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Revision #1 to TR15-061 | 9th January 2017 10:35

Arithmetic Cryptography

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Revision #1
Authors: Benny Applebaum, Jonathan Avron, Christina Brzuska
Accepted on: 9th January 2017 10:35
Downloads: 130
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Abstract:

We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a black-box use of the underlying field, and the adversary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions.

We prove several positive and negative results in this model for various cryptographic tasks.
On the positive side, we show that, under coding-related intractability assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, additively homomorphic encryption, and secure computation with low online complexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard Boolean model and the arithmetic model, and explain, in retrospect, some of the limitations of previous constructions.



Changes to previous version:

Fixed some typos. Extended the lower-bound against a homomorphic encryption to hold against a weaker form of homomorphism.


Paper:

TR15-061 | 14th April 2015 13:02

Arithmetic Cryptography


Abstract:

We study the possibility of computing cryptographic primitives in a fully-black-box arithmetic model over a finite field F. In this model, the input to a cryptographic primitive (e.g., encryption scheme) is given as a sequence of field elements, the honest parties are implemented by arithmetic circuits which make only a black-box use of the underlying field, and the adversary has a full (non-black-box) access to the field. This model captures many standard information-theoretic constructions.

We prove several positive and negative results in this model for various cryptographic tasks.
On the positive side, we show that, under reasonable assumptions, computational primitives like commitment schemes, public-key encryption, oblivious transfer, and general secure two-party computation can be implemented in this model. On the negative side, we prove that garbled circuits, multiplicative-homomorphic encryption, and secure computation with low online complexity cannot be achieved in this model. Our results reveal a qualitative difference between the standard Boolean model and the arithmetic model, and explain, in retrospect, some of the limitations of previous constructions.



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