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Verifiable Capacity-Bound Functions: A New Primitive from Kolmogorov Complexity

(Revisiting Space-Based Security in the Adaptive Setting)

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Public-Key Cryptography – PKC 2023 (PKC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13941))

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Abstract

We initiate the study of verifiable capacity-bound function (VCBF). The main VCBF property imposes a strict lower bound on the number of bits read from memory during evaluation (referred to as minimum capacity). No adversary, even with unbounded computational resources, should produce an output without spending this minimum memory capacity. Moreover, a VCBF allows for an efficient public verification process: Given a proof of correctness, checking the validity of the output takes significantly fewer memory resources, sublinear in the target minimum capacity. Finally, it achieves soundness, i.e., no computationally bounded adversary can produce a proof that passes verification for a false output. With these properties, we believe a VCBF can be viewed as a “space” analog of a verifiable delay function. We then propose the first VCBF construction relying on evaluating a degree-\(d\) polynomial f from \(\mathbb {F}_p[x]\) at a random point. We leverage ideas from Kolmogorov complexity to prove that sampling f from a large set (i.e., for high-enough d) ensures that evaluation must entail reading a number of bits proportional to the size of its coefficients. Moreover, our construction benefits from existing verifiable polynomial evaluation schemes to realize our efficient verification requirements. In practice, for a field of order \(O(2^\lambda )\) our VCBF achieves \(O((d+1)\lambda )\) minimum capacity, whereas verification requires just \(O(\lambda )\). The minimum capacity of our VCBF construction holds against adversaries that perform a constant number of random memory accesses during evaluation. This poses the natural question of whether a VCBF with high minimum capacity guarantees exists when dealing with adversaries that perform non-constant (e.g., polynomial) number of random accesses.

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Notes

  1. 1.

    We stress that, in the setting of memory-hard functions, the term “memory” is used to denote the number of memory blocks required to correctly evaluate (in a given time) the function. This differs from the VCBF objective of forcing the evaluator to read a fixed number of distinct bits (requiring n memory blocks of size w on evaluation does not imply reading nw distinct bits since multiple memory blocks may present a redundant pattern that may be compressed).

  2. 2.

    We stress that memory-hard functions present a time/space trade-off on evaluation that varies according to the notion of memory hardness considered (e.g., time-space complexity [49], cumulative space complexity [6], sustained space complexity [4]).

  3. 3.

    Considering unbounded adversaries is fundamental in order to capture the (concrete) strict lower bound on the number of distinct bits read that a VCBF must guarantee (i.e., a VCBF does not present any time/bits read trade-off). We provide a more detailed discussion in Sect. 4 and Remark 2.

  4. 4.

    For example, a particular (hard to guess) compressible pattern may be revealed after the polynomial coefficients are chosen. Note that this may happen (with a certain probability) even if the polynomial is sampled at random.

  5. 5.

    This can also be seen by observing that the Rényi family of entropies is equivalent to Shannon entropy when considering uniform distributions (as considered in this work, e.g., polynomial’s coefficients are sampled at random).

  6. 6.

    The challenge \(x= \textsf{H}(s,t)\) has this format since smart contracts cannot generate secret randomness to sample a random challenge.

  7. 7.

    We explicitly detached y from its proof \(\pi _y\). Several works define the output of the computation algorithm \(\textsf{Compute}\) as a singleton \(\sigma _y\) (the encoding of the output y) defined as \(\sigma _y = (y, \pi _y)\).

  8. 8.

    As we will discuss later, Kolmogorov Complexity considers constant-size Turing machines. This requires the use of a self-delimiting code to encode multiple inputs.

  9. 9.

    Note that not all binary strings are valid Turing machines.

  10. 10.

    The constant \(c_\textsf{T}\) corresponds to the self-delimiting description of the Turing machine \(\textsf{T}\).

  11. 11.

    Observe that \(\tau _{x,r}\) can be fetched from \(\tau \) in an adaptive fashion according to the challenge x and randomness r.

  12. 12.

    Without loss of generality, we assume the adversary reads exactly \(m\) bits since the higher the number of bits read, the higher the probability to compute the correct output \(y = \textsf{Eval}(\textsf{ek}, x)\).

  13. 13.

    Without loss of generality, we assume that reading the first \(m\) bits of \(\tau \) requires the adversary to perform a random access to the first index of \(\tau \).

  14. 14.

    Observe that \(|\textsf{vk}| + |x| + |y| + |\pi | \in o(m)\) (i.e., \(\textsf{vk},\pi ,y,x\) are “succinct”) is necessary to obtain a capacity-efficient verification of \(o(m)\). This is because \(\textsf{vk},\pi ,y,x\) are part of the verification algorithm \(\textsf{Verify}\) of VCBF.

  15. 15.

    In the verification, \(O(\lambda )\) is for reading a constant number of group elements of order p of size at most \(\lambda +1\). In the evaluation, \(O((d+1)\lambda ) = O(\lambda ^{c+1})\) is for the \(d\) coefficients \((a_0, \ldots , a_d) \in \mathbb {F}^{d+1}_p\) of the polynomial \(f(X)\in \mathbb {F}_p[x]\).

  16. 16.

    Note that the polynomial \(f_a(X)\) is RND-incompressible with overwhelming probability since it is sampled at random. This follows by leveraging Theorems 3 and 4.

  17. 17.

    We stress that the memory size n does not need to be super-polynomial (in the security parameter) in order to consider a VCBF secure. Indeed, in a scenario in which a machine has at most \(n = \lambda ^s \in \textsf{poly}\) bits of free memory (for a positive constant s), it is enough to show that the VCBF satisfies \((\epsilon , m, {\ell _{rnd}}, \lambda ^s)\)-min-capacity where \(\epsilon \) is the target advantage.

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Acknowledgments

We thank Irene Giacomelli and Luca Nizzardo for helpful discussions.

The authors were partially supported by Protocol Labs under the RFP-009 on Proof of Space and Useful Space. In addition, the second author was supported by the National Key R &D Program of China 2021YFB3100100 and CAS Project for Young Scientists in Basic Research Grant YSBR-035, the third author was supported by the Carlsberg Foundation under the Semper Ardens Research Project CF18-112 (BCM), and the fourth author was supported by Hong Kong Research Grants Council under grant GRF-16200721.

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Authors and Affiliations

  1. George Mason University, Virginia, USA

    Giuseppe Ateniese

  2. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Long Chen

  3. Aarhus University, Aarhus, Denmark

    Danilo Francati

  4. Hong Kong University of Science and Technology, Kowloon, Hong Kong

    Dimitrios Papadopoulos

  5. The University of Sydney, Sydney, Australia

    Qiang Tang

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  1. Giuseppe Ateniese
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Correspondence to Danilo Francati .

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    Alexandra Boldyreva

  2. Georgia Institute of Technology, Atlanta, GA, USA

    Vladimir Kolesnikov

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Ateniese, G., Chen, L., Francati, D., Papadopoulos, D., Tang, Q. (2023). Verifiable Capacity-Bound Functions: A New Primitive from Kolmogorov Complexity. In: Boldyreva, A., Kolesnikov, V. (eds) Public-Key Cryptography – PKC 2023. PKC 2023. Lecture Notes in Computer Science, vol 13941. Springer, Cham. https://doi.org/10.1007/978-3-031-31371-4_3

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