Unavoidable Regularities in Long Words with Bounded Number of Symbol Occurrences

Abstract

Traditionally in combinatorics on words one studies unavoidable regularities that appear in sufficiently long strings over a fixed size alphabet. Inspired by permutation problems originating from information security, another viewpoint is taken in this paper. We focus on combinatorial properties of long words in which the number of occurrences of any symbol is restritced by a fixed given constant. More precisely, we show that for all positive integers m and q there exists the least positive integer N(m,q) which is smaller than $m^{2^{q-1}}$ and satifies the following: If α is a word such that

(i) |alph(α)| ≥ N(m,q) (i.e., the cardinality of the alphabet of α is at least N(m,q)); and

(ii) |α| _a  ≤ q for each a ∈ alph(α) (i.e., the number of occurrences of any symbol of alph(α) in α is at most q),

then there exist a set A ⊆ alph(α) of cardinality |A| = m, an integer p ∈ {1,2,…,q}, and permutations σ ₁,σ ₂,…,σ _p : {1,2,…,m} → {1,2,…,m} for which

$$ \pi_A(\alpha)\in a_{\sigma_1(1)}^+\cdots a_{\sigma_1(m)}^+a_{\sigma_2(1)}^+\cdots a_{\sigma_2(m)}^+\cdots a_{\sigma_p(1)}^+\cdots a_{\sigma_p(m)}^+\ \ . $$

Here A = {a ₁,a ₂,…,a _m } and π _A is the projection morphism from alph(α)^∗ into A ^∗. Finally, we demonstrate how problems such as the one above are connected to constructing multicollision attacks on so called generalized iterated hash functions.