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  1. // Minimax polynomials computed using the work `Minimax Approximation of Sign Function by Composite Polynomial for Homomorphic Comparison`
  2. // of Lee et al. (https://eprint.iacr.org/2020/834).
  3. // Polynomials approximate the function sign: y = {-1 if -1 <= x < 0; 0 if x = 0; 1 if 0 < x <= 1}
  4. // in the interval [-1, 2^-a] U [2^-a, 1] with at least 30 bits of precision for y.
  5. // Polynomials are in Chebyshev basis and pre-scaled for the interval [-1, 1].
  6. // The maximum degree of polynomials is set to be 31.
  7. var SignPolys30 = map[int][][]float64{
  8. // Index of the map corresponds to the value `a` of the interval [-1, 2^-a] U [2^-a, 1].
  9. // Polynomials need to be evaluated sequentially: P(x) = p1(p0(x))
  10. 1: {
  11. {0, 1.230715957236511, 0, -0.328672378079633, 0, 0.121808003592360, 0, -0.037834448510340},
  12. {0, 1.212183873699944, 0, -0.271054028639235, 0, 0.070572640722558, 0, -0.012840242010512, 0, 0.001137756235760},
  13. },
  14. 2: {
  15. {0, 1.164368936676107, 0, -0.357742904412620, 0, 0.181242878840233, 0, -0.140411865249790},
  16. {0, 1.240379148451449, 0, -0.334945349347038, 0, 0.130856109146218, 0, -0.048072443154541, 0, 0.014719627092851, 0, -0.003431551092576, 0, 0.000536606293880, 0, -0.000042148241220},
  17. },
  18. 3: {
  19. {0, 1.001809440865455, 0, -0.327687163887422, 0, 0.190040622880240, 0, -0.286696324936789},
  20. {0, 1.261349722036804, 0, -0.389983227421617, 0, 0.201128820624648, 0, -0.114227992120608, 0, 0.065154686407366, 0, -0.035908549758244, 0, 0.018691354657832, 0, -0.009029822781752, 0, 0.003981510904052, 0, -0.001572235275963, 0, 0.000542626651334, 0, -0.000158064512767, 0, 0.000036742183313, 0, -0.000006132214650, 0, 0.000000561403579},
  21. },
  22. 4: {
  23. {0, 1.017844517280021, 0, -0.337576702926976, 0, 0.200540323037181, 0, -0.141169766010582, 0, 0.107768588452570, 0, -0.086277334684927, 0, 0.071342986189055, 0, -0.232634798746130},
  24. {0, 1.260863879303054, 0, -0.388631257147277, 0, 0.199191797334966, 0, -0.112071418902792, 0, 0.063120689653712, 0, -0.034233227595526, 0, 0.017472531074394, 0, -0.008245095680968, 0, 0.003536419251999, 0, -0.001352217102914, 0, 0.000449580103411, 0, -0.000125410221696, 0, 0.000027718048840, 0, -0.000004359367717, 0, 0.000000371527703},
  25. },
  26. 5: {
  27. {0, 1.008956633753181, 0, -0.335905514190525, 0, 0.201050028180481, 0, -0.143083644899948, 0, 0.110751794669743, 0, -0.090077520790836, 0, 0.075686596146289, 0, -0.065072622880857, 0, 0.056910389448104, 0, -0.050434592351999, 0, 0.045173725884113, 0, -0.040823015818841, 0, 0.037178626203318, 0, -0.034101506474008, 0, 0.223790623121785},
  28. {0, 1.261183726994639, 0, -0.389520828660227, 0, 0.200464942837797, 0, -0.113486541519283, 0, 0.064452414425541, 0, -0.035327010496721, 0, 0.018265503610338, 0, -0.008753503261469, 0, 0.003823350661710, 0, -0.001493223333382, 0, 0.000508802721767, 0, -0.000146026381338, 0, 0.000033360931537, 0, -0.000005455193401, 0, 0.000000487215668},
  29. },
  30. 6: {
  31. {0, 0.726381449578772, 0, -0.244326831713059, 0, 0.149426705113034, 0, -0.110154647371620, 0, 0.089915604796240, 0, -0.471507777576047},
  32. {0, 1.174492998551771, 0, -0.381940077363467, 0, 0.217988156846625, 0, -0.144247329681216, 0, 0.101043328009418, 0, -0.072215761265054, 0, 0.104878684901923},
  33. {0, 1.240250424900085, 0, -0.334631816128700, 0, 0.130514322336486, 0, -0.047824635032254, 0, 0.014592993570397, 0, -0.003386896021114, 0, 0.000526696506776, 0, -0.000041090843361},
  34. },
  35. 7: {
  36. {0, 0.690661297942282, 0, -0.231947821488028, 0, 0.141340228890329, 0, -0.103476779467329, 0, 0.083416610669587, 0, -0.071765491762600, 0, 0.491771955215759},
  37. {0, 1.056466095354053, 0, -0.348766438876165, 0, 0.205275925909256, 0, -0.142513174903208, 0, 0.106826945477050, 0, -0.083672108662060, 0, 0.206382755701074},
  38. {0, 1.257474242377992, 0, -0.379286371284680, 0, 0.186051636162112, 0, -0.097850850927366, 0, 0.050211779570038, 0, -0.024104809197767, 0, 0.010529369863956, 0, -0.004081726718521, 0, 0.001364829481455, 0, -0.000378842654248, 0, 0.000082235879722, 0, -0.000012501497117, 0, 0.000001009273437},
  39. },
  40. 8: {
  41. {0, 0.668212066029525, 0, -0.224091102537420, 0, 0.136134876615670, 0, -0.099145755348325, 0, 0.079261469185326, 0, -0.067305871205980, 0, 0.059830081596145, 0, -0.504058215644626},
  42. {0, 0.987144595531492, 0, -0.328146765524437, 0, 0.195829442420418, 0, -0.138787305075312, 0, 0.106882434462648, 0, -0.086461501481394, 0, 0.072299438745648, 0, -0.061994432370733, 0, 0.253234093291670},
  43. {0, 1.262606534930775, 0, -0.393507866232512, 0, 0.206258718133986, 0, -0.120078425087308, 0, 0.070857929142172, 0, -0.040812231418863, 0, 0.022457586964304, 0, -0.011623753913614, 0, 0.005580785421028, 0, -0.002449198680127, 0, 0.000965328035767, 0, -0.000333808654337, 0, 0.000097888331081, 0, -0.000023039741705, 0, 0.000003921791215, 0, -0.000000369726044},
  44. },
  45. 9: {
  46. {0, 0.654727532791956, 0, -0.219318303871991, 0, 0.132915643722334, 0, -0.096422374388037, 0, 0.076632150004477, 0, -0.064518654930197, 0, 0.056647493902729, 0, -0.051466250346325, 0, 0.510802763115053},
  47. {0, 0.996387357222740, 0, -0.331819935269768, 0, 0.198723009120481, 0, -0.141553446924650, 0, 0.109696619900665, 0, -0.089349692707187, 0, 0.075204997118521, 0, -0.064786755370178, 0, 0.056785467508955, 0, -0.050444334031389, 0, 0.045296381483515, 0, -0.041038852876463, 0, 0.037468040544350, 0, -0.034443285825705, 0, 0.031866158976499, 0, -0.232687625779271},
  48. {0, 1.262318433983681, 0, -0.392699312450817, 0, 0.205080180588373, 0, -0.118731376485093, 0, 0.069540893989413, 0, -0.039675659460785, 0, 0.021580791806741, 0, -0.011016766249940, 0, 0.005204356415222, 0, -0.002241430840437, 0, 0.000864461323961, 0, -0.000291545493433, 0, 0.000083062788037, 0, -0.000018905767192, 0, 0.000003093505150, 0, -0.000000278016179},
  49. },
  50. 10: {
  51. {0, 0.642698789498412, 0, -0.220996742368238, 0, 0.142113260813862, 0, -0.558031671551598},
  52. {0, 0.660611667803980, 0, -0.226789704885651, 0, 0.145364092529353, 0, -0.545162151180102},
  53. {0, 0.993013485493232, 0, -0.330550467520342, 0, 0.197790340132398, 0, -0.140709239261830, 0, 0.108863262283247, 0, -0.088498033526641, 0, 0.074326562586834, 0, -0.063885061360168, 0, 0.055873209738816, 0, -0.049543109713346, 0, 0.044438024835666, 0, -0.040268334971186, 0, 0.239149361283320},
  54. {0, 1.262424466311866, 0, -0.392996716610726, 0, 0.205513160669785, 0, -0.119225380566572, 0, 0.070022725640653, 0, -0.040090194750010, 0, 0.021899379588896, 0, -0.011236327532568, 0, 0.005339798781807, 0, -0.002315725783297, 0, 0.000900271224464, 0, -0.000306425022372, 0, 0.000088231851391, 0, -0.000020330723507, 0, 0.000003375129111, 0, -0.000000308673228},
  55. },
  56. 11: {
  57. {0, 0.640865083906812, 0, -0.220401985432950, 0, 0.141777181832223, 0, -0.559346542808667},
  58. {0, 0.654577894053452, 0, -0.220796511273563, 0, 0.135804114172649, 0, -0.100974802569385, 0, 0.083376387962380, 0, -0.525277529217805},
  59. {0, 0.986228163074576, 0, -0.328473520274695, 0, 0.196763121171769, 0, -0.140204882362033, 0, 0.108700952174814, 0, -0.088589433233487, 0, 0.074616867609206, 0, -0.064332783822398, 0, 0.056441092633073, 0, -0.050192777627160, 0, 0.045125709046865, 0, -0.040940322248659, 0, 0.037435123259998, 0, -0.034471065855417, 0, 0.031950957420626, 0, -0.240734678213049},
  60. {0, 1.262643465459329, 0, -0.393611620414160, 0, 0.206410268032610, 0, -0.120252196568017, 0, 0.071028559095950, 0, -0.040960283449194, 0, 0.022572559766690, 0, -0.011703978405083, 0, 0.005631000139090, 0, -0.002477213687517, 0, 0.000979098204278, 0, -0.000339661191709, 0, 0.000099975388692, 0, -0.000023632931855, 0, 0.000004043358472, 0, -0.000000383563287},
  61. },
  62. 12: {
  63. {0, 0.639947273256350, 0, -0.220104177092685, 0, 0.141608740973841, 0, -0.560004490250139},
  64. {0, 0.654584045645880, 0, -0.218780045709349, 0, 0.131980375227423, 0, -0.095053120719488, 0, 0.074768114859719, 0, -0.062069191043826, 0, 0.053480247687574, 0, -0.047387087190060, 0, 0.042944662136949, 0, -0.039675618922313, 0, 0.037299888139999, 0, -0.504217064865696},
  65. {0, 0.996922402618758, 0, -0.331996115802234, 0, 0.198826117224273, 0, -0.141624311855138, 0, 0.109748853274129, 0, -0.089389483349600, 0, 0.075235678867234, 0, -0.064810329392659, 0, 0.056803228811117, 0, -0.050457167839676, 0, 0.045304918037909, 0, -0.041043554378867, 0, 0.037469252669701, 0, -0.034441269346385, 0, 0.031861109472422, 0, -0.232263501314077},
  66. {0, 1.262301316605727, 0, -0.392651319974378, 0, 0.205010365903669, 0, -0.118651818386092, 0, 0.069463422611866, 0, -0.039609146025919, 0, 0.021529802520405, 0, -0.010981731851816, 0, 0.005182821023928, 0, -0.002229666574221, 0, 0.000858818007172, 0, -0.000289213517617, 0, 0.000082257834956, 0, -0.000018685515993, 0, 0.000003050359401, 0, -0.000000273370210},
  67. },
  68. 13: {
  69. {0, 0.638318754112657, 0, -0.215438538027740, 0, 0.132664918648966, 0, -0.098816160336827, 0, 0.081787830172421, 0, -0.537383140621793},
  70. {0, 0.655462685386661, 0, -0.218808592359105, 0, 0.131673369681911, 0, -0.094473748998731, 0, 0.073924548274812, 0, -0.060949671795410, 0, 0.052059942712402, 0, -0.045628737876099, 0, 0.040796663848179, 0, -0.037068108055776, 0, 0.034138443911833, 0, -0.031811530146058, 0, 0.029956991341525, 0, -0.028486723816671, 0, 0.027341446561453, 0, -0.498717158908141},
  71. {0, 1.010465628604333, 0, -0.336453457202503, 0, 0.201432086549165, 0, -0.143412472888098, 0, 0.111063794496537, 0, -0.090387901168407, 0, 0.076002005891980, 0, -0.065395269794535, 0, 0.057239617194349, 0, -0.050767471236796, 0, 0.045505153597172, 0, -0.041145489455405, 0, 0.037481707613122, 0, -0.034370916538843, 0, 0.031712970295347, 0, -0.221516884780931},
  72. {0, 1.261868062339837, 0, -0.391438359527687, 0, 0.203251020081091, 0, -0.116655760790723, 0, 0.067531222101515, 0, -0.037962656365043, 0, 0.020279097754233, 0, -0.010131678504651, 0, 0.004666904428748, 0, -0.001951950604400, 0, 0.000727831500518, 0, -0.000236125052764, 0, 0.000064335671693, 0, -0.000013905921645, 0, 0.000002141770905, 0, -0.000000179006708},
  73. },
  74. 14: {
  75. {0, 0.637576111870883, 0, -0.213245866151900, 0, 0.128826542383904, 0, -0.092987396247310, 0, 0.073368476768871, 0, -0.061154635695925, 0, 0.052966818471630, 0, -0.047241603407196, 0, 0.043168143152544, 0, -0.040299134355799, 0, 0.519022543210298},
  76. {0, 0.654589649969365, 0, -0.218518032880790, 0, 0.131499578176444, 0, -0.094350197743447, 0, 0.073829063684993, 0, -0.060872177507756, 0, 0.051995015679394, 0, -0.045573127055303, 0, 0.040748268466209, 0, -0.037025493591438, 0, 0.034100588501818, 0, -0.031777681150077, 0, 0.029926579126707, 0, -0.028459307234974, 0, 0.027316676861157, 0, -0.499391210553605},
  77. {0, 0.998389489973883, 0, -0.332479167660715, 0, 0.199108779378863, 0, -0.141818538794209, 0, 0.109891968562548, 0, -0.089498457292982, 0, 0.075319653464343, 0, -0.064874792542844, 0, 0.056851732199511, 0, -0.050492139611256, 0, 0.045328087230752, 0, -0.041056189323915, 0, 0.037472302217400, 0, -0.034435449558609, 0, 0.031846957638035, 0, -0.231100391153813},
  78. {0, 1.262254381380674, 0, -0.392519753592677, 0, 0.204819055775996, 0, -0.118433945769332, 0, 0.069251444166255, 0, -0.039427346199200, 0, 0.021390617235967, 0, -0.010886248075517, 0, 0.005124235529722, 0, -0.002197730947941, 0, 0.000843536208820, 0, -0.000282916576337, 0, 0.000080091381009, 0, -0.000018094994454, 0, 0.000002935202420, 0, -0.000000261038343},
  79. },
  80. 15: {
  81. {0, 0.639143667396106, 0, -0.219843360491696, 0, 0.141461136197384, 0, -0.560580472560256},
  82. {0, 0.638602415179474, 0, -0.214285905116315, 0, 0.130330235107518, 0, -0.095085193527156, 0, 0.076194247904786, 0, -0.064892234527663, 0, 0.057888451844479, 0, -0.526463145589198},
  83. {0, 0.655443895675853, 0, -0.219865199363447, 0, 0.133635925620570, 0, -0.097400726683380, 0, 0.077946709164915, 0, -0.066275006500616, 0, 0.059005017154137, 0, -0.513726948953007},
  84. {0, 0.988088374172314, 0, -0.329047115572072, 0, 0.197052141349875, 0, -0.140353406963885, 0, 0.108758147651108, 0, -0.088578979808761, 0, 0.074553056523499, 0, -0.064225979649770, 0, 0.056300106833424, 0, -0.050026155560178, 0, 0.044942621110162, 0, -0.040751363151694, 0, 0.037253203458720, 0, -0.034312560659208, 0, 0.240347910266465},
  85. {0, 1.262583474731583, 0, -0.393443092487346, 0, 0.206164142338243, 0, -0.119970045548806, 0, 0.070751593342818, 0, -0.040720058892656, 0, 0.022386096544097, 0, -0.011573943315544, 0, 0.005549661373598, 0, -0.002431869286335, 0, 0.000956829928023, 0, -0.000330206517222, 0, 0.000096607777904, 0, -0.000022677102356, 0, 0.000003847794218, 0, -0.000000361348432},
  86. },
  87. 16: {
  88. {0, 0.637426993373543, 0, -0.214374460444656, 0, 0.131004995233488, 0, -0.096323755308684, 0, 0.078099096730906, 0, -0.067676885636216, 0, 0.531844016051619},
  89. {0, 0.638493330853731, 0, -0.214249748471858, 0, 0.130308789437489, 0, -0.095070145771197, 0, 0.076182831099345, 0, -0.064883194937397, 0, 0.057881117433507, 0, -0.526545577788866},
  90. {0, 0.654082271088070, 0, -0.219414334875428, 0, 0.133369059310351, 0, -0.097214092723422, 0, 0.077805785590786, 0, -0.066164171735482, 0, 0.058915924762854, 0, -0.514757376132574},
  91. {0, 0.985420449925055, 0, -0.328207367646763, 0, 0.196607130692998, 0, -0.140097424392860, 0, 0.108621482262274, 0, -0.088528612075227, 0, 0.074569666004294, 0, -0.064296184724809, 0, 0.056413145408968, 0, -0.050172150622581, 0, 0.045111455869189, 0, -0.040931750441882, 0, 0.037431716449928, 0, -0.034472435883403, 0, 0.031956814419766, 0, -0.241373982799490},
  92. {0, 1.262669308477342, 0, -0.393684239594678, 0, 0.206516383392582, 0, -0.120373946428589, 0, 0.071148207169768, 0, -0.041064208928484, 0, 0.022653369136274, 0, -0.011760451388757, 0, 0.005666411999423, 0, -0.002497011650948, 0, 0.000988853139226, 0, -0.000343818850359, 0, 0.000101462873793, 0, -0.000024057321734, 0, 0.000004130727058, 0, -0.000000393563740},
  93. },
  94. 17: {
  95. {0, 0.637207707917286, 0, -0.213823603168710, 0, 0.130056006906933, 0, -0.094892751674636, 0, 0.076048214478565, 0, -0.064776578641033, 0, 0.057794579405634, 0, -0.527517030812699},
  96. {0, 0.637923301823331, 0, -0.214060805095795, 0, 0.130196716278058, 0, -0.094991502247279, 0, 0.076123158073898, 0, -0.064835940667543, 0, 0.057842769770172, 0, -0.526976321232809},
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  210. },
  211. }
Tags: Minimax Composite Polynomials
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