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We present a communication-efficient zero-knowledge proof of knowledge for the factorization of Blum integers, a special class of integers of the form n = p q , where p and q are distinct prime numbers satisfying p ≡ q ≡ 3 mod 4 and p ≃ q ≃ n . Existing protocols for proving such statements often incur significant communication costs, especially when demonstrating that p and q are of nearly equal size. We leverage the MPC-in-the-head paradigm, a cryptographic technique that transforms secure multi-party computation protocols into efficient zero-knowledge proof systems. In our protocol, the prover uses additive sharing of p and q over the integers. This approach simplifies proving the size relationship p ≃ q ≃ n and the congruence p ≡ q ≡ 3 mod 4 without requiring costly range proofs. To verify the primality of p and q , we employ the Boneh-Franklin biprimality test. Our protocol achieves a significant reduction in communication complexity. For a 2048-bit integer n and 128-bit security, we construct a proof as small as 12.3 KB, with prover and verifier computational costs comparable to existing protocols that require over 131 KB.

We present attacks on a generalized subset-sum pseudorandom generator, which was proposed by von zur Gathen and Shparlinski in 2004. Our attacks rely on a sub-quadratic algorithm for solving a vectorial variant of the 3SUM problem, which is of independent interest. The attacks presented have complexities well below the brute-force attack, making the generators vulnerable. We provide a thorough analysis of the attacks and their complexities and demonstrate their practicality through implementations and experiments.

Abstract Public-key cryptographic primitives are time consuming for resource-constrained devices. A classical problem is to securely offload group exponentiations from a (comparatively) weak device—the client—to an untrusted more powerful device—the server. A delegation protocol must usually meet two security objectives: privacy—the exponent or the base should not be revealed to a passive adversary—and verifiability—a malicious server should not be able to make the client accept an invalid value as the result of the delegated computation. Most proposed protocols relies on a secret splitting of the exponent and the base, and a considerable amount of literature has been devoted to their analysis. Recently, Su et al. (Su, Q., Zhang, R. and Xue, R. (2020) Secure outsourcing algorithms for composite modular exponentiation based on single untrusted cloud. Comput. J., 63, 1271.) and Rangasamy and Kuppusamy (Rangasamy, J. and Kuppusamy, L. (2018) Revisiting Single-Server Algorithms for Outsourcing Modular Exponentiation. In Chakraborty, D. and Iwata, T. (eds), Progress in Cryptology - INDOCRYPT 2018: 19th International Conference in Cryptology in India, New Delhi, India, December 912, Vol. 11356, Lecture Notes in Computer Science. Springer, Heidelberg, Germany, pp. 320. proposed outsourcing protocols for modular exponentiations. They claim that their protocols achieve security (privacy and verifiability). We show that these claims are flawed and that their schemes are broken beyond repair. They remain insecure even if one increases significantly the proposed parameters (and consequently the protocols computational and communication complexities). Our attacks rely on standard lattice-based cryptanalytic techniques, namely the Coppersmith methods to find small integer zeroes of modular multivariate polynomials and simultaneous Diophantine approximation methods for the so-called approximate greatest common divisor problem.

We consider multiparty information-theoretic private protocols, and specifically their randomness complexity. The randomness complexity of private protocols is of interest both because random bits are considered a scarce resource and because of the relation between that complexity measure and other complexity measures of boolean functions such as the circuit size or the sensitivity of the function being computed [Kushilevitz, Ostrovsky, and Rosén, J. Comput. Syst. Sci., 58 (1999), pp. 129--136] and [Gál and Rosén, SIAM J. Comput., 31 (2002), pp. 1424--1437]. More concretely, we consider the randomness complexity of the basic Boolean function \tt and, that serves as a building block in the design of many private protocols. We show that \tt and cannot be privately computed using a single random bit, thus giving the first nontrivial lower bound on the 1-private randomness complexity of an explicit Boolean function, $f: \{0,1\}^n \rightarrow \{0,1\}$. We further show that and, on any number of inputs $n$ (one input bit per player), can be privately computed using 8 random bits (and 7 random bits in the special case of $n=3$ players), improving the upper bound of 73 random bits implicit in [Kushilevitz, Ostrovsky, and Rosén, J. Comput. Syst. Sci., 58 (1999), pp. 129--136]. Together with our lower bound, we thus approach the exact determination of the randomness complexity of \tt and. To the best of our knowledge, the exact randomness complexity of private computation is not known for any explicit function (except for \tt xor, which is 1-random, and for several degenerate functions).

Internet-of-Things (IoT) devices have grown in popularity over the past few years. The RSA public-key cryptographic primitive is time consuming for resource-constrained IoT. Recently, Zhang et al. proposed a two-party outsourcing protocol between a client and a server for RSA decryption in IoT. It relies on the Chinese remainder theorem as proposed by Quisquater and Couvreur in 1982 and is very efficient. We show that their protocol does not achieve the claimed security guarantees: 1) the (secret) decryption exponent, the plaintext, and the factorization of the RSA modulus are revealed to a passive adversary and 2) a malicious server can make the client accept an (invalid) value of its choice as the result of the delegated computation.

We analyze the security and the efficiency of interactive protocols where a client wants to delegate the computation of an RSA signature given a public key, a public message and the secret signing exponent. We consider several protocols where the secret exponent is split using some algebraic decomposition. We first provide an exhaustive analysis of the delegation protocols in which the client outsources a single RSA exponentiation to the server. We then revisit the security of the protocols RSA-S1 and RSA-S2 that were proposed by Matsumoto, Kato and Imai in 1988. We present an improved lattice-based attack on RSA-S1 and we propose a simple variant of this protocol that provides better efficiency for the same security level. Eventually, we present the first attacks on the protocol RSA-S2 that employs the Chinese Remainder Theorem to speed up the client’s computation. The efficiency of our (heuristic) attacks has been validated experimentally.

Anonymous credential systems enable users to authenticate themselves in a privacy-preserving manner. At the conference ESORICS 2016, Kaaniche and Laurent presented an anonymous certification scheme based on a new attribute based signature. In this note, we provide several attacks on their scheme.

A pseudo-random number generator (PRNG) is a deterministic algorithm that produces numbers whose distribution is indistinguishable from uniform. A formal security model for PRNGs with input was proposed in 2005 by Barak and Halevi (BH). This model involves an internal state that is refreshed with a (potentially biased) external random source, and a cryptographic function that outputs random numbers from the continually internal state. In this work we extend the BH model to also include a new security property capturing how it should accumulate the entropy of the input data into the internal state after state compromise. This property states that a good PRNG should be able to eventually recover from compromise even if the entropy is injected into the system at a very slow pace, and expresses the real-life expected behavior of existing PRNG designs. Unfortunately, we show that neither the model nor the specific PRNG construction proposed by BH meet this new property, despite meeting a weaker robustness notion introduced by BH. From a practical side, we give a precise assessment of the Linux PRNGs, /dev/random and /dev/urandom. In particular, we show attacks proving that these PRNGs are not robust according to our definition, due to vulnerabilities in their entropy estimator and their internal mixing function. Finally, we propose a simple PRNG construction that is provably robust in our new and stronger adversarial model and we show that it is more efficient than the Linux PRNGs. We therefore recommend to use this construction whenever a PRNG with input is used for cryptography.

Blind signatures allow users to obtain signatures on messages hidden from the signer; moreover, the signer cannot link the resulting message/signature pair to the signing session. This paper presents blind signature schemes, in which the number of interactions between the user and the signer is min imal and whose blind signatures are short. Our schemes are defined over bilinear groups and are proved secure in the common-reference-string model without random oracles and under standard assumptions: CDH and the decision-linear assumption. (We also give variants over asymmetric groups based on similar assumptions.) The blind signatures are Waters signatures, which consist of 2 group elements. Moreover, we instantiate partially blind signatures, where the message consists of a part hidden from the signer and a commonly known public part, and schemes achieving perfect blindness. We propose new variants of blind signatures, such as signer-friendly partially blind signatures, where the public part can be chosen by the signer without prior agreement, 3-party blind signatures, as well as blind signatures on multiple aggregated messages provided by independent sources. We also extend Waters signatures to non-binary alphabets by proving a new result on the underlying hash function.

In 1998, Blaze, Bleumer and Strauss introduced a cryptographic primitive called proxy re-encryption in which a proxy can transform-without seeing the plaintext-a ciphertext encrypted under one key into an encryption of the same plaintext under another key. The concept has recently drawn renewed interest. Notably, Canetti and Hohenberger showed how to properly define (and realize) chosen-ciphertext security for the primitive. Their system is bidirectional as the translation key allows converting ciphertexts in both directions. This paper presents the first unidirectional proxy re-encryption schemes with chosen-ciphertext security in the standard model (i.e., without the random oracle idealization). The first system provably fits a unidirectional extension of the Canetti-Hohenberger security model. As a second contribution, the paper considers a more realistic adversarial model where attackers may choose dishonest users' keys on their own. It is shown how to modify the first scheme to achieve security in the latter scenario. At a moderate expense, the resulting system provides additional useful properties such as non-interactive temporary delegations. Both constructions are efficient and rely on mild complexity assumptions in bilinear groups. Like the Canetti-Hohenberger scheme, they meet a relaxed flavor of chosen-ciphertext security introduced by Canetti, Krawczyk and Nielsen.

At Crypto'07, Goyal introduced the concept of Accountable Authority Identity-Based Encryption (A-IBE) as a convenient tool to reduce the amount of trust in authorities in Identity-Based Encryption. In this model, if the Private Key Generator (PKG) maliciously re-distributes users' decryption keys, it runs the risk of being caught and prosecuted. Goyal proposed two constructions: the first one is efficient but can only trace well-formed decryption keys to their source; the second one allows tracing obfuscated decryption boxes in a model (called weak black-box model) where cheating authorities have no decryption oracle. The latter scheme is unfortunately far less efficient in terms of decryption cost and ciphertext size. The contribution of this paper is to describe a new construction that combines the efficiency of Goyal's first proposal with a simple weak black-box tracing mechanism. The proposed scheme is the first A-IBE that meets all security properties (although traceability is only guaranteed in the weak black-box model) in the adaptive-ID sense.

In 1998, Blaze, Bleumer, and Strauss suggested a cryptographic primitive termed proxy re-signature in which a proxy transforms a signature computed under Alice's secret key into one from Bob on the same message. The proxy is only semi-trusted in that it cannot learn any signing key or sign arbitrary messages on behalf of Alice or Bob. At CCS 2005, Ateniese and Hohenberger revisited this primitive by providing appropriate security definitions and efficient constructions in the random oracle model. Nonetheless, they left open the problem of constructing a multi-use unidirectional scheme where the proxy is only able to translate in one direction and signatures can be re-translated several times. This paper provides the first steps towards efficiently solving this problem, suggested for the first time 10 years ago, and presents the first multi-hop unidirectional proxy re-signature schemes. Although our proposals feature a linear signature size in the number of translations, they are the first multi-use realizations of the primitive that satisfy the requirements of the Ateniese-Hohenberger security model. The first scheme is secure in the random oracle model. Using the same underlying idea, it readily extends into a secure construction in the standard model (i.e. the security proof of which avoids resorting to the random oracle idealization). Both schemes are computationally efficient but require newly defined Diffie-Hellman-like assumptions in bilinear groups.