Non-malleable codes, introduced by Dziembowski, Pietrzak, and Wichs (ICS 2010), provide a means of encoding information such that if the encoding is tampered with, the result encodes something either identical or completely unrelated. Unlike error-correcting codes (for which the result of tampering must always be identical), non-malleable codes give guarantees even when tampering functions are allowed to change every symbol of a codeword.
In this thesis, we will provide constructions of non-malleable codes secure against a variety tampering classes with natural computational semantics:
• Bounded-Communication: Functions corresponding to 2-party protocols where each party receives half the input (respectively) and then may communicate <𝒏/4 bits before returning their (respective) half of the tampered output.
•Local Functions (Juntas):} each tampered output bit is only a function of n¹-ẟ inputs bits, where ẟ>0 is any constant (the efficiency of our code depends on ẟ). This class includes NC⁰.
•Decision Trees: each tampered output bit is a function of n¹/⁴-⁰(¹) adaptively chosen bits.
•Small-Depth Circuits: each tampered output bit is produced by a 𝒄log(n)/log log(n)-depth circuit of polynomial size, for some constant 𝒄. This class includes AC⁰.
•Low Degree Polynomials: each tampered output field element is produced by a low-degree (relative to the field size) polynomial.
•Polynomial-Size Circuit Tampering: each tampered codeword is produced by circuit of size 𝒏ᶜ where 𝒄 is any constant (the efficiency of our code depends on 𝒄). This result assumes that E is hard for exponential size nondeterministic circuits (all other results are unconditional).
We stress that our constructions are efficient (encoding and decoding can be performed in uniform polynomial time) and (with the exception of the last result, which assumes strong circuit lower bounds) enjoy unconditional, statistical security guarantees. We also illuminate some potential barriers to constructing codes for more complex computational classes from simpler assumptions.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/d8-debr-bw49 |
| Date | January 2021 |
| Creators | Ball Jr, Maynard Marshall |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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