The Understated Power of 0xf in Programming Realms

Oct 28, 2023

In the vast world of programming, there exists 0xf, a small yet powerful numerical representation that is very useful for many of our widely used algorithms. Let’s unravel why this seemingly modest number is very important in various programming scenarios:

What is it?

Specifically, 0xf is a hexadecimal representation (base 16) of the decimal number 15 and 0x is just a prefix that denotes that the following digits represent the hexadecimal number. In binary form, 0xf or specifically f is 1111 and you may already notice why it could be useful.

Where exactly it is useful?

In programming, it is commonly used for bitwise operations and bit masking to extract specific bits and perform logical operations on the lowest 4 bits of a byte. A bit confused? Let me clarify

AND Operation (&):

The operation bitwise AND is commonly used in programming. When using the & operator with a number and 0xf (15), it keeps only the lower 4 bits and sets the remaining bits to zero. For example:
```
10101101 & 00001111 = 00001101
```
As you can see, no matter which number is thrown at to this AND operation, it will only give the lower 4 bits as a result. This effectively restricts the result to a range of 0 to 15 (in decimal) or 0000 to 1111 (in binary), allowing for clear representation within a 4-bit space.

Modulus Operation (%):

In modular arithmetic or operations where we want to ensure a value falls within a certain range, performing value % 0xf ensures that the result is within the range of 0 to 15.

let someValue = 28; // Any numerical value

// Applying modulus operation with 0xf (15 in decimal)
let result = someValue % 0xf; // Restrict the value to the range from 0 to 15

console.log('Result after modulus operation:', result);

// If someValue were 28, the result would be 13, because 28 divided by 15 leaves a remainder of 13.
// If someValue were 35, the result would be 5, as 35 divided by 15 leaves a remainder of 5.

But why would we want to disregard the rest of the bits? Well, in the realm of binary notations, digits can become lengthy. 0xf however, with its concise 4 bits representation, elegantly condenses code, maintaining clarity.

In low-level programming or memory addressing, modulus operations using 0xf are used to map addresses within a specific memory block or range.

In circular systems or cyclical behaviors, such as clock representation, restricting a value to a 0-15 range (for 4 bits) can simulate circular behavior. For instance, in time representation, after 15, it resets back to 0.

A real-world example

Let me give you a real-world example of an algorithm that powers a feature that we all are using unknowingly, an authenticator (e.g. Google Authenticator)

Google Authenticator or any other 2FA apps use something called the Time-based One-Time Password (TOTP) algorithm. If you’re interested to learn more, here is an RFC about it. This algorithm basically generates a One-Time-Password and underneath, 0xf plays a crucial role.

So, this TOTP algorithm uses the HMAC-SHA-1 algorithm in order to generate a hash input and this hash value is 160 bits long. However, as you all know, we only use 6-digit codes for OTP due to usability reasons. So how do we do it? We need to truncate the 160-bit hash value.

The result of this truncate function has to be a 31-bit number. Why 31-bit you may ask. Well, it’s because of the intended range of the OTP, which is usually a 6-digit code.

10^6 = 1,000,000

And a 31-bit number (slightly below the range of a 32-bit integer), the resultant number stays within the range that is comfortably above 1,000,000.

2^31 - 1 = 2,147,483,647

Extracting 31-bit number:

To get this number, first, an offset needs to be obtained first and this is where our 0xf comes into play. Because the offset is the rightmost 4****-bit of the last byte in the hash, to get these 4 bits, 0xf is used in an AND operation.

After that, the 31-bit number is derived by taking 4 bytes (32 bits) starting from the byte at the index determined by the calculated offset.

function truncate(hash: Buffer): number {
  const offset = hash[hash.length - 1] & 0xf; // Obtaining the offset within the hash
  
    const binary =
    ((hash[offset] & 0x7f) << 24) |
    ((hash[offset + 1] & 0xff) << 16) |
    ((hash[offset + 2] & 0xff) << 8) |
    (hash[offset + 3] & 0xff);

  return binary;
}

In Conclusion:

0xf may appear unassuming, yet it boasts profound significance in the intricate world of programming. From efficiently handling bit-level operations to deftly managing states and settings, its focused approach on the lower 4 bits brings elegant efficiency to programming paradigms.

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