On the Evaluation of Powers and Monomials

Let $y_1 , \cdots ,y_p$ be monomials over the indeterminates $x_1 , \cdots ,x_q$. For every $y = (y_1 , \cdots ,y_p )$ there is some minimum number $L(y)$ of multiplications sufficient to compute $y_1 , \cdots ,y_p$ from $x_1 , \cdots ,x_q$ and the identity 1. Let $L(p,q,N)$ denote the maximum of $L(y)$ over all $y$ for which the exponent of any indeterminate in any monomial is at most $N$. We show that if $p = (N + 1^{o(q)} )$ and $q = (N + 1^{o(p)} )$, then $L(p,q,N) = \min \{ p,q\} \log N + H/\log H + o(H /\log H)$, where $H = pq\log (N + 1)$ and all logarithms have base 2

An identification scheme based on sparse polynomials

This is a preprint of a book chapter published in Lecture Notes in Computer Science,1751, Springer-Verlag, Berlin (2000). The original publication is available at www.springerlink.com.This paper gives a new example of exploiting the idea of using polynomials with restricted coefficients over finite fields and rings to construct reliable cryptosystems and identification schemes

Cryptographic applications of sparse polynomials over finite rings

This is a preprint of a book chapter published in Lecture Notes in Computer Science, 2015, Springer-Verlag, Berlin (2001). The original publication is available at www.springerlink.com.This paper gives new examples that exploit the idea of using sparse polynomials with restricted coefficients over a finite ring for designing fast, reliable cryptosystems and identification schemes

Preventing Denial of Service Attacks in IoT Networks through Verifiable Delay Functions

Permissionless distributed ledgers provide a promising approach to deal with the Internet of Things (IoT) paradigm. Since IoT devices mostly generate data transactions and micropayments, distributed ledgers that use fees to regulate the network access are not an optimal choice. In this paper, we study a feeless architecture developed by IOTA and designed specifically for the IoT. Due to the lack of fees, malicious nodes can exploit this feature to generate an unbounded number of transactions and perform a denial of service attacks. We propose to mitigate these attacks through verifiable delay functions. These functions, which are non-parallelizable, hard to compute, and easy to verify, have been formulated only recently. In our work, we design a denial of service prevention mechanism which addresses network heterogeneity, limited node computational capabilities, and hardware-specific implementation optimizations. Verifiable delay functions have mostly been studied from a theoretical point of view, but little has been done in tangible applications. Hence, this paper can be considered as a pioneer work in the field, since it builds a bridge between this theoretical mathematical framework and a real-world problem

Towards Faster Cryptosystems, II

http://www.math.missouri.edu/~bbanks/papers/index.htmlWe discuss three cryptosystems, NTRU, SPIFI , and ENROOT, that are based on the use of polynomials with restricted coefficients

The complexity of implementation of a system of monomials in two variables by composition circuits

Исследуется сложность реализации систем мономов схемами композиции. Для этой вычислительной модели установлена сложность реализации системы из p мономов от двух переменных с точностью до слагаемого порядка р. Показано, что для схем композиции, в отличие от других моделей, асимптотика роста сложности реализации системы из ограниченного числа мономов от двух переменных, вообще говоря, не определяется сложностью никакого несобственного подмножества мономов

Lempel-Ziv Parsing for Sequences of Blocks

The Lempel-Ziv parsing (LZ77) is a widely popular construction lying at the heart of many compression algorithms. These algorithms usually treat the data as a sequence of bytes, i.e., blocks of fixed length 8. Another common option is to view the data as a sequence of bits. We investigate the following natural question: what is the relationship between the LZ77 parsings of the same data interpreted as a sequence of fixed-length blocks and as a sequence of bits (or other “elementary” letters)? In this paper, we prove that, for any integer b>1, the number z of phrases in the LZ77 parsing of a string of length n and the number zb of phrases in the LZ77 parsing of the same string in which blocks of length b are interpreted as separate letters (e.g., b=8 in case of bytes) are related as zb=O(bzlognz). The bound holds for both “overlapping” and “non-overlapping” versions of LZ77. Further, we establish a tight bound zb=O(bz) for the special case when each phrase in the LZ77 parsing of the string has a “phrase-aligned” earlier occurrence (an occurrence equal to the concatenation of consecutive phrases). The latter is an important particular case of parsing produced, for instance, by grammar-based compression methods

Lempel-Ziv Parsing for Sequences of Blocks

The Lempel-Ziv parsing (LZ77) is a widely popular construction lying at the heart of many compression algorithms. These algorithms usually treat the data as a sequence of bytes, i.e., blocks of fixed length 8. Another common option is to view the data as a sequence of bits. We investigate the following natural question: what is the relationship between the LZ77 parsings of the same data interpreted as a sequence of fixed-length blocks and as a sequence of bits (or other “elementary” letters)? In this paper, we prove that, for any integer b>1, the number z of phrases in the LZ77 parsing of a string of length n and the number zb of phrases in the LZ77 parsing of the same string in which blocks of length b are interpreted as separate letters (e.g., b=8 in case of bytes) are related as zb=O(bzlognz). The bound holds for both “overlapping” and “non-overlapping” versions of LZ77. Further, we establish a tight bound zb=O(bz) for the special case when each phrase in the LZ77 parsing of the string has a “phrase-aligned” earlier occurrence (an occurrence equal to the concatenation of consecutive phrases). The latter is an important particular case of parsing produced, for instance, by grammar-based compression methods