Universal Hash Function Family Research Articles

Authentication of variable length messages in quantum key distribution

Authentication plays a critical role in the security of quantum key distribution (QKD) protocols. We propose using Polynomial Hash and its variants for authentication of variable length messages in QKD protocols. Since universal hashing is used not only for authentication in QKD but also in other steps in QKD like error correction and privacy amplification, and also in several other areas of quantum cryptography, Polynomial Hash and its variants as the most efficient universal hash function families can be used in these important steps and areas, as well. We introduce and analyze several efficient variants of Polynomial Hash and, using deep results from number theory, prove that each variant gives an ε-almost-Δ-universal family of hash functions. We also give a general method for transforming any such family to an ε-almost-strongly universal family of hash functions. The latter families can then, among other applications, be used in the Wegman–Carter MAC construction which has been shown to provide a universally composable authentication method in QKD protocols. As Polynomial Hash has found many applications, our constructions and results are potentially of interest in various areas.

Everlasting security of quantum key distribution with 1K-DWCDM and quadratic hash

Quantum key distribution (QKD) offers a very strong property called everlasting security, which says if authentication is unbroken during the execution of QKD, the generated key remains information-theoretically secure indefinitely. For this purpose, we propose the use of certain universal hashing based MACs for use in QKD, which are fast, very efficient with key material, and are shown to be highly secure. Universal hash functions are ubiquitous in computer science with many applications ranging from quantum key distribution and information security to data structures and parallel computing. In QKD, they are used at least for authentication, error correction, and privacy amplification. Using results from Cohen [Duke Math. J., 1954], we also construct some new families of $\varepsilon$-almost-$\Delta$-universal hash function families which have much better collision bounds than the well-known Polynomial Hash. Then we propose a general method for converting any such family to an $\varepsilon$-almost-strongly universal hash function family, which makes them useful in a wide range of applications, including authentication in QKD.

On an Almost-Universal Hash Function Family with Applications to Authentication and Secrecy Codes

Universal hashing, discovered by Carter and Wegman in 1979, has many important applications in computer science. MMH[Formula: see text], which was shown to be [Formula: see text]-universal by Halevi and Krawczyk in 1997, is a well-known universal hash function family. We introduce a variant of MMH[Formula: see text], that we call GRDH, where we use an arbitrary integer [Formula: see text] instead of prime [Formula: see text] and let the keys [Formula: see text] satisfy the conditions [Formula: see text] ([Formula: see text]), where [Formula: see text] are given positive divisors of [Formula: see text]. Then via connecting the universal hashing problem to the number of solutions of restricted linear congruences, we prove that the family GRDH is an [Formula: see text]-almost-[Formula: see text]-universal family of hash functions for some [Formula: see text] if and only if [Formula: see text] is odd and [Formula: see text] [Formula: see text]. Furthermore, if these conditions are satisfied then GRDH is [Formula: see text]-almost-[Formula: see text]-universal, where [Formula: see text] is the smallest prime divisor of [Formula: see text]. Finally, as an application of our results, we propose an authentication code with secrecy scheme which strongly generalizes the scheme studied by Alomair et al. [J. Math. Cryptol. 4 (2010) 121–148], and [J.UCS 15 (2009) 2937–2956].

MMH⁎ with arbitrary modulus is always almost-universal

Universal hash functions, discovered by Carter and Wegman in 1979, are of great importance in computer science with many applications. MMH⁎ is a well-known △-universal hash function family, based on the evaluation of a dot product modulo a prime. In this paper, we introduce a generalization of MMH⁎, that we call GMMH⁎, using the same construction as MMH⁎ but with an arbitrary integer modulus n>1, and show that GMMH⁎ is 1p-almost-△-universal, where p is the smallest prime divisor of n. This bound is tight.

On a family of universal hash functions

Построено новое семейство сжимающих отображений с помощью нескольких биективных преобразований и отображений с заданными свойствами. Доказывается, что все функции из данного семейства являются универсальными функциями хеширования. Приводятся явные примеры функций из рассматриваемого семейства, которые могут использоваться при построении прикладных криптографических алгоритмов.

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 .

Variationally universal hashing

The strongest well-known measure for the quality of a universal hash-function family H is its being ε-strongly universal, which measures, for randomly chosen h ∈ H , one's inability to guess h ( m ′ ) even if h ( m ) is known for some m ≠ m ′ . We give example applications in which this measure is too weak, and we introduce a stronger measure for the quality of a hash-function family, ε-variationally universal, which measures one's inability to distinguish h ( m ′ ) from a random value even if h ( m ) is known for some m ≠ m ′ . We explain the utility of this notion and provide an approach for constructing efficiently computable ε-VU hash-function families.

Bounds on the OBDD-size of integer multiplication via universal hashing

Bryant [On the complexity of VLSI implementations and graph representations of boolean functions with applications to integer multiplication, IEEE Trans. Comput. 40 (1991) 205–213] has shown that any OBDD for the function MUL n - 1 , n , i.e. the middle bit of the n-bit multiplication, requires at least 2 n / 8 nodes. In this paper a stronger lower bound of essentially 2 n / 2 / 61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128-bit multiplication circuits using OBDD -techniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDD -nodes. Further, a first non-trivial upper bound of 7 / 3 · 2 4 n / 3 for the OBDD -size of MUL n - 1 , n is provided.

Bucket Hashing and Its Application to Fast Message Authentication

We introduce a new technique for constructing a family of universal hash functions. At its center is a simple metaphor: to hash a string x , cast each of its words into a small number of buckets; xor the contents of each bucket; then collect up all the buckets' contents. Used in the context of Wegman—Carter authentication, this style of hash function provides a fast approach for software message authentication.

Communication-Space Tradeoffs for Unrestricted Protocols

This paper introduces communicating branching programs and develops a general technique for demonstrating communication-space tradeoffs for pairs of communicating branching programs. This technique is then used to prove communication-space tradeoffs for any pair of communicating branching programs that hashes according to a universal family of hash functions. Other tradeoffs follow from this result. As an example, any pair of communicating Boolean branching programs that computes matrix-vector products over ${\text{GF}}(2)$ requires communication-space product $\Omega (n^2 )$, provided the space used is $o({n / {\log n}})$. These are the first examples of communication-space tradeoffs on a completely general model of communicating processes.