Garbled circuit (GC) is one of the few promising protocols to realize general-purpose secure computation. The target computation is represented by a Boolean circuit that is subsequently transformed into a network of encrypted tables for execution. The need for distributing GCs among parties, however, requires excessive data communication, called garbling cost, which bottlenecks system performance. Due to the zero garbling cost of XOR operations, existing works reduce garbling cost by representing the target computation as the XOR-AND graph (XAG) with minimal structural multiplicative complexity (MC). Starting with a thorough study of the cipher-text efficiency of different types of logic primitives, for the first time, we propose XOR-OneHot graph (X1G) as a suitable logic representation for the generation of low-cost GCs. Our contribution includes (a) an exact algorithm to synthesize garbling-cost-optimal X1G implementations for small-scale functions and (b) a set of logic optimization algorithms customized for X1Gs, which together form a robust optimization flow that delivers high-quality X1Gs for practical functions. The effectiveness of the proposals is evidenced by comprehensive evaluations: compared with the state of the art, 7.34%, 26.14%, 13.51%, and 4.34% reductions in garbling costs are achieved on average for the involved benchmark suites, respectively, with reasonable runtime overheads.