Adding draft for Pollard's (p-1) factorization algorithm.

src/questions/allquestions.c

@@ -20,5 +20,5 @@ void register_all_questions(void)
   LIST_INIT(&questions);
 
   REGISTER_QUESTION(ExampleQuestion);
-
+  REGISTER_QUESTION(PollardQuestion);
 }

src/questions/pollard.c

@@ -0,0 +1,84 @@
+/**
+ * \file pollard.c
+ *
+ * \brief Pollard's (p-1) factorization algorithm.
+ *
+ * This file contains an implementations of Pollard's (p-1) algorithm, used to
+ * attack the public modulus of RSA.
+ *
+ * Consider the public modulus N = pq. Now,
+ *  (p-1) = q₀ᵉ⁰q₁ᵉ¹… qₖᵉᵏ .  q₀ᵉ⁰ < q₁ᵉ¹ < … < qₖᵉᵏ ≤ B
+ * implies that  (p-1) | B! , since all factors of (p-1) belongs to {1, …, B}.
+ * Consider a ≡ 2^(B!) (mod N)
+ *   a = 2^(B!) + kN  = 2^(B!) + kqp → a ≡ 2^(B!) (mod p)
+ * Since
+ * <pre>
+ *
+ *   ⎧ 2ᵖ⁻¹ ≡ 1 (mod p)                              ⎧ p | (a-1)
+ *   ⎨                  →  a ≡ 2^(B!) ≡ 1 (mod p) →  ⎨           → p | gcd(a-1, N)
+ *   ⎩ p-1 | B!                                      ⎩ p | N
+ *
+ * </pre>
+ * And gcd(a-1, N) is a non-trivial factor of N, unless a = 1.
+ */
+
+#include <openssl/x509.h>
+
+#include "questions.h"
+
+int pollard1_question_setup(void)
+{
+  return 0;
+}
+
+int pollard1_question_teardown(void)
+{
+  return 0;
+}
+
+
+int pollard1_question_test(X509 *cert)
+{
+  return 0;
+}
+
+
+/**
+ * \brief Pollard (p-1) factorization.
+ *
+ * Trivially the algorithm computes a = 2^(B!) (mod N), and then verifies that
+ * gcd(a-1, N) is a nontrivial factor of N.
+ *
+ * According to Wikipedia™,
+ * « By Dixon's theorem, the probability that the largest factor of such a
+ * number is less than (p − 1)^ε is roughly ε^(−ε); so there is a probability of
+ * about 3^(−3) = 1/27 that a B value of n^(1/6) will yield a factorisation.»
+ *
+ */
+int pollard1_question_ask(X509 *cert)
+{
+  RSA *rsa;
+  BIGNUM *a, *B;
+  BIGNUM *n;
+
+  rsa = X509_get_pubkey(cert)->pkey.rsa;
+  n = rsa->n;
+  a = BN_new();
+  B = BN_new();
+
+  BN_dec2bn(&a, "2");
+
+  BN_free(a);
+  BN_free(B);
+
+  return 0;
+}
+
+
+struct qa_question PollardQuestion = {
+  .name = "Pollard's (p-1) factorization",
+  .setup = pollard1_question_setup,
+  .teardown = pollard1_question_teardown,
+  .test = pollard1_question_test,
+  .ask = pollard1_question_ask,
+};