characteristic, Discrete Math., 267 (2003), 95-112.
doi: 10.1016/S0012-365X(02)00606-4. |
[16] |
H. Dobbertin,
Almost perfect nonlinear power functions on $GF(2^n)$: The Welch case, IEEE Trans. Inform. Theory, 45 (1999), 1271-1275.
doi: 10.1109/18.761283. |
[17] |
Y. Edel, G. Kyureghyan and A. Pott,
A new APN functions which is not equivalent to a power mapping, IEEE Trans. Inform. Theory, 52 (2006), 744-747.
doi: 10.1109/TIT.2005.862128. |
[18] |
T. Helleseth, C. M. Rong and D. Sandberg,
New families of almost perfect nonlinear power mapping, IEEE Trans. Inform. Theory, 45 (1999), 475-485.
doi: 10.1109/18.748997. |
[19] |
T. Helleseth and D. Sandberg,
Some power mappings with low differential uniformity, Appl. Algebra Engrg. Comm. Comput., 8 (1997), 363-370.
doi: 10.1007/s002000050073. |
[20] |
T. Helleseth,
Some results about the cross-correlation function between two maximal linear sequences, Discrete Math., 16 (1976), 209-232.
doi: 10.1016/0012-365X(76)90100-X. |
[21] |
T. Jakobsen and L. R. Knudsen,
The interpolation attack on block ciphers, Fast Software Encryption - FSE, Lecture Notes in Comput. Sci., Springer, Berlin, 1267 (1997), 28-40.
doi: 10.1007/BFb0052332. |
[22] |
P. V. Kumar and O. Moreno,
Prime-phase sequences with periodic correlation properties better than binary sequences, IEEE Trans. Inform. Theory, 37 (1991), 603-616.
doi: 10.1109/18.79916. |
[23] |
R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[24] |
S. X. Ma, H. L. Zhang, W. D. Jin and X. H. Niu,
A new family of optimal ternary cyclic codes, IEICE Trans. Fund., E97 (2014), 690-693.
doi: 10.1587/transfun.E97.A.690. |
[25] |
G. J. Ness and T. Helleseth,
A new family of ternary almost perfect nonlinear mappings, IEEE Trans. Inform. Theory, 53 (2007), 2581-2586.
doi: 10.1109/TIT.2007.899508. |
[26] |
A. Pott,
Almost perfect and planar functions, Des. Codes Cryptogr., 78 (2016), 141-195.
doi: 10.1007/s10623-015-0151-x. |
[27] |
H. Trachtenberg, On the Cross-Correlation Functions of Maximal Linear Sequences, Ph. D. thesis, University of Southern California, 1970. Google Scholar |
[28] |
M. S. Xiong and H. D. Yan,
A note on the differential spectrum of a differentially 4-uniform power function, Finite Fields Appl., 48 (2017), 117-125.
doi: 10.1016/j.ffa.2017.07.008. |
[29] |
M. S. Xiong, H. D. Yan and P. Z. Yuan,
On a conjecture of differentially 8-uniform power functions, Des. Codes Cryptogr., 86 (2018), 1601-1621.
doi: 10.1007/s10623-017-0416-7. |
[30] |
G. K. Xu, X. W. Cao and S. D. Xu,
Several classes of polynomials with low differential uniformity over finite fields of odd characteristic, Appl. Algebra Engrg. Comm. Comput., 27 (2016), 91-103.
doi: 10.1007/s00200-015-0272-5. |
[31] |
H. D. Yan and D. C. Han, A class of 3-uniform ternary power function and related codes, IEICE Trans. Fund., E102-A (2019), 849-853. Google Scholar |
[32] |
H. D. Yan, Z. C. Zhou, J. Weng, J. M. Wen, T. Helleseth and Q. Wang,
Differencial spectrum of Kasami power permutation over odd characteristic finite fields, IEEE Trans. Inform. Theory, 65 (2019), 6819-6826.
doi: 10.1109/TIT.2019.2910070. |
[33] |
X. Y. Zeng, L. Hu, W. F. Jiang, Q. Yue and X. W. Cao,
The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.
doi: 10.1016/j.ffa.2009.12.001. |
[34] |
Z. B. Zha and X. L. Wang,
Almost perfect nonlinear power functions in odd characteristic, IEEE Trans. Inform. Theory, 57 (2011), 4826-4832.
doi: 10.1109/TIT.2011.2145130. |
[35] |
Z. B. Zha and X. L. Wang,
Power functions with low uniformity on odd characteristic finite fields, Sci. China Math., 53 (2010), 1931-1940.
doi: 10.1007/s11425-010-3149-x. |
[36] |
Z. C. Zhou and C. S. Ding,
A class of three-weight codes, Finite Fields Appl., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
condition | references | ||||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [28] | |||
2 | 6 | [4] | |||
2 | [5] | ||||
2 | [29] | ||||
2 | [29] | ||||
3 | [31] | ||||
odd | [10] | ||||
odd | [10] | ||||
odd | [32] | ||||
odd | This paper |
condition | references | ||||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [3] | |||
2 | 4 | [28] | |||
2 | 6 | [4] | |||
2 | [5] | ||||
2 | [29] | ||||
2 | [29] | ||||
3 | [31] | ||||
odd | [10] | ||||
odd | [10] | ||||
odd | [32] | ||||
odd | This paper |
3 | |||||
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