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