I don't really know how to phrase this question, so I'll just mention the idea. This is partially reminiscent (at least in spirit to white box crypto): Can one derive arithmetic operations from a number? Is there a formula to do that?

So, the simplest example would be:

a = 4
ops = get_operations(a)

/* output:
* b = 2
* c = 2
* return b + c

The idea is to simply have a way to "derive" a mathematical operation from a value. There are obviously more than "one" operation that could be derived from the above example. The system does not have to be complete.

The flow of such a "derivation" algorithm in my head is as such:

  1. Get input (k)
  2. Randomly choose an arithmetic operand (P) where y = k (P) x and store the operand (P)
  3. Pick a random number (x) and store its value
  4. Solve for (y) and store its value
  5. Now, we have a combination of (P), (x), and (y) that, when solved, would yield the correct input (k).

My question is this: did someone research this and make a way better algorithm that what I could muster up here?

  • $\begingroup$ "a way better algorithm" - define "better", that is, by what criteria are you comparing the various possible algorithms? $\endgroup$ – poncho Jun 27 '19 at 6:33
  • $\begingroup$ "way better" means if there is a name for this topic and has it been researched before by professional mathematicians $\endgroup$ – jimmytann Jun 27 '19 at 6:45

Is there a formula to do that?

Your example looks like a problem about Partitions to me. A partition is a way of representing a positive number as a sum of $n$ positive integers, for example:

The partitions of the number $5$ are:

  • $5 \space (+0)$
  • $4 + 1$
  • $3 + 2$
  • $3 + 1 + 1$
  • $2 + 2 + 1$
  • $2 + 1 + 1 + 1$
  • $1 + 1 + 1 + 1 + 1$

Since partitions consider every possibility valid, the number for possible ways of any integer $n$ can grow very fast, i.e. the number of possible partitions of the number $100$ is $190,569,292$.

There exist some formulas (some more complicated or faster than others and some are only approximations).

In your example you only have an operator once, so here $4 + 1$ and $3 + 2$ would be a possiblity.

Since you mentioned "more than one operation", a possibility would also be to represent numbers as their prime factors, for example:

The prime factorization of the number 22 is:

  • $2 \times 11$

There's always only one possiblity to represent a number as it's prime factors (if you don't care about the way you write it ($2 \times 11$ is the same as $11 \times 2$)).

There are a lot of factoring algorithms, since this is still a unsolved problem (Does P = NP ?) and this problem is important for cryptography.

I assume that a problem occurs if you'd try to do this with the operators of division and subtraction, for example:

There are infinite possibilities to represent a number as a subtraction, for example:

For the number $5$:

  • $6 - 1$
  • $7 - 2$
  • $8 - 3$
  • $9 - 4$
  • $\ldots$
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I could think of two things: Tweakable Block Cipher, and Tweakable Compression Function.

  1. Tweakable Block Cipher:

More info here.

A Tweakable Block Cipher is defined with the prototype: $f : D \times K \times T \rightarrow D $ where $D$ is the block of data to be encrypted/decrypted with the key $K$ using the block cipher function $f$, that can be efficiently parameterized with $T$.

  1. Tweakable Compression Function:

I assume you mean binary operations when you say operation (binary in the sense that there are 2 operands).

Basically, we just build a compression function from a tweakable block cipher using Davies–Meyer construct or any other method to turn a block cipher into a compression function.

$g : D \times D \times T \rightarrow D$

f <- blockcipher(d, key, tweak);
g <- compressionfunc(a, b, tweak){ return blockcipher(a, b, tweak) xor a; }
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