A *cryptographic fuzzy hash* with respect to a distance metric ```
D: {0,1}^* x
{0,1}^* -> [0,1]
```

is a tuple `(H, S)`

of functions satisfying the following
properties.

`H: {0,1}^* -> {0,1}^256`

.`S: {0,1}^256 x {0,1}^256 -> [0,1]`

.`H`

is*preimage resistant*; that is, for a randomly chosen string`y`

, it is hard to compute a string`x`

such that`H(x)=y`

.`(H, S)`

is*fuzzy*with respect to`D`

; that is, for randomly chosen`x1`

and`x2`

, it holds that`D(x1, x2)`

and`S(H(x1), H(x2))`

are about the same.`(H, S)`

is*tamper resistant*with respect to`D`

; that is, for a randomly chosen string`x`

, there is not efficient modification`M`

such that`D(x, M(x))`

is small but`S(H(x), H(M(x)))`

is large.

**Note.** A pedantic person, who for once is not me, can write the above
definition more formally using two epsilons.

### Explicit Examples (or Why I Care)

#### 1. Fingerprint Authentication

One can think of such a scheme being used to hash fingerprints. Currently, to the best of my knowledge, there is no open standard for fingerprint hashing. If such a scheme existed, we could use that. This is more generally applicable to biometric authentication.

**Note.** Fingerprint authentication is easier as we can assume that the noise
or the “fuzziness” is random and not adversarial. If we assume this, then we
can use ideas from error correcting codes—namely, *decoding* error correcting
codes—to construct a fuzzy hash. The key idea is to decode the input (under a
suitable error correcting code) before hashing it with a cryptographic hash.
These ideas, to the best of my knowledge, were first discussed in Juels and
Wattenberg (1999).

#### 2. CSAM Detection

Cloudflare talks about this in their blog post on CSAM scanning in which they say the following.

The challenge was if criminal producers and distributors of CSAM knew exactly how such tools worked then they might be able to craft how they alter their images in order to beat it.

Looks like a classic case of security by obscurity to me. An explicit construction would help us here.

#### 3. The Right to Be Forgotten

The GDPR includes a Right to Erasure but that doesn’t stop people from re-uploading content you deleted. Suppose you uploaded some rude pictures of yourself as a public post on Facebook, and then deleted them. There does not seem to be an easy way for Facebook to prevent someone else uploading the same images back. Indeed, the threat model seems similar to the one for CSAM.

### Literature Review

None of the articles I could find on the internet (e.g., TLSH, ssdeep, sdhash, and Nilsimsa) seem to talking about this definition or these properties explicitly. In fact, I found an article

Using Randomization to Attack Similarity Digests; Jonathan Oliver, Scott Forman, and Chun Cheng; DOI: 10.1007/978-3-662-45670-5_19

which shows how most fuzzy hashes fail under the naive random noise attack.

### Simple Constructions for Restricted Cases

- If you only care about bitflip errors, then piecewise hashing with a cryptographic hash works.

### Moving Forward

It would be nice to formally define an attack model and construct a cryptographic fuzzy hash for it.