## Converting from binary to integer

The following is an updated and edited version of my posts to this sage-support thread.

**Problem**

You have a bitstring as output by

sage.crypto.stream.blum_blum_shub

and you want to convert that bitstring to an integer. Or in general, you want to convert a bit vector to its integer representation.

**Solution**

Here are two ways, assuming that you want the bits in little-endian order, i.e. you read the bits from right to left in increasing order of powers of 2.

sage: version() 'Sage Version 4.5.3, Release Date: 2010-09-04' sage: from sage.crypto.stream import blum_blum_shub sage: b = blum_blum_shub(length=6, lbound=10**4, ubound=10**5); b 100110 sage: type(b) <class 'sage.monoids.string_monoid_element.StringMonoidElement'> sage: # read in little-endian order sage: # conversion using Python's built-in int() sage: int(str(b), base=2) 38 sage: # conversion using Sage's built-in Integer() sage: Integer(str(b), base=2) 38

Now assume you read the bitstring as output by blum_blum_shub in big-endian order, i.e. from left to right in increasing order of powers of 2. You simply convert the bitstring to a string, reverse that string, and apply any of the above two methods.

sage: # reversing a string sage: str(b) '100110' sage: str(b)[::-1] '011001' sage: # read in big-endian order sage: int(str(b)[::-1], base=2) 25 sage: Integer(str(b)[::-1], base=2) 25

Or you can do as follows:

sage: b = "100110" sage: sum(Integer(i) * (2^Integer(e)) for e, i in enumerate(b)) 25 sage: sum(Integer(i) * (2^Integer(e)) for e, i in enumerate(b[::-1])) 38

Another way is to use Horner’s method. Here’s a Sage function that computes the integer representation of a bit vector read using big-endian order. A usage example is also shown.

sage: def horner(A, x0): ... # Evaluate the polynomial P(x) at x = x_0. ... # ... # INPUT ... # ... # - A -- list of coefficients of P where A[i] is the coefficient of ... # x_i. ... # - x0 -- the value x_0 at which to evaluate P(x). ... # ... # OUTPUT ... # ... # An evaluation of P(x) using Horner's method. ... i = len(A) - 1 ... b = A[i] ... i -= 1 ... while i >= 0: ... b = b*x0 + A[i] ... i -= 1 ... return b sage: A = [1, 0, 0, 1, 1, 0] sage: horner(A, 2) 25

As an exercise, modify the function horner to output the integer representation of a bit vector that is read using little-endian order.

If b is a binary string, such as

‘011001’

then the command

ZZ(‘0b’+b)

returns the value 25. If you want the other order, then

ZZ(‘0b’+b[::-1])

returns 38. These are both very concise.