Revision #2 Authors: Nir Bitansky, Chethan Kamath, Omer Paneth, Ron Rothblum, Prashant Nalini Vasudevan

Accepted on: 25th July 2023 10:55

Downloads: 101

Keywords:

Batch proofs are proof systems that convince a verifier that $x_1,\dots, x_t \in L$, for some $NP$ language $L$, with communication that is much shorter than sending the $t$ witnesses. In the case of statistical soundness (where the cheating prover is unbounded but honest prover is efficient), interactive batch proofs are known for $UP$, the class of unique witness $NP$ languages. In the case of computational soundness (aka arguments, where both honest and dishonest provers are efficient), non-interactive solutions are now known for all of $NP$, assuming standard cryptographic assumptions. We study the necessary conditions for the existence of batch proofs in these two settings. Our main results are as follows.

1. Statistical Soundness: the existence of a statistically-sound batch proof for $L$ implies that $L$ has a statistically witness indistinguishable ($SWI$) proof, with inverse polynomial $SWI$ error, and a non-uniform honest prover. The implication is unconditional for public-coin protocols and relies on one-way functions in the private-coin case.

This poses a barrier for achieving batch proofs beyond $UP$ (where witness indistinguishability is trivial). In particular, assuming that $NP$ does not have $SWI$ proofs, batch proofs for all of $NP$ do not exist. This motivates further study of the complexity class $SWI$, which, in contrast to the related class $SZK$, has been largely left unexplored.

2. Computational Soundness: the existence of batch arguments ($BARG$s) for $NP$, together with one-way functions, implies the existence of statistical zero-knowledge ($SZK$) arguments for $NP$ with roughly the same number of rounds, an inverse polynomial zero-knowledge error, and non-uniform honest prover.

Thus, constant-round interactive $BARG$s from one-way functions would yield constant-round $SZK$ arguments from one-way functions. This would be surprising as $SZK$ arguments are currently only known assuming constant-round statistically-hiding commitments (which in turn are unlikely to follow from one-way functions).

3. Non-interactive: the existence of non-interactive $BARG$s for $NP$ and one-way functions, implies non-interactive statistical zero-knowledge arguments ($NISZKA$) for $NP$, with negligible soundness error, inverse polynomial zero-knowledge error, and non-uniform honest prover. Assuming also lossy public-key encryption, the statistical zero-knowledge error can be made negligible. We further show that $BARG$s satisfying a notion of honest somewhere extractability imply lossy public key encryption.

All of our results stem from a common framework showing how to transform a batch protocol for a language $L$ into an $SWI$ protocol for $L$.

Revised statement and proof of Theorems 3.10 and 3.14

Revision #1 Authors: Nir Bitansky, Chethan Kamath, Omer Paneth, Ron Rothblum, Prashant Nalini Vasudevan

Accepted on: 26th June 2023 22:02

Downloads: 71

Keywords:

Batch proofs are proof systems that convince a verifier that $x_1,\dots,x_t \in L$, for some $NP$ language $L$, with communication that is much shorter than sending the $t$ witnesses. In the case of statistical soundness (where the cheating prover is unbounded but the honest prover is efficient given the witnesses), interactive batch proofs are known for $UP$, the class of unique witness $NP$ languages. In the case of computational soundness (a.k.a. arguments, where both honest and dishonest provers are efficient), non-interactive solutions are now known for all of $NP$, assuming standard cryptographic assumptions. We study the necessary conditions for the existence of batch proofs in these two settings. Our main results are as follows.

1. Statistical Soundness: the existence of a statistically-sound batch proof for $L$ implies that $L$ has a statistically witness indistinguishable ($SWI$) proof, with inverse polynomial $SWI$ error, and a non-uniform honest prover. The implication is unconditional for obtaining honest-verifier $SWI$ or for obtaining full-fledged $SWI$ from public-coin protocols, whereas for private-coin protocols full-fledged $SWI$ is obtained assuming one-way functions.

This poses a barrier for achieving batch proofs beyond $UP$ (where witness indistinguishability is trivial). In particular, assuming that $NP$ does not have $SWI$ proofs, batch proofs for all of $NP$ do not exist.

2. Computational Soundness: the existence of batch arguments ($BARG$s) for $NP$, together with one-way functions, implies the existence of statistical zero-knowledge ($SZK$) arguments for $NP$ with roughly the same number of rounds, an inverse polynomial zero-knowledge error, and non-uniform honest prover.

Thus, constant-round interactive $BARG$s from one-way functions would yield constant-round $SZK$ arguments from one-way functions. This would be surprising as $SZK$ arguments are currently only known assuming constant-round statistically-hiding commitments (which in turn are unlikely to follow from one-way functions).

3. Non-interactive: the existence of non-interactive $BARG$s for $NP$ and one-way functions, implies non-interactive statistical zero-knowledge arguments ($NISZKA$) for $NP$, with negligible soundness error, inverse polynomial zero-knowledge error, and non-uniform honest prover. Assuming also lossy public-key encryption, the statistical zero-knowledge error can be made negligible and the honest prover can be made uniform.

All of our results stem from a common framework showing how to transform a batch protocol for a language $L$ into an $SWI$ protocol for $L$.

The third result has been improved upon in this version.

TR23-077 Authors: Nir Bitansky, Chethan Kamath, Omer Paneth, Ron Rothblum, Prashant Nalini Vasudevan

Publication: 27th May 2023 11:16

Downloads: 170

Keywords:

Batch proofs are proof systems that convince a verifier that $x_1,\dots, x_t \in L$, for some $NP$ language $L$, with communication that is much shorter than sending the $t$ witnesses. In the case of statistical soundness (where the cheating prover is unbounded but honest prover is efficient), interactive batch proofs are known for $UP$, the class of unique witness $NP$ languages. In the case of computational soundness (aka arguments, where both honest and dishonest provers are efficient), non-interactive solutions are now known for all of $NP$, assuming standard cryptographic assumptions. We study the necessary conditions for the existence of batch proofs in these two settings. Our main results are as follows.

1. Statistical Soundness: the existence of a statistically-sound batch proof for $L$ implies that $L$ has a statistically witness indistinguishable ($SWI$) proof, with inverse polynomial $SWI$ error, and a non-uniform honest prover. The implication is unconditional for public-coin protocols and relies on one-way functions in the private-coin case.

This poses a barrier for achieving batch proofs beyond $UP$ (where witness indistinguishability is trivial). In particular, assuming that $NP$ does not have $SWI$ proofs, batch proofs for all of $NP$ do not exist. This motivates further study of the complexity class $SWI$, which, in contrast to the related class $SZK$, has been largely left unexplored.

2. Computational Soundness: the existence of batch arguments ($BARG$s) for $NP$, together with one-way functions, implies the existence of statistical zero-knowledge ($SZK$) arguments for $NP$ with roughly the same number of rounds, an inverse polynomial zero-knowledge error, and non-uniform honest prover.

Thus, constant-round interactive $BARG$s from one-way functions would yield constant-round $SZK$ arguments from one-way functions. This would be surprising as $SZK$ arguments are currently only known assuming constant-round statistically-hiding commitments (which in turn are unlikely to follow from one-way functions).

3. Non-interactive: the existence of non-interactive $BARG$s for $NP$ and one-way functions, implies non-interactive statistical zero-knowledge arguments ($NISZKA$) for $NP$, with negligible soundness error, inverse polynomial zero-knowledge error, and non-uniform honest prover. Assuming also lossy public-key encryption, the statistical zero-knowledge error can be made negligible. We further show that $BARG$s satisfying a notion of honest somewhere extractability imply lossy public key encryption.

All of our results stem from a common framework showing how to transform a batch protocol for a language $L$ into an $SWI$ protocol for $L$.