Large-scale decentralized learning frameworks such as federated learning (FL), require both communication efficiency and strong data security, motivating the study of secure aggregation (SA). While information-theoretic SA is well understood in centralized and fully connected networks, its extension to decentralized networks with limited local connectivity remains largely unexplored. This paper introduces \emph{topological secure aggregation} (TSA), which studies one-shot, information-theoretically secure aggregation of neighboring users' inputs over arbitrary network topologies. We develop a unified linear design framework that characterizes TSA achievability through the spectral properties of the communication graph, specifically the kernel of a diagonally modulated adjacency matrix. For several representative classes of $d$-regular graphs including ring, prism and complete topologies, we establish the optimal communication and secret key rate region. In particular, to securely compute one symbol of the neighborhood sum, each user must (i) store at least one key symbol, (ii) broadcast at least one message symbol, and (iii) collectively, all users must hold at least $d$ i.i.d. key symbols. Notably, this total key requirement depends only on the \emph{neighborhood size} $d$, independent of the network size, revealing a fundamental limit of SA in decentralized networks with limited local connectivity.

In decentralized federated learning (FL), multiple clients collaboratively learn a shared machine learning (ML) model by leveraging their privately held datasets distributed across the network, through interactive exchange of intermediate model updates. To ensure data security, cryptographic techniques are commonly employed to protect model updates during aggregation. Despite growing interest in secure aggregation, existing works predominantly focus on protocol design and computational guarantees, with limited understanding of the fundamental information-theoretic limits of such systems. Moreover, optimal bounds on communication and key usage remain unknown in decentralized settings, where no central aggregator is available. Motivated by these gaps, we study the problem of decentralized secure aggregation (DSA) from an information-theoretic perspective. Specifically, we consider a network ofKfully-connected users, each holding a private input—an abstraction of local training data—who aim to securely compute the sum of all inputs. The security constraint requires that no user learns anything beyond the input sum, even when colluding with up toTother users. We characterize the optimal rate region, which specifies the minimum achievable communication and secret key rates for DSA. In particular, we show that to securely compute one symbol of the desired input sum, each user must (i) transmit at least one symbol to others, (ii) hold at least one symbol of secret key, and (iii) all users must collectively hold no fewer thanK−1independent key symbols. Our results establish the fundamental performance limits of DSA, providing insights for the design of provably secure and communication-efficient protocols in decentralized learning.

In the problem of conditional disclosure of secrets (CDS), two parties, Alice and Bob, each has an input and shares a common secret. Their goal is to reveal the secret to a third party, Carol, as efficiently as possible, only if the inputs of Alice and Bob satisfy a certain functional relation$f$. To prevent leakage of the secret to Carol when the input combination is unqualified, both Alice and Bob introduce noise. This work aims to determine the noise capacity, defined as the maximum number of secret bits that can be securely revealed to Carol, normalized by the total number of independent noise bits held jointly by Alice and Bob. Our contributions are twofold. First, we establish the necessary and sufficient conditions under which the CDS noise capacity attains its maximum value of 1. Second, in addition to the above best-case scenarios, we derive an upper bound on the linear noise capacity for any CDS instance. In particular, this upper bound is equal to$(\rho-1)(d-1) /(\rho d-1)$, where$\rho$is the covering parameter of the graph representation of$f$, and$d$is the number of unqualified edges in residing unqualified path.

Motivated by federated learning (FL), secure aggregation (SA) aims to securely compute, as efficiently as possible, the sum of a set of inputs distributed across many users. To understand the impact of network topology, hierarchical secure aggregation (HSA) investigated the communication and secret key generation efficiency in a 3-layer relay network, where clusters of users are connected to the aggregation server through an intermediate layer of relays. Due to the pre-aggregation of the messages at the relays, HSA reduces the communication burden on the relay-to-server links and is able to support a large number of users. However, as the number of users increases, a practical challenge arises from heterogeneous security requirements–for example, users in different clusters may require varying levels of input protection. Motivated by this, we study weakly-secure HSA (WS-HSA) with collusion resilience, where instead of protecting all the inputs from any set of colluding users, only the inputs belonging to a predefined collection of user groups (referred to as security input sets) need to be protected against another predefined collection of user groups (referred to as collusion sets). Since the security input sets and collusion sets can be arbitrarily defined, our formulation offers a flexible framework for addressing heterogeneous security requirements in HSA. We characterize the optimal total key rate, i.e., the total number of independent key symbols required to ensure both server and relay security, for a broad range of parameter configurations. For the remaining cases, we establish lower and upper bounds on the optimal key rate, providing constant-factor gap optimality guarantees.

In secure summation,Kusers, each holds an input, wish to compute the sum of the inputs at a server without revealing any information aboutall the inputseven if the server may collude withan arbitrary subset of users. In this work, we relax the security and colluding constraints, where the set of inputs whose information is prohibited from leakage is from a predetermined collection of sets (e.g., any set of up toSinputs) and the set of colluding users is from another predetermined collection of sets (e.g., any set of up toTusers). For arbitrary collection of security input sets and colluding user sets, we characterize the optimal randomness assumption, i.e., the minimum number of key bits that need to be held by the users, per input bit, for weakly secure summation to be feasible, which generally involves solving a linear program.

A storage code over a graph maps$K$independent source symbols, each of$L_{w}$bits, to$N$coded symbols, each of$L_{v}$bits, such that each coded symbol is stored in a node of the graph and each edge of the graph is associated with one source symbol. From a pair of nodes connected by an edge, the source symbol that is associated with the edge can be decoded. The ratio$L_{w}/L_{v}$is called the symbol rate of a storage code and the highest symbol rate is called the capacity. We show that the three highest capacity values of storage codes over graphs are$2, 3/2, 4/3$. We characterize all graphs over which the storage code capacity is 2 and$3/2$, and for capacity value of$4/3$, necessary condition and sufficient condition (that do not match) on the graphs are given.

A storage code over a graph maps K independent source symbols, each of Lwbits, to N coded symbols, each of Lvbits, such that each coded symbol is stored in a node of the graph and each edge of the graph is associated with one source symbol. From a pair of nodes connected by an edge, the source symbol that is associated with the edge can be decoded. The ratio Lw/Lvis called the symbol rate of a storage code and the highest symbol rate is called the capacity. We show that the three highest capacity values of storage codes over graphs are 2, 3/2, 4/3. We characterize all graphs over which the storage code capacity is 2 and 3/2, and for capacity value of 4/3, necessary condition and sufficient condition (that do not match) on the graphs are given.

In secure summation, K users, each holds an input, wish to compute the sum of the inputs at a server without revealing any information about all the inputs even if the server may collude with an arbitrary subset of users. In this work, we relax the security and colluding constraints, where the set of inputs whose information is prohibited from leakage is from a predetermined collection of sets (e.g., any set of up to S inputs) and the set of colluding users is from another predetermined collection of sets (e.g., any set of up to T users). For arbitrary collection of security input sets and colluding user sets, we characterize the optimal randomness assumption, i.e., the minimum number of key bits that need to be held by the users, per input bit, for weakly secure summation to be feasible, which generally involves solving a linear program.

Conditional disclosure of secrets (CDS) is the problem of disclosing as efficiently as possible, one secret from Alice and Bob to Carol if and only if the inputs at Alice and Bob satisfy some function. The information theoretic capacity of CDS is the maximum number of bits of the secret that can be securely disclosed per bit of total communication from Alice and Bob to Carol. All CDS instances, where the capacity is the highest and is equal to 1/2, are recently characterized through a noise and signal alignment approach and are described using a graph representation of the function. In this work, we go beyond the best case scenarios and further develop the alignment approach to characterize the linear capacity of a class of CDS instances to be$(\rho -1)/(2\rho)$, where$\rho $is a newly introduced and highly specific covering parameter of the graph representation of the function.

A secure storage code maps$K$source symbols, each of$L_{w}$bits, to$N$coded symbols, each of$L_{v}$bits, such that each coded symbol is stored in a node of a graph (one may view a node as a server). Each edge of the graph is either associated with$D$of the$K$source symbols such that from the pair of nodes connected by the edge, we can decode the$D$source symbols and learn no information about the remaining$K-D$source symbols; or the edge is associated with no source symbols such that from the pair of nodes connected by the edge, nothing about the$K$source symbols is revealed. The ratio$L_{w}/L_{v}$is called the symbol rate of a secure storage code and the highest possible symbol rate is called the capacity. We characterize all graphs over which the capacity of a secure storage code is equal to 1, when$D = 1$. This result is generalized to$D> 1$, i.e., we characterize all graphs over which the capacity of a secure storage code is equal to$1/D$under a mild condition that for any node, the source symbols associated with each of its connected edges do not include a common element. Further, we characterize all graphs over which the capacity of a secure storage code is equal to$2/D$.

In the conditional disclosure of secrets (CDS) problem, Alice and Bob (each holds an input and a common secret) wish to disclose, as efficiently as possible, the secret to Carol if and only if their inputs satisfy some function. The capacity of CDS is the maximum number of bits of the secret that can be securely disclosed per bit of total communication. We characterize the necessary and sufficient condition for the extreme case where the capacity of CDS is the highest and is equal to$1/2$. For the simplest instance where the capacity is smaller than$1/2$, we show that the linear capacity is$2/5$.

Conditional disclosure of secrets (CDS) is the problem of disclosing as efficiently as possible, one secret from Alice and Bob to Carol if and only if the inputs at Alice and Bob satisfy some function f. The information theoretic capacity of CDS is the maximum number of bits of the secret that can be securely disclosed per bit of total communication. All CDS instances, where the capacity is the highest and is equal to 1/2, are recently characterized through a noise and signal alignment approach and are described using a graph representation of the function f, Gf. In this work, we go beyond the best case scenarios and further develop the alignment approach to characterize the linear capacity of a class of CDS instances to be (p- 1)/(2p), where$p$is a covering parameter of Gf.

In the conditional disclosure of secrets (CDS) problem, Alice and Bob (each holds an input and a common secret) wish to disclose, as efficiently as possible, the secret to Carol if and only if their inputs satisfy some function. The capacity of CDS is the maximum number of bits of the secret that can be securely disclosed per bit of total communication. We characterize the necessary and sufficient condition for the extreme case where the capacity of CDS is the highest and is equal to 1/2.