22 research outputs found
SL2 homomorphic hash functions: Worst case to average case reduction and short collision search
We study homomorphic hash functions into SL(2,q), the 2x2 matrices with determinant 1 over the
field with elements.
Modulo a well supported number theoretic hypothesis, which holds in particular for concrete
homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions:
upto a logarithmic factor, a random homomorphism is as secure as _any_ concrete homomorphism.
For a family of homomorphisms containing several concrete proposals in the literature,
we prove that collisions of length O(log(q)) can be found in running time O(sqrt(q)).
For general homomorphisms we offer an algorithm that, heuristically and according to experiments,
in running time O(sqrt(q)) finds collisions of length O(log(q)) for q even, and length O(log^2(q)/loglog(q))$ for arbitrary q.
While exponetial time, our algorithms are faster in practice than all earlier generic algorithms,
and produce much shorter collisions
Cayley Graphs of Semigroups and Applications to Hashing
In 1994, Tillich and Zemor proposed a scheme for a family of hash functions that uses products of matrices in groups of the form . In 2009, Grassl et al. developed an attack to obtain collisions for palindromic bit strings by exploring a connection between the Tillich-Zemor functions and maximal length chains in the Euclidean algorithm for polynomials over .
In this work, we present a new proposal for hash functions based on Cayley graphs of semigroups. In our proposed hash function, the noncommutative semigroup of linear functions under composition is considered as platform for the scheme. We will also discuss its efficiency, pseudorandomness and security features.
Furthermore, we generalized the Fit-Florea and Matula\u27s algorithm (2004) that finds the discrete logarithm in the multiplicative group of integers modulo by establishing a connection between semi-primitive roots modulo where and the logarithmic base used in the algorithm
Compositions of linear functions and applications to hashing
Cayley hash functions are based on a simple idea of using a pair of
(semi)group elements, and , to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the
natural way, by using multiplication of elements in the (semi)group.
In this paper, we focus on hashing with linear functions of one
variable over F_p. The corresponding hash functions are very efficient. In particular, we show that hashing a bit string of length n with our method requires, in general, at most 2n multiplications in F_p, but
with particular pairs of linear functions that we suggest, one does
not need to perform any multiplications at all. We also give explicit lower bounds on the length of collisions for hash functions corresponding to these particular pairs of linear functions over F_p
Cayley hashing with cookies
Cayley hash functions are based on a simple idea of using a pair of semigroup elements, and , to hash the 0 and 1 bit, respectively, and then to hash an arbitrary bit string in the natural way, by using multiplication of elements in the semigroup. The main advantage of Cayley hash functions compared to, say, hash functions in the SHA family is that when an already hashed document is amended, one does not have to hash the whole amended document all over again, but rather hash just the amended part and then multiply the result by the hash of the original document. Some authors argued that this may be a security hazard, specifically that this property may facilitate finding a second preimage by splitting a long bit string into shorter pieces. In this paper, we offer a way to get rid of this alleged disadvantage and keep the advantages at the same time. We call this method ``Cayley hashing with cookies using terminology borrowed from the theory of random walks in a random environment. For the platform semigroup, we use matrices over
Applications of Finite non-Abelian Simple Groups to Cryptography in the Quantum Era
The theory of finite simple groups is a (rather unexplored) area likely to provide interesting computational problems and modelling tools useful in a cryptographic context. In this note, we review some applications of finite non-abelian simple groups to cryptography and discuss different scenarios in which this theory is clearly central, providing the relevant definitions to make the material accessible to both cryptographers and group theorists, in the hope of stimulating further interaction between these two (non-disjoint) communities. In particular, we look at constructions based on various group-theoretic factorization problems, review group theoretical hash functions, and discuss fully homomorphic encryption using simple groups. The Hidden Subgroup Problem is also briefly discussed in this context
Securing Update Propagation with Homomorphic Hashing
In database replication, ensuring consistency when propagating updates is a
challenging and extensively studied problem. However, the problem of securing
update propagation against malicious adversaries has received less attention in
the literature. This consideration becomes especially relevant when sending
updates across a large network of untrusted peers.
In this paper we formalize the problem of secure update propagation and
propose a system that allows a centralized distributor to propagate signed
updates across a network while adding minimal overhead to each transaction.
We show that our system is secure (in the random oracle model) against an
attacker who can maliciously modify any update and its signature. Our approach
relies on the use of a cryptographic primitive known as homomorphic
hashing, introduced by Bellare, Goldreich, and Goldwasser.
We make our study of secure update propagation concrete with an instantiation of
the lattice-based homomorphic hash LtHash of Bellare and Miccancio. We
provide a detailed security analysis of the collision resistance of LtHash,
and we implement Lthash using a selection of parameters that gives at least
200 bits of security. Our implementation has been deployed to secure update
propagation in production at Facebook, and is included in the Folly open-source
library
International Symposium on Mathematics, Quantum Theory, and Cryptography
This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography
International Symposium on Mathematics, Quantum Theory, and Cryptography
This open access book presents selected papers from International Symposium on Mathematics, Quantum Theory, and Cryptography (MQC), which was held on September 25-27, 2019 in Fukuoka, Japan. The international symposium MQC addresses the mathematics and quantum theory underlying secure modeling of the post quantum cryptography including e.g. mathematical study of the light-matter interaction models as well as quantum computing. The security of the most widely used RSA cryptosystem is based on the difficulty of factoring large integers. However, in 1994 Shor proposed a quantum polynomial time algorithm for factoring integers, and the RSA cryptosystem is no longer secure in the quantum computing model. This vulnerability has prompted research into post-quantum cryptography using alternative mathematical problems that are secure in the era of quantum computers. In this regard, the National Institute of Standards and Technology (NIST) began to standardize post-quantum cryptography in 2016. This book is suitable for postgraduate students in mathematics and computer science, as well as for experts in industry working on post-quantum cryptography