BigInt

BigInt is Botan’s implementation of a multiple-precision integer. Thanks to C++’s operator overloading features, using BigInt is often quite similar to using a native integer type. The number of functions related to BigInt is quite large. You can find most of them in botan/bigint.h and botan/numthry.h.

Encoding Functions

These transform the normal representation of a BigInt into some other form, such as a decimal string:

secure_vector<uint8_t> BigInt::encode(const BigInt &n, Encoding enc = Binary)

This function encodes the BigInt n into a memory vector. Encoding is an enum that has values Binary, Decimal, and Hexadecimal.

BigInt BigInt::decode(const std::vector<uint8_t> &vec, Encoding enc)

Decode the integer from vec using the encoding specified.

These functions are static member functions, so they would be called like this:

BigInt n1 = ...; // some number
secure_vector<uint8_t> n1_encoded = BigInt::encode(n1);
BigInt n2 = BigInt::decode(n1_encoded);
assert(n1 == n2);

There are also C++-style I/O operators defined for use with BigInt. The input operator understands negative numbers and hexadecimal numbers (marked with a leading “0x”). The ‘-‘ must come before the “0x” marker. The output operator will never adorn the output; for example, when printing a hexadecimal number, there will not be a leading “0x” (though a leading ‘-‘ will be printed if the number is negative). If you want such things, you’ll have to do them yourself.

BigInt has constructors that can create a BigInt from an unsigned integer or a string. You can also decode an array (a byte pointer plus a length) into a BigInt using a constructor.

Number Theory

Number theoretic functions available include:

BigInt gcd(BigInt x, BigInt y)

Returns the greatest common divisor of x and y

BigInt lcm(BigInt x, BigInt y)

Returns an integer z which is the smallest integer such that z % x == 0 and z % y == 0

BigInt inverse_mod(BigInt x, BigInt m)

Returns the modular inverse of x modulo m, that is, an integer y such that (x*y) % m == 1. If no such y exists, returns zero.

BigInt power_mod(BigInt b, BigInt x, BigInt m)

Returns b to the xth power modulo m. If you are doing many exponentiations with a single fixed modulus, it is faster to use a Power_Mod implementation.

BigInt ressol(BigInt x, BigInt p)

Returns the square root modulo a prime, that is, returns a number y such that (y*y) % p == x. Returns -1 if no such integer exists.

bool is_prime(BigInt n, RandomNumberGenerator &rng, size_t prob = 56, double is_random = false)

Test n for primality using a probablistic algorithm (Miller-Rabin). With this algorithm, there is some non-zero probability that true will be returned even if n is actually composite. Modifying prob allows you to decrease the chance of such a false positive, at the cost of increased runtime. Sufficient tests will be run such that the chance n is composite is no more than 1 in 2prob. Set is_random to true if (and only if) n was randomly chosen (ie, there is no danger it was chosen maliciously) as far fewer tests are needed in that case.

bool quick_check_prime(BigInt n, RandomNumberGenerator &rng)
bool check_prime(BigInt n, RandomNumberGenerator &rng)
bool verify_prime(BigInt n, RandomNumberGenerator &rng)

Three variations on is_prime, with probabilities set to 32, 56, and 80 respectively.

BigInt random_prime(RandomNumberGenerator &rng, size_t bits, BigInt coprime = 1, size_t equiv = 1, size_t equiv_mod = 2)

Return a random prime number of bits bits long that is relatively prime to coprime, and equivalent to equiv modulo equiv_mod.