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+/**
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+ * \file dixon.c
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+ * \brief An implementation of Dixon's Attack using bignums.
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+ *
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+ * Given a non-empty set B of primes, called factor-base, and
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+ * given a non-empty set of random numbers R, s.t. ∀ r ∈ R, s ≡ r² (mod N) is B-smooth.
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+ *
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+ * Try to find
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+ * U ⊂ R | (Πᵤ rᵢ)² ≡ Π s (mod N) and by defining x ≝ Πᵤ rᵢ, y ≝ √(Π s)
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+ * x² ≡ y² (mod N)
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+ * Asserting that x ≢ y (mod N) we claim to have found the two non-trivial
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+ * factors of N by computing gcd(x+y, N) and gcd(x-y, N).
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+ *
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+ * \note N = pq is assumed to be the public modulus,
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+ * while e, d . ed ≡ 1 (mod φ(N)) are respectively the public and the private
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+ * exponent.
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+ */
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+
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+#include <assert.h>
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+#include <stdlib.h>
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+#include <strings.h>
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+
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+#include <openssl/bn.h>
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+
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+#include "qa/questions/qarith.h"
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+#include "qa/questions/qstrings.h"
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+#include "qa/questions/questions.h"
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+
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+#define EPOCHS 100
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+#define REPOP_EPOCHS 50
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+#define BPOOL_EXTEND_STEP 42
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+#define BPOOL_STARTING_BITS 7
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+#define RPOOL_EXTEND_STEP 42
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+#define U_SIZE 10
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+
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+#define qa_rand rand
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+
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+static BIGNUM* zero;
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+
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+/**
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+ * \struct dixon_number_t
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+ * \brief Auxiliary structure holding informations for R_pool.
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+ */
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+typedef struct dixon_number {
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+ BIGNUM *r; /**< the random number which have been chosen */
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+ BIGNUM *s; /**< s ≡ r² (mod N) */
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+ BIGNUM **v; /**< a cached vectors holding the exponents for the prime
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+ * factorization of s. */
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+} dixon_number_t;
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+
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+/** Pool of random numbers, i.e. the set R. */
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+dixon_number_t *R_pool = NULL;
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+
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+static size_t R_size = 0;
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+
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+/** Pool of prime numbers, i.e. B, the factor base. */
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+static BIGNUM** B_pool = NULL;
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+static size_t B_size = 0;
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+
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+
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+/**
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+ * \brief Extends the factor base, and then adjusts R_pool
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+ *
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+ */
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+static void extend_B_pool(int max_bits)
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+{
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+ size_t i, j, old_B_size;
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+ int bits;
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+
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+ old_B_size = B_size;
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+ B_size += BPOOL_EXTEND_STEP;
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+ /* check for size_t overflow */
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+ assert(old_B_size < B_size);
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+
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+ B_pool = realloc(B_pool, B_size * sizeof(BIGNUM*));
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+
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+ for (i=old_B_size; i!=B_size; i++) {
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+ bits = 1 + qa_rand() % max_bits;
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+ B_pool[i] = BN_generate_prime(NULL, bits, 0, NULL, NULL, NULL, NULL);
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+ }
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+ /* reallocate space for vectors in R_pool */
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+ for (i=0; i!=R_size; i++) {
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+ R_pool[i].v = realloc(R_pool[i].v, sizeof(BIGNUM*) * B_size);
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+ for (j=old_B_size; j!=B_size; j++) R_pool[i].v[j] = NULL;
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+ }
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+}
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+
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+#define B_pool_free() free(B_pool)
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+
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+/**
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+ * We have two possible choices here, for generating a valid random rumber
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+ * satisfying Dixon's theorem requirements.
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+ *
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+ * Alg. 1 - 1. Start by generating a random r such that r > √N,
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+ * 2. Calculate s ≡ r² (mod N)
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+ * 3. Factorize s using B and see if that's B-smooth
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+ * This algorithm shall have complexity O(k + N² + |B|lg N)
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+ *
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+ * Alg. 2 - 1. Generate the random exponents for s, {e₀, e₁, …, eₘ} where m = |B|
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+ * 2. From the generated exponents, calculate s = p₀^e₀·p₁^e₁·…·pₘ^eₘ
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+ * knowing that s < N
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+ * 3. Find an r = √(s + tN) , t ∈ {1..N-1}
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+ * This algorithm shall have complexity O(k|B| + (N-1)lg N)
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+ */
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+static void extend_R_pool(BIGNUM* N)
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+{
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+ const size_t old_R_size = R_size;
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+ size_t i, j;
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+ int e_bits;
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+ BN_CTX *ctx = BN_CTX_new();
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+ BIGNUM
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+ *e,
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+ *tmp = BN_new(),
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+ *rem = BN_new(),
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+ *t = BN_new();
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+ dixon_number_t *d;
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+
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+ R_size += RPOOL_EXTEND_STEP;
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+ /* size_t overflow */
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+ assert(R_size > old_R_size);
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+ R_pool = realloc(R_pool, sizeof(dixon_number_t));
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+ /*
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+ * XXX. There is much more to think about this.
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+ * We are trying to generate some random exponents e₀…eₖ such that s < N .
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+ * Hence, log(N) = ae₀ + be₁ + … + leₖ
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+ */
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+ e_bits = BN_num_bits(N) / 5;
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+
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+ for (i=old_R_size; i!= R_size; i++) {
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+ d = &R_pool[i];
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+ d->s = BN_new();
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+ d->r = BN_new();
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+
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+ /* generate exponents and calculate s */
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+ for (j=0; j != B_size && BN_cmp(N, d->s) == 1; j++) {
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+ e = d->v[j] = BN_new();
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+ /* XXX. better check for error here. */
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+ BN_pseudo_rand(e, e_bits, -1, 0);
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+ BN_exp(tmp, B_pool[j], e, ctx);
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+ BN_mul(d->s, tmp, d->s, ctx);
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+ }
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+
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+ /* Find an r = √(s + tN) , t ∈ {1..N-1} */
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+ BN_sqr(tmp, N, ctx);
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+ BN_one(t);
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+ for (BN_add(t, t, N); BN_cmp(tmp, t) == 1; BN_add(t, t, N))
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+ if (BN_sqrtmod(d->r, rem, t, ctx)) break;
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+ }
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+
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+
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+ BN_CTX_free(ctx);
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+ BN_free(rem);
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+ BN_free(tmp);
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+ BN_free(t);
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+
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+}
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+
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+
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+#define R_pool_free() free(R_pool)
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+
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+int dixon_question_setup(void)
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+{
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+ extern BIGNUM* zero;
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+ zero = BN_new();
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+ BN_zero(zero);
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+
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+ extend_B_pool(BPOOL_STARTING_BITS);
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+ return 1;
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+}
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+
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+int dixon_question_teardown(void) {
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+ BN_free(zero);
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+
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+ B_pool_free();
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+ R_pool_free();
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+ return 0;
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+}
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+
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+
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+RSA* dixon_question_ask_rsa(const RSA *rsa) {
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+ /* key data */
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+ RSA *ret = NULL;
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+ BIGNUM
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+ *n,
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+ *p, *q;
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+ /* x, y */
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+ BIGNUM
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+ *x, *x2,
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+ *y, *y2;
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+ BN_CTX *ctx;
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+ /* U ⊆ R */
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+ ssize_t *U_bucket;
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+ /* internal data */
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+ int epoch;
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+ BIGNUM *tmp;
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+ char *even_powers;
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+ size_t i, j, k;
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+
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+ n = rsa->n;
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+ U_bucket = malloc(sizeof(ssize_t) * U_SIZE);
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+ even_powers = malloc(sizeof(char) * B_size);
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+ ctx = BN_CTX_new();
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+ x = BN_new();
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+ y = BN_new();
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+ x2 = BN_new();
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+ y2 = BN_new();
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+ tmp = BN_new();
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+
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+ /* mainloop: iterate until a key is found, or convergence. */
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+ for (epoch=0; epoch < EPOCHS; epoch++) {
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+ /* depending on the epoch, populate R_pool and B_pool */
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+ if (epoch % REPOP_EPOCHS) extend_R_pool(n);
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+
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+ /* reset variables */
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+ for (i=0; i!=U_SIZE; i++) U_bucket[i] = -1;
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+ bzero(even_powers, B_size * sizeof(char));
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+ j = 0;
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+
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+ /* choose a subset of R such that the product of primes can be squared */
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+ do {
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+ for (i=0; i!=B_size && j < U_SIZE; i++) {
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+ /* choose whether to take or not R_pool[i] */
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+ if (qa_rand() % 2) continue;
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+
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+ /* add the number */
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+ U_bucket[j++] = i;
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+ for (k=0; k!=B_size; k++)
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+ even_powers[k] ^= BN_is_odd(R_pool[i].v[j]);
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+ }
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+ } while (!is_vzero(even_powers, B_size * sizeof(char)));
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+
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+ /* let x = Πᵢ rᵢ , y² = Πᵢ sᵢ */
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+ BN_one(x);
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+ BN_one(y2);
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+ for (i=0; i != U_SIZE; i++) {
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+ if (U_bucket[i] == -1) continue;
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+
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+ j = U_bucket[i];
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+ BN_mul(x, x, R_pool[j].r, ctx);
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+ BN_mul(y2, y2, R_pool[j].s, ctx);
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+ }
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+ /* retrieve x² from x */
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+ BN_sqr(x2, x, ctx);
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+ /* retrieve y from y² */
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+ /* test: shall *always* be a perfect square */
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+ if (!BN_sqrtmod(y, tmp, y2, ctx)) continue;
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+ /* test: assert that x ≡ y (mod N) */
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+ if (!BN_cmp(x, y)) continue;
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+
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+ /* p, q found :) */
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+ ret = RSA_new();
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+ ret->e = rsa->e;
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+ ret->n = rsa->n;
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+ ret->p = p = BN_new();
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+ ret->q = q = BN_new();
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+
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+ BN_uadd(tmp, x, y);
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+ BN_gcd(p, tmp, n, ctx);
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+ assert(!BN_is_one(p) && BN_cmp(p, n));
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+ BN_usub(tmp, x, y);
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+ BN_gcd(q, tmp, n, ctx);
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+ assert(!BN_is_one(q) && BN_cmp(q, n));
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+ }
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+
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+ BN_free(x);
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+ BN_free(x2);
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+ BN_free(y);
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+ BN_free(y2);
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+ free(U_bucket);
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+ free(even_powers);
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+
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+ return ret;
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+}
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+
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+qa_question_t DixonQuestion = {
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+ .name = "dixon",
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+ .pretty_name = "Dixon's Factorization",
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+ .setup = dixon_question_setup,
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+ .teardown = dixon_question_teardown,
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+ .test = NULL,
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+ .ask_rsa = dixon_question_ask_rsa,
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+ .ask_crt = NULL
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+};
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