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Optimal Monotone Encodings Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists
 

Summary: Optimal Monotone Encodings
Noga Alon
Rani Hod
Abstract
Moran, Naor and Segev have asked what is the minimal r = r(n, k) for which there exists
an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size
up to k of {1, 2, . . . , n} to r bits. Monotone encodings are relevant to the study of tamper-
proof data structures and arise also in the design of broadcast schemes in certain communication
networks. To answer this question, we develop a relaxation of k-superimposed families, which we
call -fraction k-multi-user tracing ((k, )-FUT families). We show that r(n, k) = (k log(n/k))
by proving tight asymptotic lower and upper bounds on the size of (k, )-FUT families and by
constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit
construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an
additive constant.
1 Introduction
In their pursuit of history-independent schemes that use a write-once memory, motivated by cryp-
tographic applications, Moran et al. [14] have considered monotone injective functions that map
subsets of size up to k of [n] into 2[r] (all subsets of [r]), henceforth called (n, k)-monotone encodings
of length r, or ME(n, k, r). They have shown the existence of an (n, k)-monotone encoding of length
O(k log n log(n/k)) and raised the question of determining the minimal r = r(n, k) for which an

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics