    doi: 10.3934/amc.2022007
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## Computing square roots faster than the Tonelli-Shanks/Bernstein algorithm

 Applied Statistics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata, India 700108

*Corresponding author: Palash Sarkar

Received  July 2021 Revised  November 2021 Early access February 2022

Let $p$ be a prime such that $p = 1+2^nm$, where $n\geq 1$ and $m$ is odd. Given a square $u$ in $\mathbb{Z}_p$ and a non-square $z$ in $\mathbb{Z}_p$, we describe an algorithm to compute a square root of $u$ which requires $\mathfrak{T}+O(n^{3/2})$ operations (i.e., squarings and multiplications), where $\mathfrak{T}$ is the number of operations required to exponentiate an element of $\mathbb{Z}_p$ to the power $(m-1)/2$. This improves upon the Tonelli-Shanks (TS) algorithm which requires $\mathfrak{T}+O(n^{2})$ operations. Bernstein had proposed a table look-up based variant of the TS algorithm which requires $\mathfrak{T}+O((n/w)^{2})$ operations and $O(2^wn/w)$ storage, where $w$ is a parameter. A table look-up variant of the new algorithm requires $\mathfrak{T}+O((n/w)^{3/2})$ operations and the same storage. In concrete terms, the new algorithm is shown to require significantly fewer operations for particular values of $n$.

Citation: Palash Sarkar. Computing square roots faster than the Tonelli-Shanks/Bernstein algorithm. Advances in Mathematics of Communications, doi: 10.3934/amc.2022007
##### References:
  L. M. Adleman, K. L. Manders and G. L. Miller, On taking roots in finite fields, in 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), IEEE Computer Society, (1977), 175–178.  A. O. L. Atkin, Probabilistic primality testing, in INRIA Res. Rep., (1992), 159–163. E. Bach and J. Shallit, Algorithmic Number Theory Volume 1, Efficient Algorithms, Foundations of Computing Series. MIT Press, Cambridge, MA, 1996.  D. J. Bernstein, Faster square roots in annoying finite fields, https://cr.yp.to/papers.html#sqroot, 2001. D. J. Bernstein., Pippenger's exponentiation algorithm., https://cr.yp.to/papers.html#pippenger, 2002. Z. Cao, Q. Sha and X. Fan, Adleman-Manders-Miller root extraction method revisited, Lecture Notes in Computer Science, 7537 (2011), 77-85. M. Cipolla, Un metodo per la risolutione della congruenza di secondo grado, Rendiconto dell'Accademia Scienze Fisiche e Matematiche, Napoli, Series 3, (1903), 154–163. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1993. doi: 10.1007/978-3-662-02945-9.   F. Kong, Z. Cai, J. Yu and D. Li, Improved generalized Atkin algorithm for computing square roots in finite fields, Inform. Process. Lett., 98 (2006), 1-5.  doi: 10.1016/j.ipl.2005.11.015.   D. H. Lehmer, Computer technology applied to the theory of numbers, MAA Studies in Number Theory, Prentice-Hall, Englewood Cliffs, N. J., 6 (1969), 117–151.  S. Lindhurst, An analysis of Shanks algorithm for computing square roots in finite fields, CRM Proceedings and Lecture Notes, 19 (1999), 231-242.  S. Müller, On the computation of square roots in finite fields, Des. Codes Cryptogr., 31 (2004), 301-312.  doi: 10.1023/B:DESI.0000015890.44831.e2.   A. S. Rotaru and S. Iftene, A complete generalization of Atkin's square root algorithm, Fund. Inform., 125 (2013), 71-94.  doi: 10.3233/FI-2013-853.   D. Shanks, Five number-theoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium, Utilitas Mathematica, 7 (1973), 51–70.  A. Tonelli, Bemerkung über die auflösung quadratischer congruenzen, Göttinger Nachrichten, (1891), 344–346. show all references

##### References:
  L. M. Adleman, K. L. Manders and G. L. Miller, On taking roots in finite fields, in 18th Annual Symposium on Foundations of Computer Science (Providence, R.I., 1977), IEEE Computer Society, (1977), 175–178.  A. O. L. Atkin, Probabilistic primality testing, in INRIA Res. Rep., (1992), 159–163. E. Bach and J. Shallit, Algorithmic Number Theory Volume 1, Efficient Algorithms, Foundations of Computing Series. MIT Press, Cambridge, MA, 1996.  D. J. Bernstein, Faster square roots in annoying finite fields, https://cr.yp.to/papers.html#sqroot, 2001. D. J. Bernstein., Pippenger's exponentiation algorithm., https://cr.yp.to/papers.html#pippenger, 2002. Z. Cao, Q. Sha and X. Fan, Adleman-Manders-Miller root extraction method revisited, Lecture Notes in Computer Science, 7537 (2011), 77-85. M. Cipolla, Un metodo per la risolutione della congruenza di secondo grado, Rendiconto dell'Accademia Scienze Fisiche e Matematiche, Napoli, Series 3, (1903), 154–163. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag Berlin Heidelberg, 1993. doi: 10.1007/978-3-662-02945-9.   F. Kong, Z. Cai, J. Yu and D. Li, Improved generalized Atkin algorithm for computing square roots in finite fields, Inform. Process. Lett., 98 (2006), 1-5.  doi: 10.1016/j.ipl.2005.11.015.   D. H. Lehmer, Computer technology applied to the theory of numbers, MAA Studies in Number Theory, Prentice-Hall, Englewood Cliffs, N. J., 6 (1969), 117–151.  S. Lindhurst, An analysis of Shanks algorithm for computing square roots in finite fields, CRM Proceedings and Lecture Notes, 19 (1999), 231-242.  S. Müller, On the computation of square roots in finite fields, Des. Codes Cryptogr., 31 (2004), 301-312.  doi: 10.1023/B:DESI.0000015890.44831.e2.   A. S. Rotaru and S. Iftene, A complete generalization of Atkin's square root algorithm, Fund. Inform., 125 (2013), 71-94.  doi: 10.3233/FI-2013-853.   D. Shanks, Five number-theoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics, Congressus Numerantium, Utilitas Mathematica, 7 (1973), 51–70.  A. Tonelli, Bemerkung über die auflösung quadratischer congruenzen, Göttinger Nachrichten, (1891), 344–346. Comparison of the average case number of operations required by the new algorithm and the TS algorithm
 $n$ #TS new algorithm $k$ ops #tot 16 91 3 26.0[S]+26.0[M] 52.0 32 311 5 66.5[S]+68.0[M] 134.5 48 659 6 121.5[S]+114.0[M] 235.5 64 1135 7 181.0[S]+170.0[M] 351.0 80 1739 8 246.5[S]+231.5[M] 478.0 96 2471 9 313.5[S]+300.0[M] 613.5
 $n$ #TS new algorithm $k$ ops #tot 16 91 3 26.0[S]+26.0[M] 52.0 32 311 5 66.5[S]+68.0[M] 134.5 48 659 6 121.5[S]+114.0[M] 235.5 64 1135 7 181.0[S]+170.0[M] 351.0 80 1739 8 246.5[S]+231.5[M] 478.0 96 2471 9 313.5[S]+300.0[M] 613.5
Comparison of Bernstein's table look-up method with the new table look-up method
 $w=2$ $w=4$ $n=96$ (94[S]+1178[M],1272,49) (92[S]+302[M],394,25) (6,182[S]+338[M],520,49) (4,170[S]+122[M],292,25) $n=128$ (126[S]+2082[M],2208,65) (124[S]+530[M],654,33) (9,239[S]+514[M],753,65) (4,228[S]+194[M],422,33) $n=256$ (254[S]+8252[M],8512,129) (252[S]+2082[M],2334,65) (14,519[S]+1484[M],2003,129) (8,484[S]+514[M],998,65) $n=512$ (510[S]+32898[M],33408,257) (508[S]+8258[M],8766,129) (17,991[S]+4098[M],5089,257) (8,972[S]+1538[M],2501,129) $n=1024$ (1022[S]+131330[M],132352,513) (1020[S]+32898[M],33918,257) (19,2087[S]+11801[M],13888,513) (16,1996[S]+4098[M],6094,257) $w=6$ $w=8$ $n=96$ (90[S]+138[M],228,17) (88[S]+80[M],168,13) (2,146[S]+82[M],228,17) (1,100[S]+80[M],180,13) $n=128$ (122[S]+255[M],377,42) (120[S]+138[M],258,17) (3,235[S]+115[M],350,29) (1,136[S]+138[M],274,17) $n=256$ (250[S]+948[M],1198,84) (248[S]+530[M],778,33) (4,469[S]+332[M],801,53) (4,448[S]+194[M],642,33) $n=512$ (506[S]+3743[M],4249,170) (504[S]+2082[M],2586,65) (5,1057[S]+955[M],2012,103) (8,960[S]+514[M],1474,65) $n=1024$ (1018[S]+14708[M],15726,340) (1016[S]+8258[M],9274,129) (16,2241[S]+2378[M],4619,188) (8,1928[S]+1538[M],3466,129)
 $w=2$ $w=4$ $n=96$ (94[S]+1178[M],1272,49) (92[S]+302[M],394,25) (6,182[S]+338[M],520,49) (4,170[S]+122[M],292,25) $n=128$ (126[S]+2082[M],2208,65) (124[S]+530[M],654,33) (9,239[S]+514[M],753,65) (4,228[S]+194[M],422,33) $n=256$ (254[S]+8252[M],8512,129) (252[S]+2082[M],2334,65) (14,519[S]+1484[M],2003,129) (8,484[S]+514[M],998,65) $n=512$ (510[S]+32898[M],33408,257) (508[S]+8258[M],8766,129) (17,991[S]+4098[M],5089,257) (8,972[S]+1538[M],2501,129) $n=1024$ (1022[S]+131330[M],132352,513) (1020[S]+32898[M],33918,257) (19,2087[S]+11801[M],13888,513) (16,1996[S]+4098[M],6094,257) $w=6$ $w=8$ $n=96$ (90[S]+138[M],228,17) (88[S]+80[M],168,13) (2,146[S]+82[M],228,17) (1,100[S]+80[M],180,13) $n=128$ (122[S]+255[M],377,42) (120[S]+138[M],258,17) (3,235[S]+115[M],350,29) (1,136[S]+138[M],274,17) $n=256$ (250[S]+948[M],1198,84) (248[S]+530[M],778,33) (4,469[S]+332[M],801,53) (4,448[S]+194[M],642,33) $n=512$ (506[S]+3743[M],4249,170) (504[S]+2082[M],2586,65) (5,1057[S]+955[M],2012,103) (8,960[S]+514[M],1474,65) $n=1024$ (1018[S]+14708[M],15726,340) (1016[S]+8258[M],9274,129) (16,2241[S]+2378[M],4619,188) (8,1928[S]+1538[M],3466,129)
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