/*
* mpint_i.h: definitions used internally by the bignum code, and
* also a few other vaguely-bignum-like places.
*/
/* ----------------------------------------------------------------------
* The assorted conditional definitions of BignumInt and multiply
* macros used throughout the bignum code to treat numbers as arrays
* of the most conveniently sized word for the target machine.
* Exported so that other code (e.g. poly1305) can use it too.
*
* This code must export, in whatever ifdef branch it ends up in:
*
* - two types: 'BignumInt' and 'BignumCarry'. BignumInt is an
* unsigned integer type which will be used as the base word size
* for all bignum operations. BignumCarry is an unsigned integer
* type used to hold the carry flag taken as input and output by
* the BignumADC macro (see below).
*
* - five constant macros:
* + BIGNUM_INT_BITS, the number of bits in BignumInt,
* + BIGNUM_INT_BYTES, the number of bytes that works out to
* + BIGNUM_TOP_BIT, the BignumInt value consisting of only the top bit
* + BIGNUM_INT_MASK, the BignumInt value with all bits set
* + BIGNUM_INT_BITS_BITS, log to the base 2 of BIGNUM_INT_BITS.
*
* - four statement macros: BignumADC, BignumMUL, BignumMULADD,
* BignumMULADD2. These do various kinds of multi-word arithmetic,
* and all produce two output values.
* * BignumADC(ret,retc,a,b,c) takes input BignumInt values a,b
* and a BignumCarry c, and outputs a BignumInt ret = a+b+c and
* a BignumCarry retc which is the carry off the top of that
* addition.
* * BignumMUL(rh,rl,a,b) returns the two halves of the
* double-width product a*b.
* * BignumMULADD(rh,rl,a,b,addend) returns the two halves of the
* double-width value a*b + addend.
* * BignumMULADD2(rh,rl,a,b,addend1,addend2) returns the two
* halves of the double-width value a*b + addend1 + addend2.
*
* Every branch of the main ifdef below defines the type BignumInt and
* the value BIGNUM_INT_BITS_BITS. The other constant macros are
* filled in by common code further down.
*
* Most branches also define a macro DEFINE_BIGNUMDBLINT containing a
* typedef statement which declares a type _twice_ the length of a
* BignumInt. This causes the common code further down to produce a
* default implementation of the four statement macros in terms of
* that double-width type, and also to defined BignumCarry to be
* BignumInt.
*
* However, if a particular compile target does not have a type twice
* the length of the BignumInt you want to use but it does provide
* some alternative means of doing add-with-carry and double-word
* multiply, then the ifdef branch in question can just define
* BignumCarry and the four statement macros itself, and that's fine
* too.
*/
/* You can lower the BignumInt size by defining BIGNUM_OVERRIDE on the
* command line to be your chosen max value of BIGNUM_INT_BITS_BITS */
#if defined BIGNUM_OVERRIDE
#define BB_OK(b) ((b) <= BIGNUM_OVERRIDE)
#else
#define BB_OK(b) (1)
#endif
#if defined __SIZEOF_INT128__ && BB_OK(6)
/*
* 64-bit BignumInt using gcc/clang style 128-bit BignumDblInt.
*
* gcc and clang both provide a __uint128_t type on 64-bit targets
* (and, when they do, indicate its presence by the above macro),
* using the same 'two machine registers' kind of code generation
* that 32-bit targets use for 64-bit ints.
*/
typedef unsigned long long BignumInt;
#define BIGNUM_INT_BITS_BITS 6
#define DEFINE_BIGNUMDBLINT typedef __uint128_t BignumDblInt
#elif defined _MSC_VER && defined _M_AMD64 && BB_OK(6)
/*
* 64-bit BignumInt, using Visual Studio x86-64 compiler intrinsics.
*
* 64-bit Visual Studio doesn't provide very much in the way of help
* here: there's no int128 type, and also no inline assembler giving
* us direct access to the x86-64 MUL or ADC instructions. However,
* there are compiler intrinsics giving us that access, so we can
* use those - though it turns out we have to be a little careful,
* since they seem to generate wrong code if their pointer-typed
* output parameters alias their inputs. Hence all the internal temp
* variables inside the macros.
*/
#include <intrin.h>
typedef unsigned char BignumCarry; /* the type _addcarry_u64 likes to use */
typedef unsigned __int64 BignumInt;
#define BIGNUM_INT_BITS_BITS 6
#define BignumADC(ret, retc, a, b, c) do \
{ \
BignumInt ADC_tmp; \
(retc) = _addcarry_u64(c, a, b, &ADC_tmp); \
(ret) = ADC_tmp; \
} while (0)
#define BignumMUL(rh, rl, a, b) do \
{ \
BignumInt MULADD_hi; \
(rl) = _umul128(a, b, &MULADD_hi); \
(rh) = MULADD_hi; \
} while (0)
#define BignumMULADD(rh, rl, a, b, addend) do \
{ \
BignumInt MULADD_lo, MULADD_hi; \
MULADD_lo = _umul128(a, b, &MULADD_hi); \
MULADD_hi += _addcarry_u64(0, MULADD_lo, (addend), &(rl)); \
(rh) = MULADD_hi; \
} while (0)
#define BignumMULADD2(rh, rl, a, b, addend1, addend2) do \
{ \
BignumInt MULADD_lo1, MULADD_lo2, MULADD_hi; \
MULADD_lo1 = _umul128(a, b, &MULADD_hi); \
MULADD_hi += _addcarry_u64(0, MULADD_lo1, (addend1), &MULADD_lo2); \
MULADD_hi += _addcarry_u64(0, MULADD_lo2, (addend2), &(rl)); \
(rh) = MULADD_hi; \
} while (0)
#elif (defined __GNUC__ || defined _LLP64 || __STDC__ >= 199901L) && BB_OK(5)
/* 32-bit BignumInt, using C99 unsigned long long as BignumDblInt */
typedef unsigned int BignumInt;
#define BIGNUM_INT_BITS_BITS 5
#define DEFINE_BIGNUMDBLINT typedef unsigned long long BignumDblInt
#elif defined _MSC_VER && BB_OK(5)
/* 32-bit BignumInt, using Visual Studio __int64 as BignumDblInt */
typedef unsigned int BignumInt;
#define BIGNUM_INT_BITS_BITS 5
#define DEFINE_BIGNUMDBLINT typedef unsigned __int64 BignumDblInt
#elif defined _LP64 && BB_OK(5)
/*
* 32-bit BignumInt, using unsigned long itself as BignumDblInt.
*
* Only for platforms where long is 64 bits, of course.
*/
typedef unsigned int BignumInt;
#define BIGNUM_INT_BITS_BITS 5
#define DEFINE_BIGNUMDBLINT typedef unsigned long BignumDblInt
#elif BB_OK(4)
/*
* 16-bit BignumInt, using unsigned long as BignumDblInt.
*
* This is the final fallback for real emergencies: C89 guarantees
* unsigned short/long to be at least the required sizes, so this
* should work on any C implementation at all. But it'll be
* noticeably slow, so if you find yourself in this case you
* probably want to move heaven and earth to find an alternative!
*/
typedef unsigned short BignumInt;
#define BIGNUM_INT_BITS_BITS 4
#define DEFINE_BIGNUMDBLINT typedef unsigned long BignumDblInt
#else
/* Should only get here if BB_OK(4) evaluated false, i.e. the
* command line defined BIGNUM_OVERRIDE to an absurdly small
* value. */
#error Must define BIGNUM_OVERRIDE to at least 4
#endif
#undef BB_OK
/*
* Common code across all branches of that ifdef: define all the
* easy constant macros in terms of BIGNUM_INT_BITS_BITS.
*/
#define BIGNUM_INT_BITS (1 << BIGNUM_INT_BITS_BITS)
#define BIGNUM_INT_BYTES (BIGNUM_INT_BITS / 8)
#define BIGNUM_TOP_BIT (((BignumInt)1) << (BIGNUM_INT_BITS-1))
#define BIGNUM_INT_MASK (BIGNUM_TOP_BIT | (BIGNUM_TOP_BIT-1))
/*
* Just occasionally, we might need a GET_nnBIT_xSB_FIRST macro to
* operate on whatever BignumInt is.
*/
#if BIGNUM_INT_BITS_BITS == 4
#define GET_BIGNUMINT_MSB_FIRST GET_16BIT_MSB_FIRST
#define GET_BIGNUMINT_LSB_FIRST GET_16BIT_LSB_FIRST
#define PUT_BIGNUMINT_MSB_FIRST PUT_16BIT_MSB_FIRST
#define PUT_BIGNUMINT_LSB_FIRST PUT_16BIT_LSB_FIRST
#elif BIGNUM_INT_BITS_BITS == 5
#define GET_BIGNUMINT_MSB_FIRST GET_32BIT_MSB_FIRST
#define GET_BIGNUMINT_LSB_FIRST GET_32BIT_LSB_FIRST
#define PUT_BIGNUMINT_MSB_FIRST PUT_32BIT_MSB_FIRST
#define PUT_BIGNUMINT_LSB_FIRST PUT_32BIT_LSB_FIRST
#elif BIGNUM_INT_BITS_BITS == 6
#define GET_BIGNUMINT_MSB_FIRST GET_64BIT_MSB_FIRST
#define GET_BIGNUMINT_LSB_FIRST GET_64BIT_LSB_FIRST
#define PUT_BIGNUMINT_MSB_FIRST PUT_64BIT_MSB_FIRST
#define PUT_BIGNUMINT_LSB_FIRST PUT_64BIT_LSB_FIRST
#else
#error Ran out of options for GET_BIGNUMINT_xSB_FIRST
#endif
/*
* Common code across _most_ branches of the ifdef: define a set of
* statement macros in terms of the BignumDblInt type provided. In
* this case, we also define BignumCarry to be the same thing as
* BignumInt, for simplicity.
*/
#ifdef DEFINE_BIGNUMDBLINT
typedef BignumInt BignumCarry;
#define BignumADC(ret, retc, a, b, c) do \
{ \
DEFINE_BIGNUMDBLINT; \
BignumDblInt ADC_temp = (BignumInt)(a); \
ADC_temp += (BignumInt)(b); \
ADC_temp += (c); \
(ret) = (BignumInt)ADC_temp; \
(retc) = (BignumCarry)(ADC_temp >> BIGNUM_INT_BITS); \
} while (0)
#define BignumMUL(rh, rl, a, b) do \
{ \
DEFINE_BIGNUMDBLINT; \
BignumDblInt MUL_temp = (BignumInt)(a); \
MUL_temp *= (BignumInt)(b); \
(rh) = (BignumInt)(MUL_temp >> BIGNUM_INT_BITS); \
(rl) = (BignumInt)(MUL_temp); \
} while (0)
#define BignumMULADD(rh, rl, a, b, addend) do \
{ \
DEFINE_BIGNUMDBLINT; \
BignumDblInt MUL_temp = (BignumInt)(a); \
MUL_temp *= (BignumInt)(b); \
MUL_temp += (BignumInt)(addend); \
(rh) = (BignumInt)(MUL_temp >> BIGNUM_INT_BITS); \
(rl) = (BignumInt)(MUL_temp); \
} while (0)
#define BignumMULADD2(rh, rl, a, b, addend1, addend2) do \
{ \
DEFINE_BIGNUMDBLINT; \
BignumDblInt MUL_temp = (BignumInt)(a); \
MUL_temp *= (BignumInt)(b); \
MUL_temp += (BignumInt)(addend1); \
MUL_temp += (BignumInt)(addend2); \
(rh) = (BignumInt)(MUL_temp >> BIGNUM_INT_BITS); \
(rl) = (BignumInt)(MUL_temp); \
} while (0)
#endif /* DEFINE_BIGNUMDBLINT */
/* ----------------------------------------------------------------------
* Data structures used inside mpint.c.
*/
struct mp_int {
size_t nw;
BignumInt *w;
};
struct MontyContext {
/*
* The actual modulus.
*/
mp_int *m;
/*
* Montgomery multiplication works by selecting a value r > m,
* coprime to m, which is really easy to divide by. In binary
* arithmetic, that means making it a power of 2; in fact we make
* it a whole number of BignumInt.
*
* We don't store r directly as an mp_int (there's no need). But
* its value is 2^rbits; we also store rw = rbits/BIGNUM_INT_BITS
* (the corresponding word offset within an mp_int).
*
* pw is the number of words needed to store an mp_int you're
* doing reduction on: it has to be big enough to hold the sum of
* an input value up to m^2 plus an extra addend up to m*r.
*/
size_t rbits, rw, pw;
/*
* The key step in Montgomery reduction requires the inverse of -m
* mod r.
*/
mp_int *minus_minv_mod_r;
/*
* r^1, r^2 and r^3 mod m, which are used for various purposes.
*
* (Annoyingly, this is one of the rare cases where it would have
* been nicer to have a Pascal-style 1-indexed array. I couldn't
* _quite_ bring myself to put a gratuitous zero element in here.
* So you just have to live with getting r^k by taking the [k-1]th
* element of this array.)
*/
mp_int *powers_of_r_mod_m[3];
/*
* Persistent scratch space from which monty_* functions can
* allocate storage for intermediate values.
*/
mp_int *scratch;
};
/* Functions shared between mpint.c and mpunsafe.c */
mp_int *mp_make_sized(size_t nw);