/* This Source Code Form is subject to the terms of the Mozilla Public * License, v. 2.0. If a copy of the MPL was not distributed with this * file, You can obtain one at http://mozilla.org/MPL/2.0/. */ #ifndef __ecp_fp_h_ #define __ecp_fp_h_ #include "mpi.h" #include "ecl.h" #include "ecp.h" #include #include "mpi-priv.h" #ifdef ECL_DEBUG #include #endif /* Largest number of doubles to store one reduced number in floating * point. Used for memory allocation on the stack. */ #define ECFP_MAXDOUBLES 10 /* For debugging purposes */ #ifndef ECL_DEBUG #define ECFP_ASSERT(x) #else #define ECFP_ASSERT(x) assert(x) #endif /* ECFP_Ti = 2^(i*24) Define as preprocessor constants so we can use in * multiple static constants */ #define ECFP_T0 1.0 #define ECFP_T1 16777216.0 #define ECFP_T2 281474976710656.0 #define ECFP_T3 4722366482869645213696.0 #define ECFP_T4 79228162514264337593543950336.0 #define ECFP_T5 1329227995784915872903807060280344576.0 #define ECFP_T6 22300745198530623141535718272648361505980416.0 #define ECFP_T7 374144419156711147060143317175368453031918731001856.0 #define ECFP_T8 6277101735386680763835789423207666416102355444464034512896.0 #define ECFP_T9 105312291668557186697918027683670432318895095400549111254310977536.0 #define ECFP_T10 1766847064778384329583297500742918515827483896875618958121606201292619776.0 #define ECFP_T11 29642774844752946028434172162224104410437116074403984394101141506025761187823616.0 #define ECFP_T12 497323236409786642155382248146820840100456150797347717440463976893159497012533375533056.0 #define ECFP_T13 8343699359066055009355553539724812947666814540455674882605631280555545803830627148527195652096.0 #define ECFP_T14 139984046386112763159840142535527767382602843577165595931249318810236991948760059086304843329475444736.0 #define ECFP_T15 2348542582773833227889480596789337027375682548908319870707290971532209025114608443463698998384768703031934976.0 #define ECFP_T16 39402006196394479212279040100143613805079739270465446667948293404245\ 721771497210611414266254884915640806627990306816.0 #define ECFP_T17 66105596879024859895191530803277103982840468296428121928464879527440\ 5791236311345825189210439715284847591212025023358304256.0 #define ECFP_T18 11090678776483259438313656736572334813745748301503266300681918322458\ 485231222502492159897624416558312389564843845614287315896631296.0 #define ECFP_T19 18607071341967536398062689481932916079453218833595342343206149099024\ 36577570298683715049089827234727835552055312041415509848580169253519\ 36.0 #define ECFP_TWO160 1461501637330902918203684832716283019655932542976.0 #define ECFP_TWO192 6277101735386680763835789423207666416102355444464034512896.0 #define ECFP_TWO224 26959946667150639794667015087019630673637144422540572481103610249216.0 /* Multiplicative constants */ static const double ecfp_two32 = 4294967296.0; static const double ecfp_two64 = 18446744073709551616.0; static const double ecfp_twom16 = .0000152587890625; static const double ecfp_twom128 = .00000000000000000000000000000000000000293873587705571876992184134305561419454666389193021880377187926569604314863681793212890625; static const double ecfp_twom129 = .000000000000000000000000000000000000001469367938527859384960920671527807097273331945965109401885939632848021574318408966064453125; static const double ecfp_twom160 = .0000000000000000000000000000000000000000000000006842277657836020854119773355907793609766904013068924666782559979930620520927053718196475529111921787261962890625; static const double ecfp_twom192 = .000000000000000000000000000000000000000000000000000000000159309191113245227702888039776771180559110455519261878607388585338616290151305816094308987472018268594098344692611135542392730712890625; static const double ecfp_twom224 = .00000000000000000000000000000000000000000000000000000000000000000003709206150687421385731735261547639513367564778757791002453039058917581340095629358997312082723208437536338919136001159027049567384892725385725498199462890625; /* ecfp_exp[i] = 2^(i*ECFP_DSIZE) */ static const double ecfp_exp[2 * ECFP_MAXDOUBLES] = { ECFP_T0, ECFP_T1, ECFP_T2, ECFP_T3, ECFP_T4, ECFP_T5, ECFP_T6, ECFP_T7, ECFP_T8, ECFP_T9, ECFP_T10, ECFP_T11, ECFP_T12, ECFP_T13, ECFP_T14, ECFP_T15, ECFP_T16, ECFP_T17, ECFP_T18, ECFP_T19 }; /* 1.1 * 2^52 Uses 2^52 to truncate, the .1 is an extra 2^51 to protect * the 2^52 bit, so that adding alphas to a negative number won't borrow * and empty the important 2^52 bit */ #define ECFP_ALPHABASE_53 6755399441055744.0 /* Special case: On some platforms, notably x86 Linux, there is an * extended-precision floating point representation with 64-bits of * precision in the mantissa. These extra bits of precision require a * larger value of alpha to truncate, i.e. 1.1 * 2^63. */ #define ECFP_ALPHABASE_64 13835058055282163712.0 /* * ecfp_alpha[i] = 1.5 * 2^(52 + i*ECFP_DSIZE) we add and subtract alpha * to truncate floating point numbers to a certain number of bits for * tidying */ static const double ecfp_alpha_53[2 * ECFP_MAXDOUBLES] = { ECFP_ALPHABASE_53 * ECFP_T0, ECFP_ALPHABASE_53 * ECFP_T1, ECFP_ALPHABASE_53 * ECFP_T2, ECFP_ALPHABASE_53 * ECFP_T3, ECFP_ALPHABASE_53 * ECFP_T4, ECFP_ALPHABASE_53 * ECFP_T5, ECFP_ALPHABASE_53 * ECFP_T6, ECFP_ALPHABASE_53 * ECFP_T7, ECFP_ALPHABASE_53 * ECFP_T8, ECFP_ALPHABASE_53 * ECFP_T9, ECFP_ALPHABASE_53 * ECFP_T10, ECFP_ALPHABASE_53 * ECFP_T11, ECFP_ALPHABASE_53 * ECFP_T12, ECFP_ALPHABASE_53 * ECFP_T13, ECFP_ALPHABASE_53 * ECFP_T14, ECFP_ALPHABASE_53 * ECFP_T15, ECFP_ALPHABASE_53 * ECFP_T16, ECFP_ALPHABASE_53 * ECFP_T17, ECFP_ALPHABASE_53 * ECFP_T18, ECFP_ALPHABASE_53 * ECFP_T19 }; /* * ecfp_alpha[i] = 1.5 * 2^(63 + i*ECFP_DSIZE) we add and subtract alpha * to truncate floating point numbers to a certain number of bits for * tidying */ static const double ecfp_alpha_64[2 * ECFP_MAXDOUBLES] = { ECFP_ALPHABASE_64 * ECFP_T0, ECFP_ALPHABASE_64 * ECFP_T1, ECFP_ALPHABASE_64 * ECFP_T2, ECFP_ALPHABASE_64 * ECFP_T3, ECFP_ALPHABASE_64 * ECFP_T4, ECFP_ALPHABASE_64 * ECFP_T5, ECFP_ALPHABASE_64 * ECFP_T6, ECFP_ALPHABASE_64 * ECFP_T7, ECFP_ALPHABASE_64 * ECFP_T8, ECFP_ALPHABASE_64 * ECFP_T9, ECFP_ALPHABASE_64 * ECFP_T10, ECFP_ALPHABASE_64 * ECFP_T11, ECFP_ALPHABASE_64 * ECFP_T12, ECFP_ALPHABASE_64 * ECFP_T13, ECFP_ALPHABASE_64 * ECFP_T14, ECFP_ALPHABASE_64 * ECFP_T15, ECFP_ALPHABASE_64 * ECFP_T16, ECFP_ALPHABASE_64 * ECFP_T17, ECFP_ALPHABASE_64 * ECFP_T18, ECFP_ALPHABASE_64 * ECFP_T19 }; /* 0.011111111111111111111111 (binary) = 0.5 - 2^25 (24 ones) */ #define ECFP_BETABASE 0.4999999701976776123046875 /* * We subtract beta prior to using alpha to simulate rounding down. We * make this close to 0.5 to round almost everything down, but exactly 0.5 * would cause some incorrect rounding. */ static const double ecfp_beta[2 * ECFP_MAXDOUBLES] = { ECFP_BETABASE * ECFP_T0, ECFP_BETABASE * ECFP_T1, ECFP_BETABASE * ECFP_T2, ECFP_BETABASE * ECFP_T3, ECFP_BETABASE * ECFP_T4, ECFP_BETABASE * ECFP_T5, ECFP_BETABASE * ECFP_T6, ECFP_BETABASE * ECFP_T7, ECFP_BETABASE * ECFP_T8, ECFP_BETABASE * ECFP_T9, ECFP_BETABASE * ECFP_T10, ECFP_BETABASE * ECFP_T11, ECFP_BETABASE * ECFP_T12, ECFP_BETABASE * ECFP_T13, ECFP_BETABASE * ECFP_T14, ECFP_BETABASE * ECFP_T15, ECFP_BETABASE * ECFP_T16, ECFP_BETABASE * ECFP_T17, ECFP_BETABASE * ECFP_T18, ECFP_BETABASE * ECFP_T19 }; static const double ecfp_beta_160 = ECFP_BETABASE * ECFP_TWO160; static const double ecfp_beta_192 = ECFP_BETABASE * ECFP_TWO192; static const double ecfp_beta_224 = ECFP_BETABASE * ECFP_TWO224; /* Affine EC Point. This is the basic representation (x, y) of an elliptic * curve point. */ typedef struct { double x[ECFP_MAXDOUBLES]; double y[ECFP_MAXDOUBLES]; } ecfp_aff_pt; /* Jacobian EC Point. This coordinate system uses X = x/z^2, Y = y/z^3, * which enables calculations with fewer inversions than affine * coordinates. */ typedef struct { double x[ECFP_MAXDOUBLES]; double y[ECFP_MAXDOUBLES]; double z[ECFP_MAXDOUBLES]; } ecfp_jac_pt; /* Chudnovsky Jacobian EC Point. This coordinate system is the same as * Jacobian, except it keeps z^2, z^3 for faster additions. */ typedef struct { double x[ECFP_MAXDOUBLES]; double y[ECFP_MAXDOUBLES]; double z[ECFP_MAXDOUBLES]; double z2[ECFP_MAXDOUBLES]; double z3[ECFP_MAXDOUBLES]; } ecfp_chud_pt; /* Modified Jacobian EC Point. This coordinate system is the same as * Jacobian, except it keeps a*z^4 for faster doublings. */ typedef struct { double x[ECFP_MAXDOUBLES]; double y[ECFP_MAXDOUBLES]; double z[ECFP_MAXDOUBLES]; double az4[ECFP_MAXDOUBLES]; } ecfp_jm_pt; struct EC_group_fp_str; typedef struct EC_group_fp_str EC_group_fp; struct EC_group_fp_str { int fpPrecision; /* Set to number of bits in mantissa, 53 * or 64 */ int numDoubles; int primeBitSize; int orderBitSize; int doubleBitSize; int numInts; int aIsM3; /* True if curvea == -3 (mod p), then we * can optimize doubling */ double curvea[ECFP_MAXDOUBLES]; /* Used to truncate a double to the number of bits in the curve */ double bitSize_alpha; /* Pointer to either ecfp_alpha_53 or ecfp_alpha_64 */ const double *alpha; void (*ecfp_singleReduce) (double *r, const EC_group_fp * group); void (*ecfp_reduce) (double *r, double *x, const EC_group_fp * group); /* Performs a "tidy" operation, which performs carrying, moving excess * bits from one double to the next double, so that the precision of * the doubles is reduced to the regular precision ECFP_DSIZE. This * might result in some float digits being negative. */ void (*ecfp_tidy) (double *t, const double *alpha, const EC_group_fp * group); /* Perform a point addition using coordinate system Jacobian + Affine * -> Jacobian. Input and output should be multi-precision floating * point integers. */ void (*pt_add_jac_aff) (const ecfp_jac_pt * p, const ecfp_aff_pt * q, ecfp_jac_pt * r, const EC_group_fp * group); /* Perform a point doubling in Jacobian coordinates. Input and output * should be multi-precision floating point integers. */ void (*pt_dbl_jac) (const ecfp_jac_pt * dp, ecfp_jac_pt * dr, const EC_group_fp * group); /* Perform a point addition using Jacobian coordinate system. Input * and output should be multi-precision floating point integers. */ void (*pt_add_jac) (const ecfp_jac_pt * p, const ecfp_jac_pt * q, ecfp_jac_pt * r, const EC_group_fp * group); /* Perform a point doubling in Modified Jacobian coordinates. Input * and output should be multi-precision floating point integers. */ void (*pt_dbl_jm) (const ecfp_jm_pt * p, ecfp_jm_pt * r, const EC_group_fp * group); /* Perform a point doubling using coordinates Affine -> Chudnovsky * Jacobian. Input and output should be multi-precision floating point * integers. */ void (*pt_dbl_aff2chud) (const ecfp_aff_pt * p, ecfp_chud_pt * r, const EC_group_fp * group); /* Perform a point addition using coordinates: Modified Jacobian + * Chudnovsky Jacobian -> Modified Jacobian. Input and output should * be multi-precision floating point integers. */ void (*pt_add_jm_chud) (ecfp_jm_pt * p, ecfp_chud_pt * q, ecfp_jm_pt * r, const EC_group_fp * group); /* Perform a point addition using Chudnovsky Jacobian coordinates. * Input and output should be multi-precision floating point integers. */ void (*pt_add_chud) (const ecfp_chud_pt * p, const ecfp_chud_pt * q, ecfp_chud_pt * r, const EC_group_fp * group); /* Expects out to be an array of size 16 of Chudnovsky Jacobian * points. Fills in Chudnovsky Jacobian form (x, y, z, z^2, z^3), for * -15P, -13P, -11P, -9P, -7P, -5P, -3P, -P, P, 3P, 5P, 7P, 9P, 11P, * 13P, 15P */ void (*precompute_chud) (ecfp_chud_pt * out, const ecfp_aff_pt * p, const EC_group_fp * group); /* Expects out to be an array of size 16 of Jacobian points. Fills in * Chudnovsky Jacobian form (x, y, z), for O, P, 2P, ... 15P */ void (*precompute_jac) (ecfp_jac_pt * out, const ecfp_aff_pt * p, const EC_group_fp * group); }; /* Computes r = x*y. * r must be different (point to different memory) than x and y. * Does not tidy or reduce. */ void ecfp_multiply(double *r, const double *x, const double *y); /* Performs a "tidy" operation, which performs carrying, moving excess * bits from one double to the next double, so that the precision of the * doubles is reduced to the regular precision group->doubleBitSize. This * might result in some float digits being negative. */ void ecfp_tidy(double *t, const double *alpha, const EC_group_fp * group); /* Performs tidying on only the upper float digits of a multi-precision * floating point integer, i.e. the digits beyond the regular length which * are removed in the reduction step. */ void ecfp_tidyUpper(double *t, const EC_group_fp * group); /* Performs tidying on a short multi-precision floating point integer (the * lower group->numDoubles floats). */ void ecfp_tidyShort(double *t, const EC_group_fp * group); /* Performs a more mathematically precise "tidying" so that each term is * positive. This is slower than the regular tidying, and is used for * conversion from floating point to integer. */ void ecfp_positiveTidy(double *t, const EC_group_fp * group); /* Computes R = nP where R is (rx, ry) and P is (px, py). The parameters * a, b and p are the elliptic curve coefficients and the prime that * determines the field GFp. Elliptic curve points P and R can be * identical. Uses mixed Jacobian-affine coordinates. Uses 4-bit window * method. */ mp_err ec_GFp_point_mul_jac_4w_fp(const mp_int *n, const mp_int *px, const mp_int *py, mp_int *rx, mp_int *ry, const ECGroup *ecgroup); /* Computes R = nP where R is (rx, ry) and P is the base point. The * parameters a, b and p are the elliptic curve coefficients and the prime * that determines the field GFp. Elliptic curve points P and R can be * identical. Uses mixed Jacobian-affine coordinates (Jacobian * coordinates for doubles and affine coordinates for additions; based on * recommendation from Brown et al.). Uses window NAF method (algorithm * 11) for scalar-point multiplication from Brown, Hankerson, Lopez, * Menezes. Software Implementation of the NIST Elliptic Curves Over Prime * Fields. */ mp_err ec_GFp_point_mul_wNAF_fp(const mp_int *n, const mp_int *px, const mp_int *py, mp_int *rx, mp_int *ry, const ECGroup *ecgroup); /* Uses mixed Jacobian-affine coordinates to perform a point * multiplication: R = n * P, n scalar. Uses mixed Jacobian-affine * coordinates (Jacobian coordinates for doubles and affine coordinates * for additions; based on recommendation from Brown et al.). Not very * time efficient but quite space efficient, no precomputation needed. * group contains the elliptic curve coefficients and the prime that * determines the field GFp. Elliptic curve points P and R can be * identical. Performs calculations in floating point number format, since * this is faster than the integer operations on the ULTRASPARC III. * Uses left-to-right binary method (double & add) (algorithm 9) for * scalar-point multiplication from Brown, Hankerson, Lopez, Menezes. * Software Implementation of the NIST Elliptic Curves Over Prime Fields. */ mp_err ec_GFp_pt_mul_jac_fp(const mp_int *n, const mp_int *px, const mp_int *py, mp_int *rx, mp_int *ry, const ECGroup *ecgroup); /* Cleans up extra memory allocated in ECGroup for this implementation. */ void ec_GFp_extra_free_fp(ECGroup *group); /* Converts from a floating point representation into an mp_int. Expects * that d is already reduced. */ void ecfp_fp2i(mp_int *mpout, double *d, const ECGroup *ecgroup); /* Converts from an mpint into a floating point representation. */ void ecfp_i2fp(double *out, const mp_int *x, const ECGroup *ecgroup); /* Tests what precision floating point arithmetic is set to. This should * be either a 53-bit mantissa (IEEE standard) or a 64-bit mantissa * (extended precision on x86) and sets it into the EC_group_fp. Returns * either 53 or 64 accordingly. */ int ec_set_fp_precision(EC_group_fp * group); #endif