Search

In this research paper, we address the Distinct Elements estimation problem in the context of streaming algorithms. The problem involves estimating the number of distinct elements in a given data stream $\mathcal{A} = (a_1, a_2,\ldots, a_m)$, where $a_i \in \{1, 2, \ldots, n\}$. Over the past four decades, the Distinct Elements problem has received considerable attention, theoretically and empirically, leading to the development of space-optimal algorithms. A recent sampling-based algorithm proposed by Chakraborty et al.[11] has garnered significant interest and has even attracted the attention of renowned computer scientist Donald E. Knuth, who wrote an article on the same topic [6] and called the algorithm CVM. In this paper, we thoroughly examine the algorithms (referred to as CVM1, CVM2 in [11] and DonD, DonD' in [6]. We first unify all these algorithms and call them cutoff-based algorithms. Then we provide an approximation and biasedness analysis of these algorithms., Comment: We have provided an unbiased analysis (using exactly the same idea as the previous version) for the continuous score distribution instead of the discrete version

In CRYPTO 2018, Russell et al introduced the notion of crooked indifferentiability to analyze the security of a hash function when the underlying primitive is subverted. They showed that the $n$-bit to $n$-bit function implemented using enveloped XOR construction (\textsf{EXor}) with $3n+1$ many $n$-bit functions and $3n^2$-bit random initial vectors (iv) can be proven secure asymptotically in the crooked indifferentiability setting. -We identify several major issues and gaps in the proof by Russel et al, We show that their proof can achieve security only when the adversary is restricted to make queries related to a single message. - We formalize new technique to prove crooked indifferentiability without such restrictions. Our technique can handle function dependent subversion. We apply our technique to provide a revised proof for the \textsf{EXor} construction. - We analyze crooked indifferentiability of the classical sponge construction. We show, using a simple proof idea, the sponge construction is a crooked-indifferentiable hash function using only $n$-bit random iv. This is a quadratic improvement over the {\sf EXor} construction and solves the main open problem of Russel et al.

The $r$-rounds Even-Mansour block cipher is a generalization of the well known Even-Mansour block cipher to $r$ iterations. Attacks on this construction were described by Nikoli\'c et al. and Dinur et al., for $r = 2, 3$. These attacks are only marginally better than brute force, but are based on an interesting observation (due to Nikoli\'c et al.): for a "typical" permutation $P$, the distribution of $P(x) \oplus x$ is not uniform. This naturally raises the following question. Call permutations for which the distribution of $P(x) \oplus x$ is uniform "balanced." Is there a sufficiently large family of balanced permutations, and what is the security of the resulting Even-Mansour block cipher? We show how to generate families of balanced permutations from the Luby-Rackoff construction, and use them to define a $2n$-bit block cipher from the $2$-rounds Even-Mansour scheme. We prove that this cipher is indistinguishable from a random permutation of $\{0, 1\}^{2n}$, for any adversary who has oracle access to the public permutations and to an encryption/decryption oracle, as long as the number of queries is $o (2^{n/2})$. As a practical example, we discuss the properties and the performance of a $256$-bit block cipher that is based on our construction, and uses AES as the public permutation.