There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We propose a fully functional identity-based encryption scheme (IBE). The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem. Our system is based on the Weil pairing. We give precise definitions for secure identity based encryption schemes and give several applications for such systems.

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Identity-Based Encryption from the Weil Pairing

This paper investigates a novel computational problem, namely the Composite Residuosity Class Problem, and its applications to public-key cryptography. We propose a new trapdoor mechanism and derive from this technique three encryption schemes : a trapdoor permutation and two homomorphic probabilistic encryption schemes computationally comparable to RSA. Our cryptosystems, based on usual modular arithmetics, are provably secure under appropriate assumptions in the standard model.

/pdf/public-key-cryptosystems-based-on-composite-degree-1zopwb21ba.pdf

Public-key cryptosystems based on composite degree residuosity classes

We propose a fully functional identity-based encryption (IBE) scheme. The scheme has chosen ciphertext security in the random oracle model assuming a variant of the computational Diffie--Hellman problem. Our system is based on bilinear maps between groups. The Weil pairing on elliptic curves is an example of such a map. We give precise definitions for secure IBE schemes and give several applications for such systems.

/pdf/identity-based-encryption-from-the-weil-pairing-1x4254f2zc.pdf

We introduce a short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyperelliptic curves. The signature length is half the size of a DSA signature for a similar level of security. Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel.

/pdf/short-signatures-from-the-weil-pairing-5ajupg0cpi.pdf

Short Signatures from the Weil Pairing

A central focus of modern cryptography is the construction of efficient, high-level cryptographic tools (e.g., encryption schemes) from weaker, low-level cryptographic primitives (e.g., one-way functions). Of interest are both the existence of such constructions and their efficiency.
Here, we show essentially tight lower bounds on the best possible efficiency of any black-box construction of some fundamental cryptographic tools from the most basic and widely used cryptographic primitives. Our results hold in an extension of the model introduced by Impagliazzo and Rudich and improve and extend earlier results of Kim, Simon, and Tetali. We focus on constructions of pseudorandom generators, universal one-way hash functions, and digital signatures based on one-way permutations, as well as constructions of public- and private-key encryption schemes based on trapdoor permutations. In each case, we show that any black-box construction beating our efficiency bound would yield the unconditional existence of a one-way function and thus, in particular, prove $P \neq NP$.

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Bounds on the Efficiency of Generic Cryptographic Constructions

We develop a general approach to adding a threshold functionality to a large class of (non-threshold) cryptographic schemes. A threshold functionality enables a secret key to be split into a number of shares, so that only a threshold of parties can use the key, without reconstructing the key. We begin by constructing a threshold fully-homomorphic encryption scheme (ThFHE) from the learning with errors (LWE) problem. We next introduce a new concept, called a universal thresholdizer, from which many threshold systems are possible. We show how to construct a universal thresholdizer from our ThFHE. A universal thresholdizer can be used to add threshold functionality to many systems, such as CCA-secure public-key encryption (PKE), signature schemes, pseudorandom functions, and others primitives. In particular, by applying this paradigm to a (non-threshold) lattice signature system, we obtain the first single-round threshold signature scheme from LWE.

/pdf/threshold-cryptosystems-from-threshold-fully-homomorphic-41d7xoq7t3.pdf

Threshold Cryptosystems From Threshold Fully Homomorphic Encryption.

We present lower bounds on the efficiency of constructions for Pseudo-Random Generators (PRGs) and Universal One-Way Hash Functions (UOWHFs) based on black-box access to one-way permutations. Our lower bounds are tight as they match the efficiency of known constructions. A PRG (resp. UOWHF) construction based on black-box access is a machine that is given oracle access to a permutation. Whenever the permutation is hard to invert, the construction is hard to break. In this paper we give lower bounds on the number of invocations to the oracle by the construction. If S is the assumed security of the oracle permutation /spl pi/ (i.e. no adversary of size S can invert /spl pi/ on a fraction larger than 1/S of its inputs) then a PRG (resp. UOWHF) construction that stretches (resp. compresses) its input by k bits must query /spl pi/ in q=/spl Omega/(k/log S) points. This matches known constructions. Our results are given in an extension of the Impagliazzo-Rudich model. That is, we prove that a proof of the existence of PRG (resp. UOWHF) black-box constructions that beat our lower bound would imply a proof of the unconditional existence of such construction (which would also imply P/spl ne/NP).

Lower Bounds on the Efficiency of Generic Cryptographic Constructions

Zero Knowledge Contingent Payment (ZKCP) protocols allow fair exchange of sold goods and payments over the Bitcoin network. In this paper we point out two main shortcomings of current proposals for ZKCP, and propose ways to address them. First we show an attack that allows a buyer to learn partial information about the digital good being sold, without paying for it. This break in the zero-knowledge condition of ZKCP is due to the fact that in the protocols we attack, the buyer is allowed to choose common parameters that normally should be selected by a trusted third party. We implemented and tested this attack: we present code that learns, without paying, the value of a Sudoku cell in the "Pay-to-Sudoku" ZKCP implementation. We also present ways to fix this attack that do not require a trusted third party. Second, we show that ZKCP are not suited for the purchase of digital services} rather than goods. Current constructions of ZKCP do not allow a seller to receive payments after proving that a certain service has been rendered, but only for the sale of a specific digital good. We define the notion of Zero-Knowledge Contingent Service Payment (ZKCSP) protocols and construct two new protocols, for either public or private verification. We implemented our ZKCSP protocols for Proofs of Retrievability, where a client pays the server for providing a proof that the client's data is correctly stored by the server.We also implement a secure ZKCP protocol for "Pay-to-Sudoku" via our ZKCSP protocol, which does not require a trusted third party. A side product of our implementation effort is a new optimized circuit for SHA256 with less than a quarter than the number of AND gates of the best previously publicly available one. Our new SHA256 circuit may be of independent use for circuit-based MPC and FHE protocols that require SHA256 circuits.

https://dl.acm.org/doi/pdf/10.1145/3133956.3134060

Zero-Knowledge Contingent Payments Revisited: Attacks and Payments for Services

A signer uses an undeniable signature scheme to sign his public key to thereby create an “undeniable certificate” which can be used to verify the signer's digital signature on any message signed using the signer's corresponding private key. Hence, once the undeniable certificate is received by the recipient, the recipient and the signer engage one time in a confirmation protocol or denial protocol to the satisfaction of the recipient that the undeniable certificate has in fact been signed by the signer thus certifying signer's public key. Thereafter, the recipient can use the certified public key to verify any documents signed by the signer with no further interaction with the signer. However, third parties are precluded from verifying the signer's signature since they do not possess the confirmed undeniable certificate and corresponding public key. Digital signatures can now be verified between two parties using a public key as in traditional digital signatures but which avoids verifiability by third parties.

Rosario Gennaro

Papers

Bounds on the Efficiency of Generic Cryptographic Constructions

Threshold Cryptosystems From Threshold Fully Homomorphic Encryption.

Lower Bounds on the Efficiency of Generic Cryptographic Constructions

Zero-Knowledge Contingent Payments Revisited: Attacks and Payments for Services

Undeniable certificates for digital signature verification