+ const BIGNUM *m, BN_CTX *ctx);
+int BN_div_recp(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m,
+ BN_RECP_CTX *recp, BN_CTX *ctx);
+
+/* Functions for arithmetic over binary polynomials represented by BIGNUMs.
+ *
+ * The BIGNUM::neg property of BIGNUMs representing binary polynomials is
+ * ignored.
+ *
+ * Note that input arguments are not const so that their bit arrays can
+ * be expanded to the appropriate size if needed.
+ */
+
+int BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b); /*r = a + b*/
+#define BN_GF2m_sub(r, a, b) BN_GF2m_add(r, a, b)
+int BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p); /*r=a mod p*/
+int BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const BIGNUM *p, BN_CTX *ctx); /* r = (a * b) mod p */
+int BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
+ BN_CTX *ctx); /* r = (a * a) mod p */
+int BN_GF2m_mod_inv(BIGNUM *r, const BIGNUM *b, const BIGNUM *p,
+ BN_CTX *ctx); /* r = (1 / b) mod p */
+int BN_GF2m_mod_div(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const BIGNUM *p, BN_CTX *ctx); /* r = (a / b) mod p */
+int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const BIGNUM *p, BN_CTX *ctx); /* r = (a ^ b) mod p */
+int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
+ BN_CTX *ctx); /* r = sqrt(a) mod p */
+int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
+ BN_CTX *ctx); /* r^2 + r = a mod p */
+#define BN_GF2m_cmp(a, b) BN_ucmp((a), (b))
+/* Some functions allow for representation of the irreducible polynomials
+ * as an unsigned int[], say p. The irreducible f(t) is then of the form:
+ * t^p[0] + t^p[1] + ... + t^p[k]
+ * where m = p[0] > p[1] > ... > p[k] = 0.
+ */
+int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[]);
+ /* r = a mod p */
+int BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const unsigned int p[], BN_CTX *ctx); /* r = (a * b) mod p */
+int BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[],
+ BN_CTX *ctx); /* r = (a * a) mod p */
+int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *b, const unsigned int p[],
+ BN_CTX *ctx); /* r = (1 / b) mod p */
+int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const unsigned int p[], BN_CTX *ctx); /* r = (a / b) mod p */
+int BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
+ const unsigned int p[], BN_CTX *ctx); /* r = (a ^ b) mod p */
+int BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a,
+ const unsigned int p[], BN_CTX *ctx); /* r = sqrt(a) mod p */
+int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a,
+ const unsigned int p[], BN_CTX *ctx); /* r^2 + r = a mod p */
+int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max);
+int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a);
+
+/* faster mod functions for the 'NIST primes'
+ * 0 <= a < p^2 */
+int BN_nist_mod_192(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);
+int BN_nist_mod_224(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);
+int BN_nist_mod_256(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);
+int BN_nist_mod_384(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);
+int BN_nist_mod_521(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx);
+
+const BIGNUM *BN_get0_nist_prime_192(void);
+const BIGNUM *BN_get0_nist_prime_224(void);
+const BIGNUM *BN_get0_nist_prime_256(void);
+const BIGNUM *BN_get0_nist_prime_384(void);
+const BIGNUM *BN_get0_nist_prime_521(void);
+
+/* library internal functions */
+
+#define bn_expand(a,bits) ((((((bits+BN_BITS2-1))/BN_BITS2)) <= (a)->dmax)?\
+ (a):bn_expand2((a),(bits+BN_BITS2-1)/BN_BITS2))
+#define bn_wexpand(a,words) (((words) <= (a)->dmax)?(a):bn_expand2((a),(words)))
+BIGNUM *bn_expand2(BIGNUM *a, int words);
+#ifndef OPENSSL_NO_DEPRECATED
+BIGNUM *bn_dup_expand(const BIGNUM *a, int words); /* unused */
+#endif
+
+/* Bignum consistency macros
+ * There is one "API" macro, bn_fix_top(), for stripping leading zeroes from
+ * bignum data after direct manipulations on the data. There is also an
+ * "internal" macro, bn_check_top(), for verifying that there are no leading
+ * zeroes. Unfortunately, some auditing is required due to the fact that
+ * bn_fix_top() has become an overabused duct-tape because bignum data is
+ * occasionally passed around in an inconsistent state. So the following
+ * changes have been made to sort this out;
+ * - bn_fix_top()s implementation has been moved to bn_correct_top()
+ * - if BN_DEBUG isn't defined, bn_fix_top() maps to bn_correct_top(), and
+ * bn_check_top() is as before.
+ * - if BN_DEBUG *is* defined;
+ * - bn_check_top() tries to pollute unused words even if the bignum 'top' is
+ * consistent. (ed: only if BN_DEBUG_RAND is defined)
+ * - bn_fix_top() maps to bn_check_top() rather than "fixing" anything.
+ * The idea is to have debug builds flag up inconsistent bignums when they
+ * occur. If that occurs in a bn_fix_top(), we examine the code in question; if
+ * the use of bn_fix_top() was appropriate (ie. it follows directly after code
+ * that manipulates the bignum) it is converted to bn_correct_top(), and if it
+ * was not appropriate, we convert it permanently to bn_check_top() and track
+ * down the cause of the bug. Eventually, no internal code should be using the
+ * bn_fix_top() macro. External applications and libraries should try this with
+ * their own code too, both in terms of building against the openssl headers
+ * with BN_DEBUG defined *and* linking with a version of OpenSSL built with it
+ * defined. This not only improves external code, it provides more test
+ * coverage for openssl's own code.
+ */
+
+#ifdef BN_DEBUG