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Diffstat (limited to 'moon-abe/pbc-0.5.14/arith/ternary_extension_field.c')
| -rw-r--r-- | moon-abe/pbc-0.5.14/arith/ternary_extension_field.c | 950 |
1 files changed, 950 insertions, 0 deletions
diff --git a/moon-abe/pbc-0.5.14/arith/ternary_extension_field.c b/moon-abe/pbc-0.5.14/arith/ternary_extension_field.c new file mode 100644 index 00000000..3c79e3bd --- /dev/null +++ b/moon-abe/pbc-0.5.14/arith/ternary_extension_field.c @@ -0,0 +1,950 @@ +/* $GF(3^m) = GF(3)[x]/(x^m + x^t + 2)$ + $GF(3^{2*m}) = GF(3^m)[x]/(x^2 + 1)$ + $GF(3^{3*m}) = GF(3^m)[x]/(x^3 - x -1)$ + $GF(3^{6*m}) = GF(3^{2*m})[x]/(x^3 - x -1)$ + + The "gf3_*" functions are for $GF(3)$. + The "gf3m_*" functions are for $GF(3^m)$. + The "gf32m_*" functions are for $GF(3^{2*m})$. + The "gf33m_*" functions are for $GF(3^{3*m})$ and $GF(3^{6*m})$. + + (gf3m field_t).data is a pointer of struct params + (gf3m element_t).data is a pointer of unsigned long + (gf32m element_t).data is gf32m_ptr + (gf33m element_t).data is gf33m_ptr */ + +#include <stdlib.h> +#include <string.h> +#include <stdarg.h> +#include <stdio.h> +#include <stdint.h> +#include <gmp.h> +#include "pbc_utils.h" +#include "pbc_memory.h" +#include "pbc_field.h" + +typedef unsigned long gf3; + +typedef struct { /* private data of $GF(3^m)$ */ + unsigned int len; /* the number of native machine integers required to represent one GF(3^m) element */ + unsigned int m; /* the irreducible polynomial is $x^m + x^t + 2$ */ + unsigned int t; /* the irreducible polynomial is $x^m + x^t + 2$ */ + element_ptr p; /* $p$ is the irreducible polynomial. */ +} params; + +typedef struct { + element_t _0, _1; +} gf32m_s; + +typedef gf32m_s *gf32m_ptr; + +typedef struct { + element_t _0, _1, _2; +} gf33m_s; + +typedef gf33m_s *gf33m_ptr; + +#define W (sizeof(unsigned long)*8) /* number of GF(3) elements in one processor integer */ +#define PARAM(e) ((params *)e->field->data) +#define LEN(e) (PARAM(e)->len) +#define SIZE(e) (LEN(e) * 2 * sizeof(unsigned long)) +#define DATA1(e) ((unsigned long*)e->data) +#define DATA2(e) ((unsigned long*)e->data + LEN(e)) +#define GF32M(e) ((gf32m_s *)e->data) +#define GF33M(e) ((gf33m_s *)e->data) +#define BASE(e) ((field_ptr)e->field->data) +#define print(e) {printf(#e": "); element_out_str(stdout, 10, e); printf("\n");} + +static size_t gf3m_out_str(FILE *stream, int base, element_t e) { + if (base != 10 && base != 16) + pbc_die("only support base 10 and base 16"); + size_t size = 0; + unsigned i; + unsigned long *d = DATA1(e); + for (i = 0; i < LEN(e) * 2; i++) { + if (base == 16) + size += fprintf(stream, "0x%lx,", d[i]); + else + size += fprintf(stream, "%lu,", d[i]); + } + return size; +} + +/* $a <- 0$ */ +static void gf3m_zero(element_t a) { + memset(a->data, 0, SIZE(a)); +} + +static void gf3m_init(element_t e) { + e->data = pbc_malloc(SIZE(e)); + gf3m_zero(e); +} + +static void gf3m_clear(element_t e) { + pbc_free(e->data); +} + +/* $e <- a$ */ +static void gf3m_assign(element_t e, element_t a) { + memcpy(e->data, a->data, SIZE(a)); +} + +/* $a <- a/x$. $len$ is the number of elements in $a$ */ +static void shift_down(unsigned int len, unsigned long a[]) { + unsigned long h = 0; + const unsigned long x = 1ul << (W - 1); + int i; + for (i = len - 1; i >= 0; i--) { + unsigned long l = a[i] & 1; + a[i] >>= 1; + if (h) + a[i] |= x; + h = l; + } +} + +/* $e <- e/x$ */ +static void gf3m_shift_down(element_t e) { + shift_down(LEN(e), DATA1(e)); + shift_down(LEN(e), DATA2(e)); +} + +/* $a <- a*x$. $len$ is the number of elements in $a$ */ +static void shift_up(unsigned int len, unsigned long a[]) { + unsigned long l = 0; + const unsigned long x = 1ul << (W - 1), y = x - 1; + unsigned i; + for (i = 0; i < len; i++) { + unsigned long h = a[i] & x; + a[i] = ((a[i] & y) << 1) | l; + l = h ? 1 : 0; + } +} + +/* $e <- e*x$ */ +static void gf3m_shift_up(element_t e) { + shift_up(LEN(e), DATA1(e)); + shift_up(LEN(e), DATA2(e)); +} + +/* return the coefficient of $x^pos$ in $e$ */ +static unsigned gf3m_get(element_t e, unsigned pos) { + unsigned long *a1 = DATA1(e), *a2 = DATA2(e); + unsigned x = pos / W; + unsigned long y = 1ul << (pos % W), v1 = a1[x] & y, v2 = a2[x] & y; + return v1 ? 1 : (v2 ? 2 : 0); +} + +/* set the coefficient of $x^pos$ as 1 */ +static void gf3m_set(element_t e, unsigned pos, unsigned value) { + unsigned long *a = DATA1(e); + /* assert value == 0, 1 or 2 */ + if (value == 2) + a = DATA2(e); + if (value) + a[pos / W] |= 1ul << (pos % W); +} + +/* $e <- a+b$ */ +static void gf3m_add(element_t e, element_t a, element_t b) { + unsigned long *e1 = DATA1(e), *e2 = DATA2(e), *a1 = DATA1(a), + *a2 = DATA2(a), *b1 = DATA1(b), *b2 = DATA2(b); + unsigned i; + for (i = 0; i < LEN(e); i++, e1++, e2++, a1++, a2++, b1++, b2++) { + unsigned long t = (*a1 | *a2) & (*b1 | *b2), c1 = t ^ (*a1 | *b1), c2 = + t ^ (*a2 | *b2); + *e1 = c1; + *e2 = c2; + } +} + +/* $e <- x-y$ */ +static void gf3m_sub(element_t e, element_t a, element_t b) { + unsigned long *e1 = DATA1(e), *e2 = DATA2(e), *a1 = DATA1(a), + *a2 = DATA2(a), *b1 = DATA2(b), *b2 = DATA1(b); + unsigned i; + for (i = 0; i < LEN(e); i++, e1++, e2++, a1++, a2++, b1++, b2++) { + unsigned long t = (*a1 | *a2) & (*b1 | *b2), c1 = t ^ (*a1 | *b1), c2 = + t ^ (*a2 | *b2); + *e1 = c1; + *e2 = c2; + } +} + +/* return 0 if $a == b$ in $GF(3^m)$, 1 otherwise. */ +static int gf3m_cmp(element_t a, element_t b) { + unsigned long *pa = DATA1(a), *pb = DATA1(b); + unsigned i; + for (i = 0; i < LEN(a) * 2; i++, pa++, pb++) + if (*pa != *pb) + return 1; + return 0; +} + +/* $a <- 1$ */ +static void gf3m_one(element_t a) { + gf3m_zero(a); + *DATA1(a) = 1; +} + +static int gf3m_is0(element_t e) { + unsigned i; + for (i = 0; i < LEN(e) * 2; i++) + if (DATA1(e)[i]) + return 0; + return 1; +} + +static int gf3m_is1(element_t e) { + unsigned i; + if (DATA1(e)[0] != 1) + return 0; + for (i = 1; i < LEN(e) * 2; i++) + if (DATA1(e)[i]) + return 0; + return 1; +} + +/* set $a$ to be a random element in $GF(3^m)$ */ +static void gf3m_random(element_t a) { + /* TODO: use uniform distribution? */ + params *c = PARAM(a); + unsigned rm = c->m % W; + const unsigned long i1 = ~0ul; + unsigned long i2 = (1ul << rm) - 1; + unsigned long *a1 = DATA1(a), *a2 = DATA2(a); + unsigned i; + for (i = 0; i < c->len - 1; i++, a1++, a2++) { /* TODO: if $RAND_MAX < i1$ ? */ + *a1 = rand() & i1; + *a2 = rand() & i1 & ~(*a1); /* assuring there is no bit that a1[x] & a2[x] == 1 */ + } + unsigned long x = rm ? i2 : i1; + *a1 = rand() & x; + *a2 = rand() & x & ~(*a1); +} + +static void swap(unsigned long *a, unsigned long *b) { + *a ^= *b; + *b ^= *a; + *a ^= *b; +} + +/* $y <- (-x)$ */ +static void gf3m_neg(element_t y, element_t x) { + unsigned long *a1 = DATA1(x), *a2 = DATA2(x), *c1 = DATA1(y), + *c2 = DATA2(y); + if (a1 == c1) { + unsigned i; + for (i = 0; i < LEN(y); i++, a1++, a2++) + swap(a1, a2); + } else { + memcpy(c1, a2, SIZE(y) / 2); + memcpy(c2, a1, SIZE(y) / 2); + } +} + +/* doing reduction + * The function returns the value of $a$ modulo $the irreducible trinomial$. + * $degree$ equals the degree of $a$. + * $2*len$ is the number of elements in $a$ */ +static void gf3m_reduct(element_t e, unsigned len, unsigned degree) { + // the $len$ argument exists because sometimes $len != p->len$ + params *p = PARAM(e); + unsigned old = p->len; + p->len = len; + element_t px; + element_init(px, e->field); + gf3m_set(px, degree, 1); + gf3m_set(px, degree - p->m + p->t, 1); + gf3m_set(px, degree - p->m, 2); + while (degree >= p->m) { + unsigned v = gf3m_get(e, degree); + if (v == 1) + gf3m_sub(e, e, px); + else if (v == 2) + gf3m_add(e, e, px); + degree--; + gf3m_shift_down(px); + } + element_clear(px); + p->len = old; +} + +/* doing multiplication of $n \in \{0,1,2\}$ and $a$ in $GF(3^m)$ + * The function sets $e <- n * a$. */ +static void gf3m_f1(element_t e, unsigned n, element_t a) { + /* assert $e$ is not $a$ */ + if (n == 0) + memset(DATA1(e), 0, SIZE(e)); + else if (n == 1) + memcpy(DATA1(e), DATA1(a), SIZE(e)); + else { + memcpy(DATA1(e), DATA2(a), SIZE(e) / 2); + memcpy(DATA2(e), DATA1(a), SIZE(e) / 2); + } +} + +/* $e <- e*x mod p(x)$ */ +static void gf3m_f2(element_t e) { + params *p = PARAM(e); + gf3m_shift_up(e); + unsigned v = gf3m_get(e, p->m); + if (v == 1) + gf3m_sub(e, e, p->p); + else if (v == 2) + gf3m_add(e, e, p->p); +} + +/* doing multiplication in GF(3^m) + * The function sets $e == a*b \in GF(3^m)$ */ +static void gf3m_mult(element_t e, element_ptr a, element_t b) { + params *p = PARAM(a); + element_t aa, t, c; + element_init(aa, a->field); + element_set(aa, a); + a = aa; // clone $a$ + element_init(t, a->field); + element_init(c, a->field); + unsigned i; + for (i = 0; i < p->m; i++) { + unsigned v = gf3m_get(b, i); + gf3m_f1(t, v, a); /* t == b[i]*a in GF(3^m) */ + gf3m_add(c, c, t); /* c += b[i]*a in GF(3^m) */ + gf3m_f2(a); /* a == a*x in GF(3^m) */ + } + element_set(e, c); + element_clear(t); + element_clear(c); + element_clear(aa); +} + +/* $e <- x^3$ */ +static void gf3m_cubic(element_t e, element_t x) { + /* TODO: faster algorithm */ + params *p = PARAM(x); + unsigned old = p->len; + unsigned len = (3 * p->m - 2 + W - 1) / W; /* length of $b1 */ + p->len = len; + element_t a; + element_init(a, x->field); + unsigned i; + for (i = 0; i < p->m; i++) { + p->len = old; + unsigned v = gf3m_get(x, i); + p->len = len; + gf3m_set(a, 3 * i, v); + } + gf3m_reduct(a, len, 3 * p->m - 3); + p->len = old; + memcpy(DATA1(e), DATA1(a), SIZE(e) / 2); + memcpy(DATA2(e), DATA1(a) + len, SIZE(e) / 2); + element_clear(a); +} + +/* multiplication modulo 3 of two elements in GF(3) + * for example, $mult(2,2) == 1$, and $mult(1,2) == 2$ */ +static unsigned gf3_mult(unsigned a, unsigned b) { + static const unsigned l[] = { 0, 1, 2, 0, 1 }; + return l[a * b]; +} + +static void gf3m_swap(element_t a, element_t b) { + unsigned long *p = DATA1(a); + a->data = b->data; + b->data = p; +} + +/* computing the inversion of an element $a$ in GF(3^m), i.e., $e <- a^{-1}$ + The algorithm is by Tim Kerins, Emanuel Popovici and William Marnane + in the paper of "Algorithms and Architectures for use in FPGA", + Lecture Notes in Computer Science, 2004, Volume 3203/2004, 74-83. + Note that $U$ must have an extra bit, i.e, (_m + W - 1) // W == (_m + W) // W */ +static void gf3m_invert(element_t e, element_t a) { + struct field_s *f = a->field; + params *p = PARAM(a); + unsigned lenA = p->len; + unsigned lenS = (3 * p->m + W - 1) / W; + p->len = lenS; + element_t S, R, t, U, V, t2; + element_init(S, f); + element_init(R, f); + element_init(t, f); + memcpy(DATA1(S), DATA1(p->p), lenA * sizeof(unsigned long)); /* S = p(x) */ + memcpy(DATA1(S) + lenS, DATA1(p->p) + lenA, lenA * sizeof(unsigned long)); + memcpy(DATA1(R), DATA1(a), lenA * sizeof(unsigned long)); /* R = _clone(a) */ + memcpy(DATA1(R) + lenS, DATA1(a) + lenA, lenA * sizeof(unsigned long)); + p->len = lenA; + element_init(U, f); + gf3m_one(U); + element_init(V, f); + element_init(t2, f); + unsigned d = 0, i, r_m, s_m, q, x; + for (i = 0; i < p->m * 2; i++) { + p->len = lenS; + r_m = gf3m_get(R, p->m), s_m = gf3m_get(S, p->m); + if (r_m == 0) { + gf3m_shift_up(R); /* R = xR */ + p->len = lenA; + gf3m_f2(U); /* U = xU mod p */ + d++; + } else { + q = gf3_mult(r_m, s_m); + gf3m_f1(t, q, R); + gf3m_sub(S, S, t); /* S = S-qR */ + gf3m_shift_up(S); /* S = xS */ + p->len = lenA; + gf3m_f1(t2, q, U); + gf3m_sub(V, V, t2); /* V = V-qU */ + if (d == 0) { + gf3m_swap(S, R); + gf3m_swap(U, V); + gf3m_f2(U); /* U = xU mod p*/ + d++; + } else { + x = gf3m_get(U, 0); + if (x == 1) /* assuring x|U */ + gf3m_add(U, U, p->p); + else if (x == 2) + gf3m_sub(U, U, p->p); + gf3m_shift_down(U); /* divide U by $x$ */ + d--; + } + } + } + p->len = lenS; + r_m = gf3m_get(R, p->m); /* assume r_m is not zero */ + p->len = lenA; + if (r_m == 2) + gf3m_neg(U, U); + memcpy(e->data, U->data, lenA * 2 * sizeof(unsigned long)); + element_clear(S); + element_clear(R); + element_clear(U); + element_clear(V); + element_clear(t); + element_clear(t2); +} + +static void gf3m_sqrt(element_t e, element_t a) { + field_ptr f = e->field; + mpz_t t; + mpz_init(t); // t == (field_order + 1) / 4 + mpz_set(t, f->order); + mpz_add_ui(t, t, 1); + mpz_tdiv_q_2exp(t, t, 2); + element_pow_mpz(e, a, t); + mpz_clear(t); +} + +int gf3m_to_bytes(unsigned char *d, element_ptr e) { + unsigned long *a = DATA1(e), *b = DATA2(e); + unsigned long i, j; + for (i = 0; i < LEN(e); i++, a++, b++) { + for (j = 0; j < sizeof(unsigned long) * 8; j += 8) { + *(d++) = (unsigned char) ((*a) >> j); + *(d++) = (unsigned char) ((*b) >> j); + } + } + return SIZE(e); +} + +int gf3m_from_bytes(element_ptr e, unsigned char *d) { + unsigned long *a = DATA1(e), *b = DATA2(e); + unsigned i; + int j; + for (i = 0; i < LEN(e); i++, a++, b++, d += sizeof(unsigned long) * 2) { + *a = 0, *b = 0; + j = 2 * sizeof(unsigned long) - 2; + while (j >= 0) { + *a <<= 8, *b <<= 8; + *a += d[j]; + *b += d[j + 1]; + j -= 2; + } + } + return SIZE(e); +} + +static void field_clear_gf3m(field_t f) { + params *p = f->data; + gf3m_clear(p->p); + pbc_free(p->p); + pbc_free(p); +} + +/* initialize the finite field as $GF(3^m)$, whose irreducible polynomial is with the degree of $m$ */ +void field_init_gf3m(field_t f, unsigned m, unsigned t) { + params *p = pbc_malloc(sizeof(*p)); + p->len = (m + (W - 1) + 1) / W; /* extra one bit for $_p$ */ + p->m = m; + p->t = t; + p->p = pbc_malloc(sizeof(*(p->p))); + p->p->field = f; + p->p->data = pbc_malloc(2 * sizeof(unsigned long) * p->len); + memset(p->p->data, 0, 2 * sizeof(unsigned long) * p->len); + unsigned long *p1 = p->p->data, *p2 = p1 + p->len; + p2[0] = 1; /* _p == x^m+x^t+2 */ + unsigned int p_t = p->t; + p1[p_t / W] |= 1ul << (p_t % W); + p1[m / W] |= 1ul << (m % W); + + field_init(f); + f->field_clear = field_clear_gf3m; + f->init = gf3m_init; + f->clear = gf3m_clear; + f->set = gf3m_assign; + f->set0 = gf3m_zero; + f->set1 = gf3m_one; + f->is0 = gf3m_is0; + f->is1 = gf3m_is1; + f->add = gf3m_add; + f->sub = gf3m_sub; + f->mul = gf3m_mult; + f->cubic = gf3m_cubic; + f->invert = gf3m_invert; + f->neg = gf3m_neg; + f->random = gf3m_random; + f->cmp = gf3m_cmp; + f->sqrt = gf3m_sqrt; + f->from_bytes = gf3m_from_bytes; + f->to_bytes = gf3m_to_bytes; + f->out_str = gf3m_out_str; + f->fixed_length_in_bytes = 2 * sizeof(unsigned long) * p->len; + f->data = p; + f->name = "GF(3^m)"; + + mpz_set_ui(f->order, 3); + mpz_pow_ui(f->order, f->order, p->m); +} + +static size_t gf32m_out_str(FILE *stream, int base, element_t e) { + UNUSED_VAR(base); + element_ptr e0 = GF32M(e)->_0, e1 = GF32M(e)->_1; + size_t size = 0; + size += element_out_str(stream, base, e0); + size += element_out_str(stream, base, e1); + return size; +} + +static void gf32m_init(element_t e) { + e->data = pbc_malloc(sizeof(gf32m_s)); + gf32m_ptr p = (gf32m_ptr) e->data; + field_ptr base = BASE(e); + element_init(p->_0, base); + element_init(p->_1, base); +} + +static void gf32m_clear(element_t e) { + gf32m_ptr p = (gf32m_ptr) e->data; + element_clear(p->_0); + element_clear(p->_1); + pbc_free(e->data); +} + +static void gf32m_set0(element_t e) { + element_ptr e0 = GF32M(e)->_0, e1 = GF32M(e)->_1; + element_set0(e0); + element_set0(e1); +} + +static void gf32m_set1(element_t e) { + element_ptr e0 = GF32M(e)->_0, e1 = GF32M(e)->_1; + element_set1(e0); + element_set0(e1); +} + +static int gf32m_item_count(element_t e) { + UNUSED_VAR(e); + return 2; +} + +static element_ptr gf32m_item(element_t a, int i) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1; + return i == 0 ? a0 : a1; +} + +static void gf32m_assign(element_t e, element_t a) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, e0 = GF32M(e)->_0, e1 = + GF32M(e)->_1; + element_set(e0, a0); + element_set(e1, a1); +} + +static void gf32m_random(element_t e) { + element_ptr e0 = GF32M(e)->_0, e1 = GF32M(e)->_1; + element_random(e0); + element_random(e1); +} + +/* return 0 if $a == b$, 1 otherwise */ +static int gf32m_cmp(element_t a, element_t b) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, b0 = GF32M(b)->_0, b1 = + GF32M(b)->_1; + return element_cmp(a0, b0) || element_cmp(a1, b1); +} + +/* $c <- a+b$ */ +static void gf32m_add(element_t c, element_t a, element_t b) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, b0 = GF32M(b)->_0, b1 = + GF32M(b)->_1, c0 = GF32M(c)->_0, c1 = GF32M(c)->_1; + element_add(c0, a0, b0); + element_add(c1, a1, b1); +} + +/* $c <- a-b$ */ +static void gf32m_sub(element_t c, element_t a, element_t b) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, b0 = GF32M(b)->_0, b1 = + GF32M(b)->_1, c0 = GF32M(c)->_0, c1 = GF32M(c)->_1; + element_sub(c0, a0, b0); + element_sub(c1, a1, b1); +} + +/* $c <- (-a)$ */ +static void gf32m_neg(element_t c, element_t a) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, c0 = GF32M(c)->_0, c1 = + GF32M(c)->_1; + element_neg(c0, a0); + element_neg(c1, a1); +} + +/* $e<- a*b$ */ +static void gf32m_mult(element_t e, element_t a, element_t b) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, b0 = GF32M(b)->_0, b1 = + GF32M(b)->_1, e0 = GF32M(e)->_0, e1 = GF32M(e)->_1; + field_ptr base = BASE(a); + element_t a0b0, a1b1, t0, t1, c1; + element_init(a0b0, base); + element_init(a1b1, base); + element_init(t0, base); + element_init(t1, base); + element_init(c1, base); + element_mul(a0b0, a0, b0); + element_mul(a1b1, a1, b1); + element_add(t0, a1, a0); + element_add(t1, b1, b0); + element_mul(c1, t0, t1); // c1 == (a1+a0)*(b1+b0) + element_sub(c1, c1, a1b1); + element_sub(c1, c1, a0b0); + element_ptr c0 = a0b0; + element_sub(c0, c0, a1b1); // c0 == a0*b0 - a1*b1 + element_set(e0, c0); + element_set(e1, c1); + element_clear(a0b0); + element_clear(a1b1); + element_clear(t0); + element_clear(t1); + element_clear(c1); +} + +/* $e <- a^3$ */ +static void gf32m_cubic(element_t e, element_t a) { + element_ptr a0 = GF32M(a)->_0, a1 = GF32M(a)->_1, e0 = GF32M(e)->_0, e1 = + GF32M(e)->_1; + field_ptr base = BASE(a); + element_t c0, c1; + element_init(c0, base); + element_init(c1, base); + element_cubic(c0, a0); + element_cubic(c1, a1); + element_neg(c1, c1); // c1 == -(a1^3) + element_set(e0, c0); + element_set(e1, c1); + element_clear(c0); + element_clear(c1); +} + +void field_clear_gf32m(field_t f) { + UNUSED_VAR(f); +} + +/* initialize the finite field as $base_field[x]/(x^2 + 1)$, whose base field is $b$ */ +void field_init_gf32m(field_t f, field_t b) { + field_init(f); + f->data = b; + f->field_clear = field_clear_gf32m; + f->init = gf32m_init; + f->clear = gf32m_clear; + f->set = gf32m_assign; + f->set0 = gf32m_set0; + f->set1 = gf32m_set1; + f->random = gf32m_random; + f->cmp = gf32m_cmp; + f->add = gf32m_add; + f->sub = gf32m_sub; + f->neg = gf32m_neg; + f->mul = gf32m_mult; + f->cubic = gf32m_cubic; + f->item_count = gf32m_item_count; + f->item = gf32m_item; + f->out_str = gf32m_out_str; + mpz_pow_ui(f->order, b->order, 2); + f->name = "GF(3^{2*m})"; +} + +static size_t gf33m_out_str(FILE *stream, int base, element_t e) { + UNUSED_VAR(base); + element_ptr e0 = GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + size_t size = 0; + size += element_out_str(stream, base, e0); + size += element_out_str(stream, base, e1); + size += element_out_str(stream, base, e2); + return size; +} + +static void gf33m_init(element_t e) { + e->data = pbc_malloc(sizeof(gf33m_s)); + gf33m_ptr p = (gf33m_ptr) e->data; + field_ptr base = BASE(e); + element_init(p->_0, base); + element_init(p->_1, base); + element_init(p->_2, base); +} + +static void gf33m_clear(element_t e) { + gf33m_ptr p = (gf33m_ptr) e->data; + element_clear(p->_0); + element_clear(p->_1); + element_clear(p->_2); + pbc_free(e->data); +} + +static void gf33m_set0(element_t e) { + element_ptr e0 = GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + element_set0(e0); + element_set0(e1); + element_set0(e2); +} + +static void gf33m_set1(element_t e) { + element_ptr e0 = GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + element_set1(e0); + element_set0(e1); + element_set0(e2); +} + +static int gf33m_item_count(element_t e) { + UNUSED_VAR(e); + return 3; +} + +static element_ptr gf33m_item(element_t a, int i) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2; + return i == 0 ? a0 : (i == 1 ? a1 : a2); +} + +static void gf33m_assign(element_t e, element_t a) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, e0 = + GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + element_set(e0, a0); + element_set(e1, a1); + element_set(e2, a2); +} + +static void gf33m_random(element_t e) { + element_ptr e0 = GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + element_random(e0); + element_random(e1); + element_random(e2); +} + +/* return 0 if $a == b$, 1 otherwise */ +static int gf33m_cmp(element_t a, element_t b) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, b0 = + GF33M(b)->_0, b1 = GF33M(b)->_1, b2 = GF33M(b)->_2; + return element_cmp(a0, b0) || element_cmp(a1, b1) || element_cmp(a2, b2); +} + +/* $c <- a+b$ */ +static void gf33m_add(element_t c, element_t a, element_t b) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, b0 = + GF33M(b)->_0, b1 = GF33M(b)->_1, b2 = GF33M(b)->_2, c0 = + GF33M(c)->_0, c1 = GF33M(c)->_1, c2 = GF33M(c)->_2; + element_add(c0, a0, b0); + element_add(c1, a1, b1); + element_add(c2, a2, b2); +} + +/* $c <- a-b$ */ +static void gf33m_sub(element_t c, element_t a, element_t b) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, b0 = + GF33M(b)->_0, b1 = GF33M(b)->_1, b2 = GF33M(b)->_2, c0 = + GF33M(c)->_0, c1 = GF33M(c)->_1, c2 = GF33M(c)->_2; + element_sub(c0, a0, b0); + element_sub(c1, a1, b1); + element_sub(c2, a2, b2); +} + +/* $c <- a*b$ */ +static void gf33m_mult(element_t e, element_t a, element_t b) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, b0 = + GF33M(b)->_0, b1 = GF33M(b)->_1, b2 = GF33M(b)->_2, e0 = + GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + field_ptr base = BASE(e); + element_t t0, t1, c1, a0b0, a1b1, a2b2; + element_init(t0, base); + element_init(t1, base); + element_init(c1, base); + element_init(a0b0, base); + element_init(a1b1, base); + element_init(a2b2, base); + element_mul(a0b0, a0, b0); + element_mul(a1b1, a1, b1); + element_mul(a2b2, a2, b2); + element_ptr d0 = a0b0; + element_add(t0, a1, a0); + element_add(t1, b1, b0); + element_t d1; + element_init(d1, base); + element_mul(d1, t0, t1); + element_sub(d1, d1, a1b1); + element_sub(d1, d1, a0b0); + element_add(t0, a2, a0); + element_add(t1, b2, b0); + element_t d2; + element_init(d2, base); + element_mul(d2, t0, t1); + element_add(d2, d2, a1b1); + element_sub(d2, d2, a2b2); + element_sub(d2, d2, a0b0); + element_add(t0, a2, a1); + element_add(t1, b2, b1); + element_t d3; + element_init(d3, base); + element_mul(d3, t0, t1); + element_sub(d3, d3, a2b2); + element_sub(d3, d3, a1b1); + element_ptr d4 = a2b2; + element_add(t0, d0, d3); + element_ptr c0 = t0; + element_add(c1, d1, d3); + element_add(c1, c1, d4); + element_add(t1, d2, d4); + element_ptr c2 = t1; + element_set(e0, c0); + element_set(e1, c1); + element_set(e2, c2); + element_clear(t0); + element_clear(t1); + element_clear(c1); + element_clear(a0b0); + element_clear(a1b1); + element_clear(a2b2); + element_clear(d1); + element_clear(d2); + element_clear(d3); +} + +/* $e <- a^3$ */ +static void gf33m_cubic(element_t e, element_t a) { + field_ptr base = BASE(a); + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, e0 = + GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + element_t a03, a13, a23; + element_init(a03, base); + element_init(a13, base); + element_init(a23, base); + element_cubic(a03, a0); + element_cubic(a13, a1); + element_cubic(a23, a2); + element_add(a03, a03, a13); + element_add(a03, a03, a23); + element_ptr c0 = a03; + element_sub(a13, a13, a23); + element_ptr c1 = a13; + element_ptr c2 = a23; + element_set(e0, c0); + element_set(e1, c1); + element_set(e2, c2); + element_clear(a03); + element_clear(a13); + element_clear(a23); +} + +/* $e <- a^{-1}$ */ +static void gf33m_invert(element_t e, element_t a) { + element_ptr a0 = GF33M(a)->_0, a1 = GF33M(a)->_1, a2 = GF33M(a)->_2, e0 = + GF33M(e)->_0, e1 = GF33M(e)->_1, e2 = GF33M(e)->_2; + field_ptr base = BASE(e); + element_t a02, a12, a22; + element_init(a02, base); + element_init(a12, base); + element_init(a22, base); + element_mul(a02, a0, a0); + element_mul(a12, a1, a1); + element_mul(a22, a2, a2); + element_t v0; + element_init(v0, base); + element_sub(v0, a0, a2); // v0 == a0-a2 + element_t delta; + element_init(delta, base); + element_mul(delta, v0, a02); // delta = (a0-a2)*(a0^2), free + element_sub(v0, a1, a0); // v0 == a1-a0 + element_t c0; + element_init(c0, base); + element_mul(c0, v0, a12); // c0 == (a1-a0)*(a1^2) + element_add(delta, delta, c0); // delta = (a0-a2)*(a0^2) + (a1-a0)*(a1^2) + element_sub(v0, a2, v0); // v0 == a2-(a1-a0) = a0-a1+a2 + element_t c1; + element_init(c1, base); + element_mul(c1, v0, a22); // c1 == (a0-a1+a2)*(a2^2) + element_add(delta, delta, c1); // delta = (a0-a2)*(a0^2) + (a1-a0)*(a1^2) + (a0-a1+a2)*(a2^2) + element_invert(delta, delta); // delta = [(a0-a2)*(a0^2) + (a1-a0)*(a1^2) + (a0-a1+a2)*(a2^2)] ^ {-1} + element_add(v0, a02, a22); // v0 == a0^2+a2^2 + element_t c2; + element_init(c2, base); + element_mul(c2, a0, a2); // c2 == a0*a2 + element_sub(c0, v0, c2); // c0 == a0^2+a2^2-a0*a2 + element_add(v0, a1, a2); // v0 == a1+a2 + element_t c3; + element_init(c3, base); + element_mul(c3, a1, v0); // c3 == a1*(a1+a2) + element_sub(c0, c0, c3); // c0 == a0^2+a2^2-a0*a2-a1*(a1+a2) + element_mul(c0, c0, delta); // c0 *= delta + element_mul(c1, a0, a1); // c1 == a0*a1 + element_sub(c1, a22, c1); // c1 == a2^2-a0*a1 + element_mul(c1, c1, delta); // c1 *= delta + element_sub(c2, a12, c2); // c2 == a1^2-a0*a2 + element_sub(c2, c2, a22); // c2 == a1^2-a0*a2-a2^2 + element_mul(c2, c2, delta); // c2 *= delta + element_set(e0, c0); + element_set(e1, c1); + element_set(e2, c2); + element_clear(a02); + element_clear(a12); + element_clear(a22); + element_clear(v0); + element_clear(delta); + element_clear(c0); + element_clear(c1); + element_clear(c2); + element_clear(c3); +} + +void field_clear_gf33m(field_t f) { + UNUSED_VAR(f); +} + +/* initialize the finite field as $base_field[x]/(x^3 - x - 1)$, whose base field is $b$ */ +void field_init_gf33m(field_t f, field_t b) { + field_init(f); + f->data = b; + f->field_clear = field_clear_gf33m; + f->init = gf33m_init; + f->clear = gf33m_clear; + f->set = gf33m_assign; + f->set0 = gf33m_set0; + f->set1 = gf33m_set1; + f->random = gf33m_random; + f->cmp = gf33m_cmp; + f->add = gf33m_add; + f->sub = gf33m_sub; + f->mul = gf33m_mult; + f->cubic = gf33m_cubic; + f->invert = gf33m_invert; + f->item_count = gf33m_item_count; + f->item = gf33m_item; + f->out_str = gf33m_out_str; + mpz_pow_ui(f->order, b->order, 3); + f->name = "GF(3^{3*m})"; +} + |