# The biguint library interface

biguint is set of simple primitives performing arithmetical operations on (unsigned) integers of arbitrary length. It is nowhere near as powerful or efficient as specialized, assembly language-optimized libraries such as GMP, but it has the advantages of smallness and simplicity.

## Compiling

• Use #include <skalibs/biguint.h>

## Programming

You should refer to the skalibs/biguint.h header for the exact function prototypes.

### Definitions

• A biguint x is a pointer to an array u of uint32_t, together with an unsigned integer n called its length.
x = (2^32)^0 * u + (2^32)^1 * u + ... + (2^32)^(n-1) * u[n-1].
• Every u[i] is called a limb.
• The greatest integer i lesser than n for which u[i] is non-zero is called the order of x. The order of zero is 0.

### Basic operations

#### Creating a biguint

Just declare uint32_t x[n] ; - n being the length of the biguint. You could also allocate x in the heap, possibly using a uint32_t genalloc. In the following, a biguint is always referred to as a uint32_t * with its unsigned int length ; it must always be pre-allocated.

If an operation fails because a biguint's length n is too small to accommodate the result, the function will write the first (i.e. least significant) n limbs of the result, truncating it, then return 0 with errno set to EOVERFLOW.

#### Setting it to zero

```uint32_t *x ;
unsigned int n ;

bu_zero(x, n) ;
```

bu_zero() sets the first n limbs of x to zero.

#### Copying a biguint

```uint32_t const *x ;
unsigned int xn ;
uint32_t *y ;
unsigned int yn ;

bu_copy(y, yn, x, xn) ;
```

bu_copy() copies x to y, setting higher limbs of y to zero if needed. It then returns 1. If y is too small to contain x, the function returns 0 EOVERFLOW.

#### Calculating the order

```uint32_t const *x ;
unsigned int n ;
unsigned int r ;

r = bu_len(x, n) ;
```

bu_len() outputs the order of x of length n. 0 <= r <= n.

#### Comparing two biguints

```uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;
int r ;

r = bu_cmp(a, an, b, bn) ;
```

bu_cmp() returns -1 if a < b, 1 if a > b, and 0 if a = b.

### I/O operations

#### Writing a biguint as an array of bytes

```char *s ;
uint32_t const *x ;
unsigned int n ;

bu_pack(s, x, n) ;
bu_pack_big(s, x, n) ;
```

bu_pack() writes 4*n bytes to s. The bytes are a little-endian representation of x.
bu_pack_big() is the same, with a big-endian representation.

#### Reading a biguint from an array of bytes

```char const *s ;
uint32_t *x ;
unsigned int n ;

bu_unpack(s, x, n) ;
bu_unpack_big(s, x, n) ;
```

bu_unpack() reads 4*n little-endian bytes from s and writes them into the corresponding biguint x.
bu_unpack_big() is the same, but the bytes are interpreted as big-endian.

#### Formatting a biguint for readable output

```char *s ;
uint32_t const *x ;
unsigned int n ;

bu_fmt(s, x, n) ;
```

bu_fmt() writes x in s as a standard big-endian hexadecimal number. x is considered of length n, so 8*n bytes will be written to s, even if it x starts with zeros. bu_fmt returns the number of bytes written.

```char const *s ;
uint32_t *x ;
unsigned int xn ;
unsigned int z ;
unsigned int len ;

len = bu_scanlen(s, &z) ;
bu_scan(s, len, x, xn, z) ;
```

bu_scanlen() scans s for a biguint written as a hexadecimal number and returns the number of bytes read. The reading stops at the first byte encountered that is not in the 0-9, A-F or a-f range. The z integer then contains the number of bytes excluding leading zeros.

If x has not been allocated yet, you can use xn = bitarray_div8(z) (if you have included the bitarray.h header) and allocate x with length xn.

bu_scan() then reads len bytes from s, assuming there are z significant bytes (i.e. not leading zeros); it writes the resulting biguint into x of length xn. It returns 1, except if xn is too small, in which case it returns 0 EOVERFLOW.

### Arithmetic operations

```uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;
uint32_t *c ;
unsigned int cn ;
unsigned char carrybefore ;
unsigned char carryafter ;

bu_add(c, cn, a, an, b, bn) ;
bu_sub(c, cn, a, an, b, bn) ;
```

bu_add() adds a and b, and puts the result into c. It returns 1 unless it has to truncate it.

bu_sub() substracts b from a, and puts the result into c. If the result should be negative, then it is written as (2^32)^cn - c and the function returns 0 EOVERFLOW.

#### Multiplication

```uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;
uint32_t *c ;
unsigned int cn ;

bu_mul(c, cn, a, an, b, bn) ;
```

bu_mul() computes c=a*b. Make sure that cnbu_len(a, an) + bu_len(b, bn). If it is not the case, the result will be truncated and bu_mul will return 0 EOVERFLOW.

#### Division

```uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;
uint32_t *q ;
unsigned int qn ;
uint32_t *r ;
unsigned int rn ;

bu_div(a, an, b, bn, q, qn, r, rn) ;
bu_mod(r, rn, b, bn) ;
```

bu_div() computes q, the quotient, and r, the remainder, of a divided by b. If b is zero, it returns 0 EDOM. If qn or rn is to small to store the quotient or the remainder, it returns 0 EOVERFLOW. bu_mod() computes only the remainder, and stores it in-place.

#### GCD

```uint32_t *r ;
unsigned int rn ;
uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;

bu_gcd(r, rn, a, an, b, bn) ;
```

bu_gcd() computes the greatest common divisor between a and b, and stores it into r. It returns 1 if all went well.

Note that this function iterates on divisions, so it might use a non totally negligible amount of CPU time.

#### Left-shifts and right-shifts

```uint32_t *x ;
unsigned int xn ;
unsigned char carryafter ;
unsigned char carrybefore ;

carryafter = bu_slbc(x, xn, carrybefore) ;
carryafter = bu_srbc(x, xn, carrybefore) ;
```

bu_slbc() computes x <<= 1. The least significant bit of x is then set to carrybefore. bu_slbc() returns the previous value of x's most significant bit.
bu_srbc() computes x >>= 1. The most significant bit of x is then set to carrybefore. bu_slbc() returns the previous value of x's least significant bit.
bu_slb(x, n) and bu_srb(x, n) are macros for respectively bu_slbc(x, n, 0) and bu_srbc(x, n, 0).

#### Modular operations

```uint32_t const *a ;
unsigned int an ;
uint32_t const *b ;
unsigned int bn ;
uint32_t *c ;
unsigned int cn ;
uint32_t const *m ;
unsigned int mn ;

bu_addmod(c, cn, a, an, b, bn, m, mn) ;
bu_submod(c, cn, a, an, b, bn, m, mn) ;
bu_mulmod(c, cn, a, an, b, bn, m, mn) ;
bu_divmod(c, cn, a, an, b, bn, m, mn) ;
bu_invmod(c, cn, m, mn) ;
```

bu_addmod() computes c = (a+b) mod m.
bu_submod() computes c = (a-b) mod m.
bu_mulmod() computes c = (a*b) mod m.
a and b must already be numbers modulo m.
The functions return 1 if all went well.

bu_divmod() computes a divided by b modulo m and stores it into c.
bu_invmod() computes the inverse of c modulo m and stores it into c.
The divisor and m must be relatively prime, else those functions return 0 EDOM.