  This paper gives the definitions of an anomalous super-increasing sequence
and an anomalous subset sum separately, proves the two properties of an
anomalous super-increasing sequence, and proposes the REESSE2+ public-key
encryption scheme which includes the three algorithms for key generation,
encryption and decryption. The paper discusses the necessity and sufficiency of
the lever function for preventing the Shamir extremum attack, analyzes the
security of REESSE2+ against extracting a private key from a public key through
the exhaustive search, recovering a plaintext from a ciphertext plus a knapsack
of high density through the L3 lattice basis reduction method, and
heuristically obtaining a plaintext through the meet-in-the-middle attack or
the adaptive-chosen-ciphertext attack. The authors evaluate the time complexity
of REESSE2+ encryption and decryption algorithms, compare REESSE2+ with ECC and
NTRU, and find that the encryption speed of REESSE2+ is ten thousand times
faster than ECC and NTRU bearing the equivalent security, and the decryption
speed of REESSE2+ is roughly equivalent to ECC and NTRU respectively.
