/* * 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 */ #define BB_OK(b) (!defined BIGNUM_OVERRIDE || BIGNUM_OVERRIDE >= b) #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 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 bignum.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; };