A general architecture of an emittance exchanger (EEX) is considered, where the horizontal and longitudinal phase spaces are exchanged. A family of designs is described which can lead to extremely short final longitudinal lengths, even subfemtosecond. Using higher-order particle simulations, a preferred configuration is found, which has better compression capability and less emittance growth than the standard EEX design at high beam energy. An alternative design is also found which eliminates any final energy-phase coupling. Features of compression using an EEX are significantly different than with a chicane because the final longitudinal phase space is decoupled from the initial longitudinal phase space. Advantages of using an EEX for compression include less susceptibility to the coherent synchrotron radiation (CSR) microbunch instability, less susceptibility to bunch length broadening from CSR effects, and elimination of the initial energy-phase correlation that is needed for compression using a chicane as well as any residual energy-phase correlation after compression. A key disadvantage of using an EEX is that the final horizontal emittance tends to strongly depend on the initial bunch length and beam energy.