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Tsujii et al. (see ibid., vol.23, no.11, p.558-60, 1987) proposed a public key cryptosystem which requires only O(m2) bit operations for encryption/decryption and which also realised digital signatures; m here is the block length. The speed advantage over better known systems such as the RSA system (which takes O(m3) operations) could have been immensely significant. Unfortunately it is possible to deduce the private key of the Tsujii system from the public key in polynomial time except possibly in a tiny minority of randomly chosen cases.