Abstract
It was proved that the complexity of square root computation in the Galois field GF(3s), s = 2kr, is equal to O(M(2k)M(r)k + M(r) log2r) + 2kkr1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3s) is equal to O(M(2k)M(r)) and O(M(2k)M(r)) + r1+o(1), respectively. If the basis in the field GF(3r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2kkr1+o(1) and r1+o(1) can be omitted. For M(n) one may take n log2nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form Or (M (s) log 2s) = s (log 2s)2ψ(s).
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References
A. Torielli, “Bemerkung uber die Aulosurig quadratischer Corigruerizeri,” in Goettinger Nadir, 1891, pp. 344–346.
A. A. Bolotov, S. B. Gashkov, A. B. Prolov, and A. A. Chasovskikh, Elementary Introduction to Elliptic Cryptography, Algebraic and Algorithmic Foundations (URSS, Moscow, 2012) [in Russian].
E. Bach, “Explicit Bounds for Primality Testing and Related Problems,” Math. Comp. 22, 355 (1989).
E. Bach, “A Note to Square Roots in Finite Fields,” IEEE Trans. Inform. Theory 36, 1494 (1990).
M. Fuerer, “Faster Integer Multiplication,” SIAM J. Comput. 39 (3), 979 (2009).
D. Harvey, J. van der Hoeven, and G. Lecerf, “Faster Polynomial Multiplication over Finite Fields,” ArXive.org>cs> arXive: 1407.3361 12 Jul 2014.
S. B. Gashkov and I. S. Sergeev, “Complexity and Depth of Boolean Circuits for Multiplication and Inversion in Finite Fields of Characteristic Two,” Diskret. Matem. 25 (1), 3 (2013).
D. J. Bernstein, “Batch Binary Edwards,” in Advances in Cryptology (CRYPTO, 2009), pp. 317–336.
D. J. Bernstein, C. Chuerigsatiarisup, arid T. Lange, “Curve 41417: Karatsuba Revisited,” in Cryptographic Hardware and Embedded Systems (CHES, 2014), pp. 316–334.
S. B. Gashkov and I. S. Sergeev, “Complexity and Depth of Boolean Circuits for Multiplication and Inversion in Some Fields GF(2n),” Vestn. Mosk. Univ., Matem. Mekhan., No. 4, 3 (2009).
S. B. Gashkov and I. S. Sergeev, “Application of the Method of Additive Chains for Inversion in Finite Fields,” Diskret. Matem. 18 (4), 56 (2006).
A. A. Bolotov and S. B. Gashkov, “Fast Multiplication in Normal Bases of Finite Fields,” Diskret. Matem. 13 (3), 3 (2001).
S. B. Gashkov arid I. S. Sergeev, “Complexity of Calculations in Finite Fields,” Fund. Priklad. Matem. 17 (4), 95 (2012).
K. S. Kedlaya and C. Umaris, “Fast Polynomial Factorization and Modular Composition,” SIAM J. Comput. 40 (6), 1767 (2011).
I. B. Gashkov and V. M Sidelnikov, “Linear Ternary Quasi–Perfect Codes Correcting Two Errors,” Problemy Peredachi Inform. 22 (4), 43 (1986). [Problems Inform. Transmission 22 (4), 284 (1986)].
S. M. Dodunekov and Ya. Nilson, “Decoding Some Wonderful Ternary Codes,” Problemy Peredachi Inform. 31 (2), 36 (1995) [Problems Inform. Transmission 31 (2), 128 (1995)].
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Original Russian Text © S.B. Gashkov and I.B. Gashkov, 2018, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2018, Vol. 73, No. 5, pp. 8–14.
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Gashkov, S.B., Gashkov, I.B. Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic. Moscow Univ. Math. Bull. 73, 176–181 (2018). https://doi.org/10.3103/S0027132218050029
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DOI: https://doi.org/10.3103/S0027132218050029