American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020080

The differential spectrum of a class of power functions over finite fields

Lei Lei 1,2, , Wenli Ren 3, and  Cuiling Fan 1,,

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

Corresponding author: Cuiling Fan

Received  October 2019 Revised  January 2020 Published  April 2020

Fund Project: The work of L. Lei and C. Fan was supported by the National Natural Science Foundation of China under Grant 11971395, and partially supported by National Cryptography Development Fund under Grant MMJJ20180103. The work of W. Ren was supported by Natural Science Foundation of Shandong Province under Grant ZR2018LA001

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.

Keywords: Power permutation, differential uniformity, differential spectrum, exponential sum.
Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.
Citation: Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, doi: 10.3934/amc.2020080
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.  Google Scholar

[2]

E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Carlet and C. S. Ding, Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[26]

A. Pott, Almost perfect and planar functions, Des. Codes Cryptogr., 78 (2016), 141-195.  doi: 10.1007/s10623-015-0151-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

E. Biham and A. Shamir, Differential cryptanalysis of DES-like cryptosystems, J. Cryptology, 4 (1991), 3-72.  doi: 10.1007/BF00630563.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Carlet and C. S. Ding, Highly nonlinear mappings, J. Complexity, 20 (2004), 205-244.  doi: 10.1016/j.jco.2003.08.008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] R. Lidl and H. Niederreiter, Finite Fields, Second edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[26]

A. Pott, Almost perfect and planar functions, Des. Codes Cryptogr., 78 (2016), 141-195.  doi: 10.1007/s10623-015-0151-x.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  Some power functions $ \Pi(x) = x^d $ over $ \mathbb{F}_{p^n} $ with known differential spectrum
$ p $ $ d $ condition $ \Delta(\Pi) $ references
2 $ 2^s+1 $ $ \gcd(s,n)=2 $ 4 [3]
2 $ 2^{2s}-2^s+1 $ $ \gcd(s,n)=2 $ 4 [3]
2 $ 2^n-2 $ $ n $ even 4 [3]
2 $ 2^{2k}+2^k+1 $ $ n=4k $, $ k $ odd 4 [3]
2 $ 2^{2k}+2^k+1 $ $ n=4k $ 4 [28]
2 $ 2^t-1 $ $ t=3,n-2 $ 6 [4]
2 $ 2^t-1 $ $ t=(n-1)/2 $,$ (n+3)/2 $, $ n $ odd $ 6 $ or $ 8 $ [5]
2 $ 2^m+2^{(m+1)/2}+1 $ $ n=2m $, $ m\geq5 $ odd $ 8 $ [29]
2 $ 2^{m+1}+3 $ $ n=2m $, $ m\geq5 $ odd $ 8 $ [29]
3 $ (3^m-3)/4 $ $ n $ odd $ \leq 3 $ [31]
odd $ (p^k+1)/2 $ $ e=\gcd(n,k) $ $ (p^e-1)/2 $or $ p^e+1 $} [10]
odd $ (p^n+1)/(p^m+1) $$ +(p^n-1)/2 $} $ p \equiv 3 \; (\mathrm{mod}\; 4) $,$ n $ odd, $ m|n $} $ (p^m+1)/2 $ [10]
odd $ p^{2k}-p^k+1 $ $ n $ odd, $ \gcd(n,k)=1 $ $ p+1 $ [32]
odd $ p^{2k}-p^k+1 $ $ \gcd(n,k)=e $, $ \frac{n}{e} $ odd $ p^e+1 $ This paper
Table Options

Download as excel

$ p $ $ d $ condition $ \Delta(\Pi) $ references
2 $ 2^s+1 $ $ \gcd(s,n)=2 $ 4 [3]
2 $ 2^{2s}-2^s+1 $ $ \gcd(s,n)=2 $ 4 [3]
2 $ 2^n-2 $ $ n $ even 4 [3]
2 $ 2^{2k}+2^k+1 $ $ n=4k $, $ k $ odd 4 [3]
2 $ 2^{2k}+2^k+1 $ $ n=4k $ 4 [28]
2 $ 2^t-1 $ $ t=3,n-2 $ 6 [4]
2 $ 2^t-1 $ $ t=(n-1)/2 $,$ (n+3)/2 $, $ n $ odd $ 6 $ or $ 8 $ [5]
2 $ 2^m+2^{(m+1)/2}+1 $ $ n=2m $, $ m\geq5 $ odd $ 8 $ [29]
2 $ 2^{m+1}+3 $ $ n=2m $, $ m\geq5 $ odd $ 8 $ [29]
3 $ (3^m-3)/4 $ $ n $ odd $ \leq 3 $ [31]
odd $ (p^k+1)/2 $ $ e=\gcd(n,k) $ $ (p^e-1)/2 $or $ p^e+1 $} [10]
odd $ (p^n+1)/(p^m+1) $$ +(p^n-1)/2 $} $ p \equiv 3 \; (\mathrm{mod}\; 4) $,$ n $ odd, $ m|n $} $ (p^m+1)/2 $ [10]
odd $ p^{2k}-p^k+1 $ $ n $ odd, $ \gcd(n,k)=1 $ $ p+1 $ [32]
odd $ p^{2k}-p^k+1 $ $ \gcd(n,k)=e $, $ \frac{n}{e} $ odd $ p^e+1 $ This paper
Table 2.  Differential spectrum of some $ \Pi(x) = x^{p^{2k}-p^k+1} $ over $ \mathbb F_{p^n} $
$ p $ $ n $ $ k $ $ e $ $ \Pi(x) $ $ \mathcal{S}=\omega_0,\omega_{p^e-1},\omega_{p^e},\omega_{p^e+1} $
3 $ 5 $ $ 2 $ $ 1 $ $ x^{73} $ $ \left\{ {{\rm{152,60,1,30}}} \right\}$
$ 3 $ $ 6 $ $ 2 $ $ 2 $ $ x^{73} $ $ \left\{ {{\rm{647,45,1,36}}} \right\} $
$ 3 $ $ 9 $ $ 3 $ $ 3 $ $ x^{703} $ $\left\{ {{\rm{18953,378,1,351}}} \right\}$
$ 3 $ $ 9 $ $ 6 $ $ 3 $ $ x^{530713} $ $\left\{ {{\rm{18953,378,1,351}}} \right\}$
$ 5 $ $ 6 $ $ 2 $ $ 2 $ $ x^{601} $ $\left\{ {{\rm{14999,325,1,300}}} \right\}$
Table Options

Download as excel

$ p $ $ n $ $ k $ $ e $ $ \Pi(x) $ $ \mathcal{S}=\omega_0,\omega_{p^e-1},\omega_{p^e},\omega_{p^e+1} $
3 $ 5 $ $ 2 $ $ 1 $ $ x^{73} $ $ \left\{ {{\rm{152,60,1,30}}} \right\}$
$ 3 $ $ 6 $ $ 2 $ $ 2 $ $ x^{73} $ $ \left\{ {{\rm{647,45,1,36}}} \right\} $
$ 3 $ $ 9 $ $ 3 $ $ 3 $ $ x^{703} $ $\left\{ {{\rm{18953,378,1,351}}} \right\}$
$ 3 $ $ 9 $ $ 6 $ $ 3 $ $ x^{530713} $ $\left\{ {{\rm{18953,378,1,351}}} \right\}$
$ 5 $ $ 6 $ $ 2 $ $ 2 $ $ x^{601} $ $\left\{ {{\rm{14999,325,1,300}}} \right\}$
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