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An efficient and secure real-field public key cryptosystem (PKC) based on sparse recovery is proposed. The security of the proposed cryptosystem depends on the following facts: 1. when the measurement matrix is known, the decryption algorithm, Cross Low-dimensional Pursuit, can efficiently solve the sparse recovery problem, where the sparse vector has a relatively high proportion of nonzeros; 2. without the measurement matrix, it is NP-hard to directly solve the sparse recovery problem. The proposed PKC is novel. First, unlike the traditional PKCs that are defined in finite fields, the proposed PKC is defined in the real field. Second, unlike popular cryptosystems based on number-theoretic problems, the proposed cryptosystem is based on the sparse recovery problem.

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