International Association
for Cryptologic Research

Symmetric-key primitives designed over the prime field Fp with odd characteristics, rather than the traditional Fn2 , are becoming the most popular choice for MPC/FHE/ZK-protocols for better efficiencies. However, the security of Fp is less understood as there are highly nontrivial gaps when extending the cryptanalysis tools and experiences built on Fn2 in the past few decades to Fp.At CRYPTO 2015, Sun et al. established the links among impossible differential, zero-correlation linear, and integral cryptanalysis over Fn2 from the perspective of distinguishers. In this paper, following the definition of linear correlations over Fp by Baignères, Stern and Vaudenay at SAC 2007, we successfully establish comprehensive links over Fp, by reproducing the proofs and offering alternatives when necessary. Interesting and important differences between Fp and Fn2 are observed.- Zero-correlation linear hulls can not lead to integral distinguishers for some cases over Fp, while this is always possible over Fn2proven by Sun et al..- When the newly established links are applied to GMiMC, its impossible differential, zero-correlation linear hull and integral distinguishers can be increased by up to 3 rounds for most of the cases, and even to an arbitrary number of rounds for some special and limited cases, which only appeared in Fp. It should be noted that all these distinguishers do not invalidate GMiMC’s security claims.The development of the theories over Fp behind these links, and properties identified (be it similar or different) will bring clearer and easier understanding of security of primitives in this emerging Fp field, which we believe will provide useful guides for future cryptanalysis and design.

Recent practical applications using advanced cryptographic protocols such as multi-party computation (MPC) and zero-knowledge proofs (ZKP) have prompted a range of novel symmetric primitives described over large finite fields, characterized as arithmetization-oriented (AO) ciphers. Such designs, aiming to minimize the number of multiplications over fields, have a high risk of being vulnerable to algebraic attacks, especially to the higher-order differential attack. Thus, it is significant to carefully evaluate the growth of the algebraic degree. However, degree estimation for AO ciphers has been a challenge for cryptanalysts due to the lack of general and accurate methods. In this paper, we extend the division property, a state-of-the-art frame- work for finding the upper bound of the algebraic degree over binary fields, to the scope of F2n, called general monomial prediction. It is a generic method to detect the algebraic degree for AO ciphers, even applicable to Feistel ciphers which have no better bounds than the trivial exponential one. In the general monomial prediction, our idea is to evaluate whether the polynomial representation of a block cipher contains some specific monomials. With a deep investigation of the arithmetical feature, we introduce the propagation rules of monomials for field-based operations, which can be efficiently modeled using the bit-vector theory of SMT. Then the new searching tool for degree estimation can be constructed due to the relationship between the algebraic degree and the exponents of monomials. We apply our new framework to some important AO ciphers, including Feistel MiMC, GMiMC, and MiMC. For Feistel MiMC, we show that the algebraic degree grows significantly slower than the native exponential bound. For the first time, we present a secret-key higher-order differential distinguisher for up to 124 rounds, much better than the 83-round distinguisher for Feistel MiMC permutation proposed at CRYPTO 2020. We also exhibit a full-round zero-sum distinguisher with a data complexity of 2^{251}. Our method can be further extended for the general Feistel structure with more branches and exhibit higher-order differential distinguishers against the practical instance of GMiMC for up to 50 rounds. For MiMC in SP-networks, our results correspond to the exact algebraic degree proved by Bouvier. We also point out that the number of rounds in MiMC specification is not necessary to guarantee the security against the higher-order differential attack for MiMC-like schemes with different exponents. The investigation of different exponents provides some guidance on the cipher design.

Identity-based broadcast encryption (IBBE) is an effective method to protect the data security and privacy in multi-receiver scenarios, which can make broadcast encryption more practical. This paper further expands the study of scalable revocation methodology in the setting of IBBE, where a key authority releases a key update material periodically in such a way that only non-revoked users can update their decryption keys. Following the binary tree data structure approach, a concrete instantiation of revocable IBBE scheme is proposed using asymmetric pairings of prime order bilinear groups. Moreover, this scheme can withstand decryption key exposure, which is proven to be semi-adaptively secure under chosen plaintext attacks in the standard model by reduction to static complexity assumptions. In particular, the proposed scheme is very efficient both in terms of computation costs and communication bandwidth, as the ciphertext size is constant, regardless of the number of recipients. To demonstrate the practicality, it is further implemented in Charm, a framework for rapid prototyping of cryptographic primitives.

The consensus protocol underlying Bitcoin (the blockchain) works remarkably well in practice. However proving its security in a formal setting has been an elusive goal. A recent analytical result by Pass, Seeman and shelat indicates that an idealized blockchain is indeed secure against attacks in an asynchronous network where messages are maliciously delayed by at most $$\varDelta \ll 1/np$$, with n being the number of miners and p the mining hardness. This paper improves upon the result by showing that if appropriate inconsistency tolerance is allowed the blockchain can withstand even more powerful external attacks in the honest miner setting. Specifically we prove that the blockchain is secure against long delay attacks with $$\varDelta \ge 1/np$$ in an asynchronous network.