In this paper, we propose a new method for primary decomposition of a polynomial ideal, not necessarily zero-dimensional, and report on a detailed study for its practical implementation. In our method, we introduce two key techniques, effective localization and fast elimination of redundant components, by which we can get a good performance for several examples. The performance of our method is examined by comparison with other existing methods based on practical experiments.

Among many approaches for privacy-preserving biometric authentication, we focus on the approach with homomorphic encryption, which is public key encryption supporting some operations on encrypted data. In biometric authentication, the Hamming distance is often used as a metric to compare two biometric feature vectors. In this paper, we propose an efficient method to compute the Hamming distance on encrypted data using the homomorphic encryption based on ideal lattices. In our implementation of secure Hamming distance of 2048-bit binary vectors with a lattice of 4096 dimension, encryption of a vector, secure Hamming distance, and decryption respectively take about 19.89, 18.10, and 9.08 milliseconds (ms) on an Intel Xeon X3480 at 3.07 GHz. We also propose a privacy-preserving biometric authentication protocol using our method, and compare it with related protocols. Our protocol has faster performance and shorter ciphertext size than the state-of-the-art prior work using homomorphic encryption.

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This paper proposes a new higher order differential attack. The higher order differential attack proposed at FSE'97 by Jakobsen and Knudsen used exhaustive search for recovering the last round key. Our new attack improves the complexity to the cost of solving a linear system of equations. As an example we show the higher order differential attack of a CAST cipher with 5 rounds. The required number of chosen plaintexts is 2 17 and the required complexity is less than 2 25 times the computation of the round function. Our experimental results show that the last round key of the CAST cipher with 5 rounds can be recovered in less than 15 seconds on an UltraSPARC station.

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