 
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 ksuperimposed families, which we
call fraction kmultiuser 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 historyindependent schemes that use a writeonce 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
