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Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case
Cycle structure of iterating Redei functions
1. | Institute of Mathematics, State University of Campinas, Brazil |
2. | School of Mathematics and Statistics, Carleton University, Canada |
3. | Academic Department of Mathematics, UTFPR, Brazil |
Vasiga and Shallit [
References:
[1] |
L. Blum, M. Blum and M. Shub,
A simple unpredictable pseudo-random number generator, SIAM J. Comput., 15 (1986), 364-383.
doi: 10.1137/0215025. |
[2] |
R. Brent and J. Pollard,
Factorization of the eighth Fermat number, Math. Comput., 36 (1981), 627-630.
doi: 10.2307/2007666. |
[3] |
W. Chou and I. E. Shparlinski,
On the cycle structure of repeated exponentiation modulo a prime, J. Number Theory, 107 (2004), 345-356.
doi: 10.1016/j.jnt.2004.04.005. |
[4] |
D. A. Cox, Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory and Complex Multiplication, Wiley-Interscience, 1989. |
[5] |
FIPS PUB 186-4,
Digital Signature Standard (DSS), Federal Information Processing Standards Publication, NIST, 2013. |
[6] |
R. Lidl and W. B. Müller,
Generalization of the Fibonacci pseudoprime test, Discrete Math., 92 (1991), 211-220.
doi: 10.1016/0012-365X(91)90282-7. |
[7] |
W. Meidl and A. Winterhof,
On the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions, Finite Fields Appl., 13 (2007), 628-634.
doi: 10.1016/j.ffa.2005.10.001. |
[8] |
NIST Special Publication 800-56A,
Recommendation for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography, NIST, 2007. |
[9] |
R. Nobauer,
Cryptoanalysis of the Rédei scheme, Contrib. General Algebra, 3 (1984), 255-264.
|
[10] |
R. Nobauer,
Key distribution systems based on polynomial functions and Rédei functions, Probl. Contr. Inf. Theory, 15 (1986), 91-100.
|
[11] |
R. Pieper,
Cryptanalysis of Rédei-and Dickson permutations on arbitrary finite rings, AAECC, 4 (1993), 59-76.
doi: 10.1007/BF01270400. |
[12] |
J. M. Pollard,
A monte carlo method for factorization, BIT, 15 (1975), 331-334.
|
[13] |
J. M. Pollard,
Monte carlo methods for index computation (mod $p$), Math. Comp., 32 (1978), 918-924.
doi: 10.2307/2006496. |
[14] |
C. Qureshi and D. Panario,
Rédei actions on finite fields and multiplication map in cyclic groups, SIAM J. Discrete Math., 29 (2015), 1486-1503.
doi: 10.1137/140993338. |
[15] |
A. Sakzad, M. Sadeghi and D. Panario,
Cycle structure of permutation functions over finite fields and their applications, Adv. Math. Commun., 6 (2012), 347-361.
doi: 10.3934/amc.2012.6.347. |
[16] |
M. Sha,
Digraphs from endomorphisms of finite cyclic groups, J. Combin. Math. Combin. Comp., 83 (2012), 105-120.
|
[17] |
T. Vasiga and J. Shallit,
On the iteration of certain quadratic maps over $GF(p)$, Discrete Math., 277 (2004), 219-240.
doi: 10.1016/S0012-365X(03)00158-4. |
[18] |
M. Wiener and R. Zuccherato, Faster attacks on elliptic curve cryptosystems, in Selected
Areas in Cryptography, 1998,190-200.
doi: 10.1007/3-540-48892-8_15. |
show all references
References:
[1] |
L. Blum, M. Blum and M. Shub,
A simple unpredictable pseudo-random number generator, SIAM J. Comput., 15 (1986), 364-383.
doi: 10.1137/0215025. |
[2] |
R. Brent and J. Pollard,
Factorization of the eighth Fermat number, Math. Comput., 36 (1981), 627-630.
doi: 10.2307/2007666. |
[3] |
W. Chou and I. E. Shparlinski,
On the cycle structure of repeated exponentiation modulo a prime, J. Number Theory, 107 (2004), 345-356.
doi: 10.1016/j.jnt.2004.04.005. |
[4] |
D. A. Cox, Primes of the Form $x^2+ny^2$: Fermat, Class Field Theory and Complex Multiplication, Wiley-Interscience, 1989. |
[5] |
FIPS PUB 186-4,
Digital Signature Standard (DSS), Federal Information Processing Standards Publication, NIST, 2013. |
[6] |
R. Lidl and W. B. Müller,
Generalization of the Fibonacci pseudoprime test, Discrete Math., 92 (1991), 211-220.
doi: 10.1016/0012-365X(91)90282-7. |
[7] |
W. Meidl and A. Winterhof,
On the linear complexity profile of nonlinear congruential pseudorandom number generators with Rédei functions, Finite Fields Appl., 13 (2007), 628-634.
doi: 10.1016/j.ffa.2005.10.001. |
[8] |
NIST Special Publication 800-56A,
Recommendation for Pair-Wise Key Establishment Schemes Using Discrete Logarithm Cryptography, NIST, 2007. |
[9] |
R. Nobauer,
Cryptoanalysis of the Rédei scheme, Contrib. General Algebra, 3 (1984), 255-264.
|
[10] |
R. Nobauer,
Key distribution systems based on polynomial functions and Rédei functions, Probl. Contr. Inf. Theory, 15 (1986), 91-100.
|
[11] |
R. Pieper,
Cryptanalysis of Rédei-and Dickson permutations on arbitrary finite rings, AAECC, 4 (1993), 59-76.
doi: 10.1007/BF01270400. |
[12] |
J. M. Pollard,
A monte carlo method for factorization, BIT, 15 (1975), 331-334.
|
[13] |
J. M. Pollard,
Monte carlo methods for index computation (mod $p$), Math. Comp., 32 (1978), 918-924.
doi: 10.2307/2006496. |
[14] |
C. Qureshi and D. Panario,
Rédei actions on finite fields and multiplication map in cyclic groups, SIAM J. Discrete Math., 29 (2015), 1486-1503.
doi: 10.1137/140993338. |
[15] |
A. Sakzad, M. Sadeghi and D. Panario,
Cycle structure of permutation functions over finite fields and their applications, Adv. Math. Commun., 6 (2012), 347-361.
doi: 10.3934/amc.2012.6.347. |
[16] |
M. Sha,
Digraphs from endomorphisms of finite cyclic groups, J. Combin. Math. Combin. Comp., 83 (2012), 105-120.
|
[17] |
T. Vasiga and J. Shallit,
On the iteration of certain quadratic maps over $GF(p)$, Discrete Math., 277 (2004), 219-240.
doi: 10.1016/S0012-365X(03)00158-4. |
[18] |
M. Wiener and R. Zuccherato, Faster attacks on elliptic curve cryptosystems, in Selected
Areas in Cryptography, 1998,190-200.
doi: 10.1007/3-540-48892-8_15. |
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