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In this paper,we study the RSA cryptosystem with low public key e =Nα and constrained private key d Let μ be a small integer and d ≡ dmodeμ When d =eγ is sufficiently small and 0 ≤ γ ≤μ-1/8a(1+√9 +48a),we show that RSA is vulnerable to cryptanalytic attacks based on lattice basis reduction.