TEA algorithm constant 0x9e3779b9 said to be derived from golden ratio, but the golden number is 1.618?

Question

I'm trying to understand a constant 0x9e3779b9.

What kind of data is this? It's not binary, not decimal, what is this?

It's a constant used on the TEA algorithm. It says it's derived from the golden number, but the golden number is 1.618?

Answer 1

I think this StackOverflow question answers it:

https://stackoverflow.com/questions/4948780/magic-numbers-in-boosthash-combine

Essentially, it is a magic number, derived from the Golden Ratio irrational number, using the steps:

1. phi = (1 + sqrt(5)) / 2 = 1,6180339887498948482045868343656.

2. Then after it, it is calculated 2^32 / phi which results in 2 654 435 769,4972302964775847707926.

3. Truncate it to have only its integer part 2 654 435 769

4. Convert it to hexadecimal and you will get 9E37 79B9 (On Windows Calculator choose Qword)

Being it a pre-calculated integer, instead of a taking a float every time and do the computation, you accelerate the calculus of each hash done after.

The notation 0x is for a hexadecimal number, or base 16. The benefit of a base 16 number is that each pair of digits represents one byte exactly. Given a little practice, you can almost see the bit pattern in your mind, assuming you've ever worked with binary numbers.

Answer 2

As others have said, the constant is an integer in hexadecimal form. Specifically, it is a 32-bit integer in hexadecimal form. If the constant is a signed integer, then 0x9e3779b9 is negative 1640531527 decimal in two's complement form; therefore, it may be a scaled integer fraction that has been adjusted to deal with non-integral of 2 related problems.

Two's complement negative to positive conversion in hex

0x9e3779b9 ⊕ 0xffffffff + 0x00000001 = 0x61C88647 = 1640531527 in decimal

or using the 1's complement operator ~ in the C family of languages

~0x9e3779b9 + 0x00000001 = 0x61C88647 = 1640531527 in decimal 

Two's complement negative to positive conversion in binary

10011110001101110111100110111001 ⊕ 11111111111111111111111111111111 + 00000000000000000000000000000001 =  01100001110010001000011001000111 = 1640531527 in decimal

Answer 3

The number comes from the hexadecimal representation of the golden ratio.

Just like 1/4 is 0.25 (25/100) in decimal is 0.4 (4/16) in hex, the fractional portion of the golden ratio has a different representation in hex than in decimal.

You are not dividing 0x9e3779b9 by 10^8, but 16^8 (or 2/32), and 0x9e3779b9/0x100000000 = 2654435769/4294967296 ≈ 0.6180339886

Answer 4

As others pointed out here and on https://stackoverflow.com/questions/4948780/magic-numbers-in-boosthash-combine, the number is indeed constructed from the golden ratio.

But there is another important reason why numbers such as pi, phi or e are used in cryptographic functions. Of course, in principle, any "reasonably random" bit sequence could be used as a constant. But it puts the creator of the function under suspicion of having engineered the value, e.g. in order to create a particular weakness in the algorithm (known only to him) that he can then exploit when you use his function.

By constructing the value from well-known "natural" constants, this kind of suspicion can be averted to some degree.