N.H. Rev. Stat. Ann. § 235:38

Every workman on every state aid or town aid highway or bridge project, except workmen employed by the department of transportation in its engineering force or by private contractor engaged in construction or maintenance work on such project by contract with the state, county, city or town, shall be deemed to be in the employ of the county, city or town contributing to the cost of such project and not in the employ of the state.