Programming Challenge from Kattis: Apparatus
Disclaimer: I won't try and provide a different algorithm to improve the time complexity of the approach. Instead I’ll rather focus on introducing python constructs that will cleanup the code and may help gain speed a bit.
Reading data from standard input
I had trouble running your script from both IDLE and windows command prompt. Neither of them were good at handling sys.stdin as a file. But we can improve things by using the builtin input (raw_input in Python 2) since we just want to read lines one by one.
Converting back and forth between integers and their binary representation
One of the way to improve your computation of oneBite would be to use the divmod builtin. But it is even more easier when using bin and counting the number of '1' in the returned string.
Same when computing digit: the int builtin accept a second argument which is the base in which the number is represented.
Use functions
You could make the code easier to read and understand by using functions: one to parse two lines (one photo) and one to iterate over the whole set of photos and interpret the results.
Functions will also allow you to import your file into an interactive session and test things more easily. If you want to have code at the top-level to be run when invoking your script from the command line, it is recommended to put it under an if __name__ == &quot;__main__&quot; clause.
collections
The last part of your code count the frequency for a certain possibility in a bizarre way. Time for you to learn about collections.Counter.
PEP8

Use snake_case instead of camelCase for your variable names.
Use spaces around most of your operators (frequency &gt; 1, possibilities%2 == 1, possibilities &gt; 0…)
Remove parenthesis around your tests

EAFP
While
    if not (switch in wiring.keys()):
        wiring[switch] = digit
    else:
        wiring[switch] &amp;= digit

is correct, I would first write it
    if switch in wiring:
        wiring[switch] &amp;= digit
    else:
        wiring[switch] = digit

for readability (and less computation) but then change it to
    try:
        wiring[switch] &amp;= digit
    except KeyError:
        wiring[switch] = digit

It is not necessarily faster in this case (it is when failures are much less than success) but I find it clearer.
Putting it all together
In Python 2 use raw_input, xrange and itervalues instead of input, range and values.
from math import factorial
from collections import Counter


def parse_photo(wiring):
    switches = input()
    lights = input()

    if switches.count('1') != lights.count(1):
        return False # invalid data

    # Convert binary value to integer
    lights_value = int(lights, 2)

    for switch, switch_value in enumerate(switches):
        if switch_value == &quot;1&quot;:
            try:
                wiring[switch] &amp;= lights_value
            except KeyError:
                wiring[switch] = lights_value

    return True


def wiring_possibilities():
    wiring = {}
    num_switches, num_photos = map(int, input().split())

    for _ in range(num_photos):
        if not parse_photo(wiring):
            return 0

    frequencies = Counter(wiring.values())
    if 0 in frequencies:
        # Same as if 0 in wiring.values()
        return 0

    for possibility, frequency in frequencies.items():
        switched_on_bits = bin(possibility).count('1')
        if switched_on_bits &lt; frequency:
            return 0

    return factorial(num_switches - num_photos) % 1000003


if __name__ == &quot;__main__&quot;:
    print(wiring_possibilities())