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At PKC 2010 Smart and Vercauteren presented a variant of Gentry's fully homomorphic public key encryption scheme and mentioned that the scheme could support SIMD style operations. The slow key generation process of the Smart - Vercauteren system was then addressed in a paper by Gentry and Halevi, but their key generation method appears to exclude the SIMD style operation alluded to by Smart and Vercauteren. In this paper, the authors show how parameters can be selected to enable such SIMD operations, whilst still maintaining practicality of the key generation technique of Gentry and Halevi. They thus obtain a somewhat homomorphic scheme supporting SIMD operations, and operations on large finite fields of characteristic two.
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