Preimage Sampling¶
Preimage sampling is a process used in cryptographic systems, particularly in lattice-based cryptography. It involves finding a preimage (an original input) that maps to a given output under a certain function, such that this preimage possesses specific properties or lies within a desired distribution. Here’s a detailed breakdown:
Context in Cryptography¶
-
Lattice-Based Cryptography:
- In lattice-based cryptography, a common problem is finding short vectors in a lattice that satisfy certain conditions. The problem of finding a preimage that is short and satisfies a given constraint is crucial in constructing secure cryptographic schemes like digital signatures and public-key encryption.
-
Hash Functions:
- In the context of hash functions, preimage sampling might involve finding an input that hashes to a particular output while satisfying some property, such as being within a certain range or having a specific structure.
Importance and Applications¶
-
Trapdoor Functions:
- Preimage sampling is often used with trapdoor functions. These functions are easy to compute in one direction but hard to reverse unless you have special information (the trapdoor). Preimage sampling can leverage the trapdoor to efficiently find preimages with desired properties.
-
Digital Signatures:
- In digital signature schemes like those based on the Learning With Errors (LWE) or Ring-LWE problems, preimage sampling is used to generate signatures that are both valid and short enough to ensure security and efficiency.
-
Commitment Schemes:
- Preimage sampling is used in commitment schemes to ensure that a commitment can be opened to a specific value that meets required criteria.