In this work, we introduce and formalize a new primitive, referred to as key consensus (KC), and its asymmetric variant AKC, for communicating parties reaching consensus from close values. % caused by message exchange and private computations. by some secure information exchange. Then we find efficient upper bounds for any KC and AKC schemes. The bounds are particularly instrumental in choosing parameters towards different optimization goals. KC and AKC are fundamental to lattice based cryptography, in the sense that a list of cryptographic primitives based on LWE or Ring-LWE (including key exchange, public key encryption, oblivious transfer, and more) can be modularly constructed from them. As a conceptual contribution, this much simplifies the design and analysis of these cryptosystems in the future. We then design and analyze highly practical KC and AKC schemes within a general framework. The correctness constraint on parameters of our schemes is almost the same as the efficiency upper bound. The structure generalization and tightness allow us to choose parameters towards optimal balance among security, computational cost, bandwidth, consensus range, and error rate. When applied to LWE or RLWE based cryptosystems, generally speaking, by carefully choosing parameters they can lead to more practical schemes of key exchange, CPA-secure public key encryption.
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