Logarithmic Complexity

Definition ∞ Logarithmic complexity describes an algorithm whose execution time or space requirements grow very slowly as the input size increases. Specifically, the growth rate is proportional to the logarithm of the input size. This type of complexity is highly desirable for computational efficiency in large-scale systems.
Context ∞ In the context of blockchain and digital assets, logarithmic complexity is a critical factor in the scalability and performance of protocols. Discussions often arise when analyzing the computational cost of certain cryptographic operations or data retrieval methods. The pursuit of algorithms exhibiting logarithmic complexity is a constant endeavor for developers aiming to enhance network throughput and reduce processing overhead.

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→Optimal Prover Time

→Proof Size Reduction

→Non-Interactive Arguments

Optimal Prover Time Succinct Zero-Knowledge Proofs Redefine Scalability

The Libra proof system achieves optimal linear prover time, solving the primary bottleneck of ZKPs to unlock practical, large-scale verifiable computation.

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→Homomorphic Commitment

→Data Accuracy

→Cryptographic Primitive

Separable Homomorphic Commitment Achieves Constant Overhead for Verifiable Aggregation

The new Separable Homomorphic Commitment primitive reduces client-side overhead from logarithmic to constant time for verifiable, secure data aggregation.

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→State Bloat Mitigation

→Cryptographic Primitive

→Vector Commitment Scheme

Vector Commitments Enable Sublinear State Verification for Stateless Clients

A new polynomial vector commitment scheme transforms light clients into secure, stateless verifiers, dramatically improving blockchain decentralization and user security.

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→Succinctness

→Computation Integrity

→Cryptographic Primitives

Goldwasser-Kalai-Rothblum Protocol Turbocharges Verifiable Computation Efficiency

A new proof system architecture uses the sumcheck protocol to commit only to inputs and outputs, achieving logarithmic verification time for layered computations, drastically scaling ZK-EVMs.

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→Rollup Scaling

→Verifier Complexity

→Polynomial Commitment

Logarithmic-Depth Commitments Enable Truly Stateless Blockchain Verification

A new Logarithmic-Depth Merkle-Trie Commitment scheme achieves constant-time verification, enabling light clients to securely validate state without storing it.

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→Transparent Security

→Post-Quantum Security

→Proof System Efficiency

Recursive Inner Product Arguments Enable Universal Transparent Polynomial Commitments

A novel recursive folding of polynomial commitments into Inner Product Arguments yields universal, transparent proof systems for highly scalable verifiable computation.

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→Logarithmic Complexity

→Adversarial Models

→Communication Protocols

Stochastic Networks Enable Logarithmic Broadcast and Consensus Resilience

New research reveals that distributed consensus in dynamic, unreliable networks can achieve logarithmic time complexity by embracing stochasticity, overcoming pessimistic deterministic limitations.