**Hush Functions Extended to Any Size Input versus Any Size Output**

*Gideon Samid*

**Abstract: **Traditional hush functions map a large number to a small number such that the reverse-hush has an infinity of solutions, and nonetheless a collision is hard to come by. This primitive is so abundantly useful that one is tempted to extend it such that any number large or small may be mapped to any number larger, or smaller while maintaining the above conditions. This extension would increase the flexibility of the commodity hush primitive, expand its current applications, and likely suggest new ones. Additional generality may be achieved by allowing the input to determine the computational burden, and involving Turing’s Entscheidungsproblem. We propose an algorithm where a natural number, X, is mapped to another natural number Y, referring to the mapping as a "Crypto Square", and to the reverse as "Crypto Square Root": Y = X**2|c and X = √Y|c. While the crypto-square mapping is a proper function, the square root equation has infinite solutions. There exists a deterministic solution algorithm to find any desired number of solutions to a square-root equation. This asymmetry proves itself useful, since the mapping is Z+→Z+, and hence the chance of collision for any finite size set is negligible. Unlike standard one-way functions, crypto-square shields the identity of the input (X), not by the intractability of the reverse function, but by Vernam-like equivocation per the infinity of X candidates. This prospect suggests further examination of this “square” algorithm for possible useful roles in various crypto protocols, especially protocols concerned with privacy, authentication and deniability.

**Category / Keywords: **foundations / hush functions, one-way-functions, intractability, equivocation

**Publication Info: **Paper was not published.

**Date: **received 12 Aug 2012

**Contact author: **gideon samid at case edu

**Available format(s): **PDF | BibTeX Citation

**Note: **This is a first in a series of articles designed as foundation for new conceptual financial cryptography consistent with our newly awarded digital currency patent, US PTO #8,229,859

**Version: **20120813:150616 (All versions of this report)

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