Understanding Bitwise Operators

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Recall that deep down everything on the machine is just bits. There are a whole group of "bitwise" operators that operate on those bits. If re anding 2 binary values like to re anding 2 binary values the bits inside a number, you can loop over the bits and use AND to extract each bit:. Try this in NetRun now! Remember that every hex digit represents four bits.

So if you shift a hex constant by four bits, it shifts by one entire hex digit:. If you shift a hex constant by a non-multiple of four bits, you end up interleaving the hex digits of the constant, which is confusing:. Bitwise operators make perfect sense working with hex digits, because they operate on the underlying bits of those digits: Makes values bigger, by shifting the value's bits into higher places, tacking on zeros in the vacated lower places.

Interesting facts about left re anding 2 binary values. Operator precedence is screwy for bitwise operators, so you really want to use excess parenthesis! Makes values smaller, by shifting them into lower-valued places.

Note the bits in the lowest places just "fall off the end" and vanish. Interesting facts about right shift:. I always remember it like this:. This is useful to "mask out" bits you don't want, by ANDing them with zero. Bitwise AND is a really really useful tool for extracting bits from a number--you often create a "mask" value with 1's marking the bits you want, and AND by the mask.

For example, this code figures out if bit 2 of an integer is set: Be sure to use extra parenthesis! In assembly, it's the "and" instruction. Bitwise OR is useful for sticking together bit fields you've prepared separately. Note how the low bit is 0, because both input bits are 1. The second property, that XOR by 1 inverts the value, is useful for flipping a set of bits. Output bits are 1 if the corresponding input bit is zero.

The number of leading ones depends on the size of the machine's "int". I don't use bitwise NOT very often, but it's handy for making an integer whose bits are all 1: Internally, these operators map multi-bit values to a single bit by treating zero as a zero bit, and nonzero values as a one bit.

You've got to search all the HTML pages on the re anding 2 binary values for any possible word. One way to do this is for each possible word, store a giant table of every HTML document on the net maybe 10 billion documents containing one bit per document: Re anding 2 binary values table is 10 billion bits, about 1GB uncompressed, or only a few dozen megabytes compressed. Given two search words, you can find all the pages that contain both words by ANDing both tables.

The output of the bitwise AND, where both bits are set to 1, is a new table listing the HTML pages that contain both search terms; now sort by pagerank, and you're done! Note that storing the big table by re anding 2 binary values saves a lot of space, and doing a bitwise AND instead of a regular logical AND saves a lot of time over 10x speedup in my testing!

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To truly understand how to derive IP masks and apply them to addresses, you must understand binary numbers and how to convert them to decimal. Let's start with something that we're all pretty comfortable with, namely decimal base 10 numbers. Back when we were kids, we were taught that each digit in a decimal number stood for a different power of The number , for example, is interpreted as follows:. Now this is pretty simplistic, I admit, but understanding this is the basis for understanding any numeric base.

In particular, it will help us understand binary base 2. We interpret binary numbers in exactly the same way as decimal numbers, except that each column of a binary number represents a different power of 2 rather than We can easily convert a binary number to a more understandable decimal value. Let's first review the powers of 2 we're only going to go as far as we need to for an 8-bit byte because IP addresses have 8-bit bytes.

Now, we can apply what we know about binary numbers to IP addresses and subnet masks. IP addresses are 32 bits, or four 8-bit bytes, in length. While the computer stores the IP address in binary, we typically use dotted decimal notation to write out addresses because we find it easier to read.

Dotted decimal notation lets us examine an IP address one byte at a time. Subnet masks, like the IP address itself, are 32 bits in length. With classful addressing, then, the subnet mask will have 8, 16, or 24 one bits for Class A, B, and C addresses, respectively.

In the parlance of subnet masking, these masks would be said to be 8, 16, or 24 bits in length but that is a misnomer; it really only refers to the number if one bits since masks really are always 32 bits long.

Variable length subnet masking VLSM is essential to support classless addressing. VLSM allows us to build masks that are of pretty much any length and are not restricted to the byte boundaries of classful addressing. Let's start with a simple example.

Suppose we have the Class C address In binary, the address with spaces inserted for readability is:. But how does this really work? So let's carry out that operation for the Class C address and mask above:.

Let's now try a broader example. Since masks are created by writing some number of ones followed by zeroes, an all-ones byte will have the value and an all-zeroes byte will have a value of 0, as shown above.

But a VLSM may not have a mask that falls on a byte boundary so one of the bytes may have a value other than 0 or In fact, an 8-bit byte has only eight possible subnet values as we increase the number of one bits from the left:. Variable-bit subnet masks give us a great deal of flexibility in carving out multiple subnets within the Class C space.

Suppose that we want to create eight subnetworks in the We just add 3 bits to the length of the bit subnet mask. Recall that the first 24 bits are all ones, so the first three bytes will be The fourth byte will have 3 ones in it and, therefore, a value of from the table above.

Because we used 3 bits of the final byte as a mask sometimes called a subnet ID , the host IDs are limited to 5 bits. But we still have one more significant problem to solve, namely, to identify the subnet numbers. The eight possible values of the 3-bit subnet mask are:. Therefore, the eight possible values of the final address byte are again, the spaces are only for readability:. For obvious reasons, you should always indicate the subnet mask along with the address itself, as I've done above, to avoid ambiguity; the address You can reach him at gck garykessler.