The paper presents completeness theorems for non-cryptographic fault-tolerant distributed computation, focusing on the ability to compute functions securely and resiliently in the presence of Byzantine faults. The main results are: 1. **Theorem 1**: For any probabilistic function and \( t < n/2 \), there exists a \( t \)-private protocol. 2. **Theorem 2**: There are functions for which no \( n/2 \)-private protocols exist. 3. **Theorem 3**: For every probabilistic function and \( t < n/3 \), there exists a protocol that is both \( t \)-resilient and \( t \)-private. 4. **Theorem 4**: There are functions for which no \( n/3 \)-resilient protocol exists. The paper introduces a model of computation where \( n \) processors communicate securely and aim to compute a function \( f \) from \( n \) inputs to \( n \) outputs. The protocols are designed to be \( t \)-private if no set of \( t \) players can learn more information than their inputs and outputs, and \( t \)-resilient if no set of \( t \) players can disrupt the computation. The proofs of these theorems rely on algebraic coding theory, particularly generalized BCH codes, and involve techniques such as secret sharing, error correction, and zero-knowledge proofs. The paper also discusses the practical implications and applications of these results, including their use in Byzantine agreement protocols.The paper presents completeness theorems for non-cryptographic fault-tolerant distributed computation, focusing on the ability to compute functions securely and resiliently in the presence of Byzantine faults. The main results are: 1. **Theorem 1**: For any probabilistic function and \( t < n/2 \), there exists a \( t \)-private protocol. 2. **Theorem 2**: There are functions for which no \( n/2 \)-private protocols exist. 3. **Theorem 3**: For every probabilistic function and \( t < n/3 \), there exists a protocol that is both \( t \)-resilient and \( t \)-private. 4. **Theorem 4**: There are functions for which no \( n/3 \)-resilient protocol exists. The paper introduces a model of computation where \( n \) processors communicate securely and aim to compute a function \( f \) from \( n \) inputs to \( n \) outputs. The protocols are designed to be \( t \)-private if no set of \( t \) players can learn more information than their inputs and outputs, and \( t \)-resilient if no set of \( t \) players can disrupt the computation. The proofs of these theorems rely on algebraic coding theory, particularly generalized BCH codes, and involve techniques such as secret sharing, error correction, and zero-knowledge proofs. The paper also discusses the practical implications and applications of these results, including their use in Byzantine agreement protocols.