The Kernel Matrix Diffie-Hellman Assumption

Morillo, Paz; Ràfols, Carla; Villar, Jorge L.

doi:10.1007/978-3-662-53887-6_27

Cited by 42 publications

(31 citation statements)

References 37 publications

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“…Groth and Lu proved the generic hardness of the natural translation of this assumption to symmetric groups [19]. We observe that this assumption corresponds to a kernel assumption [25], the Q m -KerMDH assumption in symmetric groups.…”

Section: Hardness Assumptionsmentioning

confidence: 87%

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Shorter Ring Signatures from Standard Assumptions

González

2019

Public-Key Cryptography – PKC 2019

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Ring signatures, introduced by Rivest, Shamir and Tauman (ASIACRYPT 2001), allow to sign a message on behalf of a set of users while guaranteeing authenticity and anonymity. Groth and Kohlweiss (EUROCRYPT 2015) and Libert et al. (EUROCRYPT 2016) constructed schemes with signatures of size logarithmic in the number of users. An even shorter ring signature, of size independent from the number of users, was recently proposed by Malavolta and Schröder (ASIACRYPT 2017). However, all these short signatures are obtained relying on strong and controversial assumptions. Namely, the former schemes are both proven secure in the random oracle model while the later requires non-falsifiable assumptions. The most efficient construction under mild assumptions remains the construction of Chandran et al. (ICALP 2007) with a signature of size Θ(√ n), where n is the number of users, and security is based on the Diffie-Hellman assumption in bilinear groups (the SXDH assumption in asymmetric bilinear groups). In this work we construct an asymptotically shorter ring signature from the hardness of the Diffie-Hellman assumption in bilinear groups. Each signature comprises Θ(3 √ n) group elements, signing a message requires computing Θ(3 √ n) exponentiations, and verifying a signature requires Θ(n 2/3) pairing operations. To the best of our knowledge, this is the first ring signature based on bilinear groups with o(√ n) signatures and sublinear verification complexity.

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Section: Hardness Assumptionsmentioning

confidence: 87%

“…Definition 2 (Kernel Diffie-Hellman Assumption in G [25]). Let gk ← Gen a (1 λ ) and D ,k a distribution over Z ×k q .…”

Section: Hardness Assumptionsmentioning

confidence: 99%

Shorter Ring Signatures from Standard Assumptions

González

2019

Public-Key Cryptography – PKC 2019

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“…We stress that our aim has been to find concrete and precise reductions and prove separations among the different classes of assumptions which encompass a significant portion of existing assumptions in the literature. Thus, our approach is different from previous works such as [BBG05,Boy08,EHK + 13,MRV16] which aimed at defining algebraic frameworks or templates in generic groups to capture some families of assumptions. The closest work to ours is Abdalla et al [ABP15], which provided an Uber assumption for decisional assumptions in cyclic groups (without a bilinear map, which invalidates many decisional assumptions).…”

Section: Introductionmentioning

confidence: 95%

Towards a Classification of Non-interactive Computational Assumptions in Cyclic Groups

Ghadafi

Groth

2017

Advances in Cryptology – ASIACRYPT 2017

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We study non-interactive computational intractability assumptions in prime-order cyclic groups. We focus on the broad class of computational assumptions, which we call target assumptions, where the adversary's goal is to compute a concrete group element and investigate the structure of this class. Our analysis identifies two families of intractability assumptions, the q-Generalized Diffie-Hellman Exponent assumptions and the q-Simple Fractional assumptions that imply all other target assumptions. These two assumptions therefore serve as Uber assumptions that can underpin all the target assumptions where the adversary has to compute specific group elements. We also study the internal hierarchy among members of these two assumption families. We provide heuristic evidence that both families are necessary to cover the full class of target assumptions, and we show that the lowest level in the q-GDHE hierarchy (the 1-GDHE assumption) is equivalent to the computational Diffie-Hellman assumption. We generalize our results to the bilinear group setting. For the base groups our results translate nicely and a similar structure of non-interactive computational assumptions emerges. We also identify Uber assumptions in the target group but this requires replacing the q-GDHE assumption with a more complicated assumption, which we call the Bilinar Gap Assumption. Our analysis can assist both cryptanalysts and cryptographers. For cryptanalysts, we propose the q-GDHE and the q-SDH assumptions are the most natural and important targets for cryptanalysis in prime-order groups. For cryptographers, we believe our classification can aid the choice of assumptions underpinning cryptographic schemes and be used as a guide to minimize the overall attack surface that different assumptions expose.

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“…The Kernel Diffie-Hellman Assumption [32] says one cannot find a non-zero vector in one of the groups which is in the co-kernel of A. We also use a generalization in bilinear groups which says one cannot find a pair of vectors in G k+1 1 × G k+1 2 such that the difference of the vector of their discrete logarithms is in the co-kernel of A.…”

Section: Cryptographic Assumptionsmentioning

confidence: 99%

Shorter Pairing-Based Arguments Under Standard Assumptions

González

Ràfols

2019

Lecture Notes in Computer Science

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This paper constructs efficient non-interactive arguments for correct evaluation of arithmetic and boolean circuits with proof size O(d) group elements, where d is the multiplicative depth of the circuit, under falsifiable assumptions. This is achieved by combining techniques from SNARKs and QA-NIZK arguments of membership in linear spaces. The first construction is very efficient (the proof size is ≈ 4d group elements and the verification cost is ≈ 4d pairings and O(n + n + d) exponentiations, where n is the size of the input and n of the output) but one type of attack can only be ruled out assuming the knowledge soundness of QA-NIZK arguments of membership in linear spaces. We give an alternative construction which replaces this assumption with a decisional assumption in bilinear groups at the cost of approximately doubling the proof size. The construction for boolean circuits can be made zero-knowledge with Groth-Sahai proofs, resulting in a NIZK argument for circuit satisfiability based on falsifiable assumptions in bilinear groups of proof size O(n + d).Our main technical tool is what we call an "argument of knowledge transfer". Given a commitment C1 and an opening x, such an argument allows to prove that some other commitment C2 opens to f (x), for some function f , even if C2 is not extractable. We construct very short, constant-size, pairing-based arguments of knowledge transfer with constant-time verification for any linear function and also for Hadamard products. These allow to transfer the knowledge of the input to lower levels of the circuit.

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The Kernel Matrix Diffie-Hellman Assumption

Cited by 42 publications

References 37 publications

Shorter Ring Signatures from Standard Assumptions

Shorter Ring Signatures from Standard Assumptions

Towards a Classification of Non-interactive Computational Assumptions in Cyclic Groups

Shorter Pairing-Based Arguments Under Standard Assumptions

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