Strong RSA assumption

Summary

In cryptography, the strong RSA assumption states that the RSA problem is intractable even when the solver is allowed to choose the public exponent e (for e ≥ 3). More specifically, given a modulus N of unknown factorization, and a ciphertext C, it is infeasible to find any pair (M, e) such that C ≡ M e mod N. The strong RSA assumption was first used for constructing signature schemes provably secure against existential forgery without resorting to the random oracle model. See also

Quadratic residuosity problem
Decisional composite residuosity assumption

Official source

https://en.wikipedia.org/wiki/Strong_RSA_assumption

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We introduce the notion of forgery-resilience for digital signature schemes, a new paradigm for digital signature schemes exhibiting desirable legislative properties. It evolves around the idea that, for any message, there can only be a unique valid signature, and exponentially many acceptable signatures, all but one of them being spurious. This primitive enables a judge to verify whether an alleged forged signature is indeed a forgery. In particular, the scheme considers an adversary who has access to a signing oracle and an oracle that solves a “hard” problem, and who tries to produce a signature that appears to be acceptable from a verifier’s point of view. However, a judge can tell apart such a spurious signature from a signature that is produced by an honest signer. This property is referred to as validatibility. Moreover, the scheme provides undeniability against malicious signers who try to fabricate spurious signatures and deny them later by showing that they are not valid. Last but not least, trustability refers to the inability of a malicious judge trying to forge a valid signature. This notion for signature schemes improves upon the notion of fail-stop signatures in different ways. For example, it is possible to sign more than one messages with forgery-resilient signatures and once a forgery is found, the credibility of a previously signed signature is not under question. A concrete instance of a forgery-resilient signature scheme is constructed based on the hardness of extracting roots of higher residues, which we show to be equivalent to the factoring assumption. In particular, using collision-free accumulators, we present a tight reduction from malicious signers to adversaries against the factoring problem. Meanwhile, a secure pseudorandom function ensures that no polynomially-bounded cheating verifier, who can still solve hard problems, is able to forge valid signatures. Security against malicious judges is based on the RSA assumption.

2012

A Fully Dynamic Universal Accumulator

Atefeh Mashatan, Serge Vaudenay

A dynamic universal accumulator is an accumulator that allows one to efficiently compute both membership and nonmembership witnesses in a dynamic way. It was first defined and instantiated by Li et al., based on the Strong RSA problem, building on the dynamic accumulator of Camenisch and Lysyanskaya. We revisit their construction and show that it does not provide efficient witness computation in certain cases and, thus, is only achieving the status of a partially dynamic universal accumulator. In particular, their scheme is not equipped with an efficient mechanism to produce non-membership witnesses for a new element, whether a newly deleted element or an element which occurs for the first time. We construct the first fully dynamic universal accumulator based on the Strong RSA assumption, building upon the construction of Li et al., by providing a new proof structure for the non-membership witnesses. In a fully dynamic universal accumulator, we require that not only one can always create a membership witness without having to use the accumulated set for a newly added element, but also one can always create non-membership witnesses for a new element, whether a newly deleted element or an element which occurs for the first time, i.e., a newcomer who is not a member, without using the accumulated set.

2013

Efficient Protocols for Set Membership and Range Proofs

Rafik Chaabouni

We consider the following problem: Given a commitment to a value σ, prove in zero-knowledge that σ belongs to some discrete set Φ. The set Φ can perhaps be a list of cities or clubs; often Φ can be a numerical range such as [1, 220]. This problem arises in e-cash systems, anonymous credential systems, and various other practical uses of zero-knowledge protocols. When using commitment schemes relying on RSA-like assumptions, there are solutions to this problem which require only a constant number of RSA-group elements to be exchanged between the prover and verifier [5, 15, 16]. However, for many commitment schemes based on bilinear group assumptions, these techniques do not work, and the best known protocols require O(k) group elements to be exchanged where k is a security parameter. In this paper,we present two new approaches to building set-membership proofs. The first is based on bilinear group assumptions. When applied to the case where Φ is a range of integers, our protocols require O(k/(log(k)−log(log(k)))) group elements to be exchanged. Not only is this result asymptotically better, but the constants are small enough to provide significant improvements even for small ranges. Indeed, for a discrete logarithm based setting, our new protocol is an order of magnitude more efficient than previously know nones. We also discuss alternative implementations of our membership proof based on the strong RSA assumption. Depending on the application, e.g., when Φ is a published set of values such a frequent flyer clubs, cities, or other ad hoc collections, these alternative also outperform prior solutions.

Springer2008