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This study demonstrates some new aspects of preboundary lengthening and preaccentual shortening on a test word banana in American English. Preboundary lengthening was found to be extended to the initial unstressed syllable beyond the main-stressed syllable, presenting more complexity than has previously been assumed. Preaccentual shortening was observed regardless of boundary strength or the stress pattern (trochaic vs iambic) of the following context word, suggesting that it operates globally at an utterance level. The locus of preaccentual shortening, however, was modulated by prosodic boundary: It is realized on the final vowel IP-finally but on the non-final stressed vowel IP-medially.

The approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.