Universal Hash Families Research Articles

Analysis of the amplitude form of the quantum hash function

In this article, the properties of quantum hash functions are further explored. Previous findings show that so-called small-bias sets (special subsets of the set of elements of a cyclic group) generate a “phase” quantum hash function. Here, it was proved that they also generate an “amplitude” quantum hash function. Namely, it turned out that constructing small-bias sets while generating amplitude quantum functions yields a well-balanced combination of the cryptographic properties of unidirectionality and collision resistance. As a corollary of the obtained theorem, a general statement about the generation of new amplitude quantum hash functions based on universal hash families and small-bias sets was proved.

Quantum hashing via ∈-universal hashing constructions and classical fingerprinting

In the paper, we define the concept of the quantum hash generator and offer design, which allows to build a large amount of different quantum hash functions. The construction is based on composition of classical ∈-universal hash family and a given family of functions-quantum hash generator.

On the relationships between perfect nonlinear functions and universal hash families

In this paper, the relationships between perfect nonlinear (in brief, PN) functions and optimal universal hash families are discussed. We point out the equivalence of constructions between them, i.e., from PN functions, one can obtain optimal universal hash families and vice versa. As an application of our construction, a message authentication code is proposed, which provides better resistance to substitution attack than a known construction given by Carlet et al. in 2006. More generally, the connections between functions with given differential uniformity and some universal hash families are studied.

A new multi-linear universal hash family

A new universal hash family is described which generalises a previously known multi-linear hash family. Messages are sequences over a finite field $${\mathbb{F}_q}$$ while keys are sequences over an extension field $${\mathbb{F}_{q^n}}$$ . A linear map $${\psi}$$ from $${\mathbb{F}_{q^n}}$$ to itself is used to compute the output digest. Of special interest is the case q = 2. For this case, we show that there is an efficient way to implement $${\psi}$$ using a tower field representation of $${\mathbb{F}_{q^n}}$$ . From a practical point of view, the focus of our constructions is small hardware and other resource constrained applications. For such platforms, our constructions compare favourably to previous work.

Practical unconditionally secure two-channel message authentication

We investigate unconditional security for message authentication protocols that are designed using two-channel cryptography. (Two-channel cryptography employs a broadband, insecure wireless channel and an authenticated, narrow-band manual channel at the same time.) We study both noninteractive message authentication protocols (NIMAPs) and interactive message authentication protocols (IMAPs) in this setting. First, we provide a new proof of nonexistence of nontrivial unconditionally secure NIMAPs. This proof consists of a combinatorial counting argument and is much shorter than the previous proof by Wang and Safavi-Naini, which was based on probability distribution arguments. We also prove a new result which holds in a weakened attack model. Further, we propose a generalization of an unconditionally secure 3-round IMAP due to Naor, Segev and Smith. The IMAP is based on two ϵ-Δ universal hash families. With a careful choice of parameters, our scheme improves that of Naor et al. Our scheme is very close to optimal for most parameter situations of practical interest. Finally, a variation of the 3-round IMAP is presented, in which only one hash family is required.

The power of primes: security of authentication based on a universal hash-function family

Message authentication codes (MACs) based on universal hash-function families are becoming increasingly popular due to their fast implementation. In this paper, we investigate a family of universal hash functions that has been appeared repeatedly in the literature and provide a detailed algebraic analysis for the security of authentication codes based on this universal hash family. In particular, the universal hash family under analysis, as appeared in the literature, uses operation in the finite field ℤ p . No previous work has studied the extension of such universal hash family when computations are performed modulo a non-prime integer n . In this work, we provide the first such analysis. We investigate the security of authentication when computations are performed over arbitrary finite integer rings ℤ n and derive an explicit relation between the prime factorization of n and the bound on the probability of successful forgery. More specifically, we show that the probability of successful forgery against authentication codes based on such a universal hash-function family is bounded by the reciprocal of the smallest prime factor of the modulus n .

A construction method for optimally universal hash families and its consequences for the existence of RBIBDs

We introduce a method for constructing optimally universal hash families and equivalently RBIBDs. As a consequence of our construction we obtain minimal optimally universal hash families, if the cardinalities of the universe and the range are powers of the same prime. A corollary of this result is that the necessary conditions for the existence of an RBIBD with parameters v , k , λ , namely v ≡ 0 ( mod k ) and λ ( v - 1 ) ≡ 0 ( mod k - 1 ) , are sufficient, if v and k are powers of the same prime. As an application of our construction, we show that the k-MAXCUT algorithm of Hofmeister and Lefmann [A combinatorial design approach to MAXCUT, Random Struct. Algorithms 9 (1996) 163–173] can be implemented such that it has a polynomial running time, in the case that the number of vertices and k are powers of the same prime.

Divide-and-concatenate: an architecture-level optimization technique for universal hash functions

The authors present an architectural optimization technique called divide-and-concatenate for hardware architectures of universal hash functions based on three observations: 1) the area of a multiplier and associated data path decreases quadratically and their speeds increase gradually as their operand size is reduced; 2) multiplication is at the core of universal hash functions and multipliers consume most of the area of universal hash function hardware; and 3) two universal hash functions are equivalent if they have the same collision-probability property. In the proposed approach, the authors divide a 2w-bit data path (with collision probability 2/sup -2w/) into two w-bit data paths (each with collision probability 2/sup -w/), apply one message word to these two w-bit data paths and concatenate their results to construct an equivalent 2w-bit data path (with a collision probability 2/sup -2w/). The divide-and-concatenate technique is complementary to all circuit-, logic-, and architecture-optimization techniques. The authors applied this technique on a linear congruential universal hash (LCH) family. When compared to the 100% overhead associated with duplicating a straightforward 32-bit LCH data path, the divide-and-concatenate approach that uses four equivalent 8-bit data paths yields a 101% increase in throughput with only 52% hardware overhead.

Almost k -Wise Independent Sample Spaces and Their Cryptologic Applications

An almost k -wise independent sample space is a small subset of m bit sequences in which any k bits are ``almost independent''. We show that this idea has close relationships with useful cryptologic notions such as multiple authentication codes (multiple A -codes), almost strongly universal hash families, almost k -resilient functions, almost correlation-immune functions, indistinguishable random variables and k -wise decorrelation bias of block ciphers. We use almost k -wise independent sample spaces to construct new efficient multiple A -codes such that the number of key bits grows linearly as a function of k (where k is the number of messages to be authenticated with a single key). This improves on the construction of Atici and Stinson \cite{AS96}, in which the number of key bits is Ω (k2) . We introduce the concepts of ? -almost k -resilient functions and almost correlation-immune functions, and give a construction for almost k -resilient functions that has parameters superior to k -resilient functions. We also point out the connection between almost k -wise independent sample spaces and pseudorandom functions that can be distinguished from truly random functions, by a distinguisher limited to k oracle queries, with only a small probability. Vaudenay \cite{Vaudenay99} has shown that such functions can be used to construct block ciphers with a small decorrelation bias. Finally, new bounds (necessary conditions) are derived for almost k -wise independent sample spaces, multiple A -codes and balanced ? -almost k -resilient functions.

Constructions of authentication codes from algebraic curves over finite fields

We present a new application of algebraic curves over finite fields to the constructions of universal hash families and unconditionally secure codes. We show that the constructions derived from the Garcia-Stichtenoth curves yield new classes of authentication codes and universal hash families which are substantially better than those previously known.

Some New Results on Key Distribution Patterns and BroadcastEncryption

This paper concerns methods by which a trusted authority can distribute keys and/or broadcast a message over a network, so that each member of a privileged subset of users can compute a specified key or decrypt the broadcast message. Moreover, this is done in such a way that no coalition is able to recover any information on a key or broadcast message they are not supposed to know. The problems are studied using the tools of information theory, so the security provided is unconditional (i.e., not based on any computational assumption). In a recent paper st95a, Stinson described a method of constructing key predistribution schemes by combining Mitchell-Piper key distribution patterns with resilient functions; and also presented a construction method for broadcast encryption schemes that combines Fiat-Naor key predistribution schemes with ideal secret sharing schemes. In this paper, we further pursue these two themes, providing several nice applications of these techniques by using combinatorial structures such as orthogonal arrays, perpendicular arrays, Steiner systems and universal hash families.

Locality-preserving hash functions for general purpose parallel computation

Consider the problem of efficiently simulating the shared-memory parallel random access machine (PRAM) model on massively parallel architectures with physically distributed memory. To prevent network congestion and memory bank contention, it may be advantageous to hash the shared memory address space. The decision on whether or not to use hashing depends on (1) the communication latency in the network and (2) the locality of memory accesses in the algorithm. We relate this decision directly to algorithmic issues by studying the complexity of hashing in the Block PRAM model of Aggarwal, Chandra, and Snir, a shared-memory model of parallel computation which accounts for communication locality. For this model, we exhibit a universal family of hash functions having optimal locality. The complexity of applying these hash functions to the shared address space of the Block PRAM (i.e., by permuting data elements) is asymptotically equivalent to the complexity of performing a square matrix transpose, and this result is best possible for all pairwise independent universal hash families. These complexity bounds provide theoretical evidence that hashing and randomized routing need not destroy communication locality, addressing an open question of Valiant.