Bit commitment using pseudorandomness

Moni Naor

Dakshita Khurana, Kabir Tomer

One-way functions are central to classical cryptography. They are necessary for the existence of non-trivial classical cryptosystems, and also sufficient to realize meaningful primitives including commitments, pseudorandom generators and digital signatures. At the same time, a mounting body of evidence suggests that assumptions even weaker than one-way functions may suffice for many cryptographic tasks of interest in a quantum world, including bit commitments and secure multi-party computation. This work studies one-way state generators [Morimae-Yamakawa, CRYPTO 2022], a natural quantum relaxation of one-way functions. Given a secret key, a one-way state generator outputs a hard to invert quantum state. A fundamental question is whether this type of quantum one-wayness suffices to realize quantum cryptography. We obtain an affirmative answer to this question, by proving that one-way state generators with pure state outputs imply quantum bit commitments and secure multiparty computation. Along the way, we use efficient shadow tomography [Huang et. al., Nature Physics 2020] to build an intermediate primitive with classical outputs, which we call a (quantum) one-way puzzle. Our main technical contribution is a proof that one-way puzzles imply quantum bit commitments. This proof develops new techniques for pseudoentropy generation [Hastad et. al., SICOMP 1999] from arbitrary distributions, which may be of independent interest.

Bruno Pasqualotto Cavalar, Eli Goldin, Matthew Gray +3 more

There is a large body of work studying what forms of computational hardness are needed to realize classical cryptography. In particular, one-way functions and pseudorandom generators can be built from each other, and thus require equivalent computational assumptions to be realized. Furthermore, the existence of either of these primitives implies that P ≠ N P , which gives a lower bound on the necessary hardness.One can also define versions of each of these primitives with quantum output: respectively one-way state generators and pseudorandom state generators. Unlike in the classical setting, it is not known whether either primitive can be built from the other. Although it has been shown that pseudorandom state generators for certain parameter regimes can be used to build one-way state generators, the implication has not been previously known in full generality. Furthermore, to the best of our knowledge, the existence of one-way state generators has no known implications in complexity theory.We show that pseudorandom states compressing n bits to log ⁡ n + 1 qubits can be used to build one-way state generators and pseudorandom states compressing n bits to ω ( log ⁡ n ) qubits are one-way state generators. This is a nearly optimal result since pseudorandom states with fewer than c log ⁡ n -qubit output can be shown to exist unconditionally. We also show that any one-way state generator can be broken by a quantum algorithm with classical access to a P P oracle.An interesting implication of our results is that a t ( n ) -copy one-way state generator exists unconditionally, for every t ( n ) = o ( n / log ⁡ n ) . This contrasts nicely with the previously known fact that O ( n ) -copy one-way state generators require computational hardness. We also outline a new route towards a black-box separation between one-way state generators and quantum bit commitments.

Moni Naor

Himanshu Tyagi, Shun Watanabe

Ivan Damgård, Bernardo David, Irene Giacomelli +1 more

We present a new compact verifiable secret sharing scheme, based on this we present the first construction of a homomorphic UC commitment scheme that requires only cheap symmetric cryptography, except for a small number of seed OTs. To commit to a k -bit string, the amortized communication cost is O ( k ) bits. Assuming a sufficiently efficient pseudorandom generator, the computational complexity is O ( k ) for the verifier and O ( k 1 + ε ) for the committer (where ε < 1 is a constant). In an alternative variant of the construction, all complexities are O ( k · polylog ( k )). Our commitment scheme extends to vectors over any finite field and is additively homomorphic. By sending one extra message, the prover can allow the verifier to also check multiplicative relations on committed strings, as well as verifying that committed vectors a , b satisfy a = φ ( b ) for a linear function φ . These properties allow us to non-interactively implement any one-sided functionality where only one party has input (this includes UC secure zero-knowledge proofs of knowledge). We also present a perfectly secure implementation of any multiparty functionality, based directly on our VSS. The communication required is proportional to a circuit implementing the functionality, up to a logarithmic factor. For a large natural class of circuits the overhead is even constant. We also improve earlier results by Ranellucci et al. on the amount of correlated randomness required for string commitments with individual opening of bits. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.