# American Institute of Mathematical Sciences

November  2014, 8(4): 459-477. doi: 10.3934/amc.2014.8.459

## Smoothness testing of polynomials over finite fields

 1 Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 2 Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada

Received  January 2014 Revised  September 2014 Published  November 2014

We present an analysis of Bernstein's batch integer smoothness test when applied to the case of polynomials over a finite field $\mathbb{F}_q.$ We compare the performance of our algorithm with the standard method based on distinct degree factorization from both an analytical and a practical point of view. Our results show that although the batch test is asymptotically better as a function of the degree of the polynomials to test for smoothness, it is unlikely to offer significant practical improvements for cases of practical interest.
Citation: Jean-François Biasse, Michael J. Jacobson, Jr.. Smoothness testing of polynomials over finite fields. Advances in Mathematics of Communications, 2014, 8 (4) : 459-477. doi: 10.3934/amc.2014.8.459
##### References:
 [1] gf2x, a C/C++ software package containing routines for fast arithmetic in $GF(2)[x]$, (multiplication, ().   Google Scholar [2] D. Bernstein, How to find smooth parts of integers,, submitted., ().   Google Scholar [3] J.-F. Biasse and M. Jacobson, Practical improvements to class group and regulator computation of real quadratic fields, in Algorithmic Number Theory (eds. G. Hanrot, F. Morain and E. Thomé), Springer-Verlag, 2010, 50-65. doi: 10.1007/978-3-642-14518-6_8.  Google Scholar [4] G. Bisson and A. Sutherland, Computing the endomorphism ring of an ordinary elliptic curve over a finite field, J. Number Theory, 113 (2011), 815-831. doi: 10.1016/j.jnt.2009.11.003.  Google Scholar [5] D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two, IEEE Trans. Inf. Theory, 30 (1984), 587-594. doi: 10.1109/TIT.1984.1056941.  Google Scholar [6] J. Detrey, P. Gaudry and M. Videau, Relation collection for the function field sieve, in 21st IEEE Int. Symp. Computer Arith. (eds. A. Nannarelli, P.-M. Seidel and P. Tang), IEEE, 2013, 201-210. doi: 10.1109/ARITH.2013.28.  Google Scholar [7] A. Enge and P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith., 102 (2002), 83-103. doi: 10.4064/aa102-1-6.  Google Scholar [8] M. Jacobson, A. Menezes and A. Stein, Solving elliptic curve discrete logarithm problems using Weil descent, J. Ramanujan Math. Soc., 16 (2001), 231-260.  Google Scholar [9] H. Lenstra, Factoring integers with elliptic curves, Ann. Math., 126 (1987), 649-673. doi: 10.2307/1971363.  Google Scholar [10] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, New York, 1986.  Google Scholar [11] A. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, in Proc. EUROCRYPT 84 Workshop Adv. Cryptology: Theory Appl. Crypt. Techn., Springer-Verlag, New York, 1985, 224-314. doi: 10.1007/3-540-39757-4_20.  Google Scholar [12] A. Schnhage and V. Strassen, Schnelle Multiplikation grosser Zahlen (in German), Computing, 7 (1971), 281-292.  Google Scholar [13] V. Shoup, NTL: A library for doing number theory,, Software, ().   Google Scholar [14] M. Velichka, M. Jacobson and A. Stein, Computing discrete logarithms in the jacobian of high-genus hyperelliptic curves over even characteristic finite fields, Math. Comp., 83 (2014), 935-963. doi: 10.1090/S0025-5718-2013-02748-2.  Google Scholar [15] J. von zur Gathen and V. Shoup, Computing Frobenius maps and factoring polynomials, Comp. Complexity, 2 (1992), 187-224. doi: 10.1007/BF01272074.  Google Scholar

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##### References:
 [1] gf2x, a C/C++ software package containing routines for fast arithmetic in $GF(2)[x]$, (multiplication, ().   Google Scholar [2] D. Bernstein, How to find smooth parts of integers,, submitted., ().   Google Scholar [3] J.-F. Biasse and M. Jacobson, Practical improvements to class group and regulator computation of real quadratic fields, in Algorithmic Number Theory (eds. G. Hanrot, F. Morain and E. Thomé), Springer-Verlag, 2010, 50-65. doi: 10.1007/978-3-642-14518-6_8.  Google Scholar [4] G. Bisson and A. Sutherland, Computing the endomorphism ring of an ordinary elliptic curve over a finite field, J. Number Theory, 113 (2011), 815-831. doi: 10.1016/j.jnt.2009.11.003.  Google Scholar [5] D. Coppersmith, Fast evaluation of logarithms in fields of characteristic two, IEEE Trans. Inf. Theory, 30 (1984), 587-594. doi: 10.1109/TIT.1984.1056941.  Google Scholar [6] J. Detrey, P. Gaudry and M. Videau, Relation collection for the function field sieve, in 21st IEEE Int. Symp. Computer Arith. (eds. A. Nannarelli, P.-M. Seidel and P. Tang), IEEE, 2013, 201-210. doi: 10.1109/ARITH.2013.28.  Google Scholar [7] A. Enge and P. Gaudry, A general framework for subexponential discrete logarithm algorithms, Acta Arith., 102 (2002), 83-103. doi: 10.4064/aa102-1-6.  Google Scholar [8] M. Jacobson, A. Menezes and A. Stein, Solving elliptic curve discrete logarithm problems using Weil descent, J. Ramanujan Math. Soc., 16 (2001), 231-260.  Google Scholar [9] H. Lenstra, Factoring integers with elliptic curves, Ann. Math., 126 (1987), 649-673. doi: 10.2307/1971363.  Google Scholar [10] R. Lidl and H. Niederreiter, Introduction to Finite Fields and their Applications, Cambridge Univ. Press, New York, 1986.  Google Scholar [11] A. Odlyzko, Discrete logarithms in finite fields and their cryptographic significance, in Proc. EUROCRYPT 84 Workshop Adv. Cryptology: Theory Appl. Crypt. Techn., Springer-Verlag, New York, 1985, 224-314. doi: 10.1007/3-540-39757-4_20.  Google Scholar [12] A. Schnhage and V. Strassen, Schnelle Multiplikation grosser Zahlen (in German), Computing, 7 (1971), 281-292.  Google Scholar [13] V. Shoup, NTL: A library for doing number theory,, Software, ().   Google Scholar [14] M. Velichka, M. Jacobson and A. Stein, Computing discrete logarithms in the jacobian of high-genus hyperelliptic curves over even characteristic finite fields, Math. Comp., 83 (2014), 935-963. doi: 10.1090/S0025-5718-2013-02748-2.  Google Scholar [15] J. von zur Gathen and V. Shoup, Computing Frobenius maps and factoring polynomials, Comp. Complexity, 2 (1992), 187-224. doi: 10.1007/BF01272074.  Google Scholar
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