Treated in this paper are one-dimensional discrete Schr dinger operators with a quasiperiodic potential, which are derived from the model proposed by Kohmoto, Kadanoff and Tang in 1983. The aim of this paper is to show the absence of point spectrum of the operators under certain conditions. 1991 Mathematics Subject Classification: 47A10, 47B39, 47B80, 47N50. l IntroductionWe consider the following discrete one-dimensional Schr dinger operators on t 2 (Z) given by (1) ( with a potential V e (n) given by(2) *Here λ is a non-zero constant, χ Α is the characteristic function of an interval A on the torus fR/Z, Φ is the canonical projection from IR onto (R/Z, and 0elR/Z. This operator was proposed by Kohmoto, Kadanoff and Tang [5] for these case of α = (J/5 -l)/2, A = Φ([1 -α, 1)) and 0 = 0. The potential V 9 (ri) with an irrational number α means a quasiperiodic one, and the operator (1) is interpreted by Luck and Petritis [8] s a model describing the phonon spectra in one-dimensional quasicrystals. In this case, S t ([9], [10]) concluded the spectrum of H 0 was a Cantor set (i.e. nowhere dense closed set without an isolated point) of zero Lebesgue measure and was purely Singular continuousi Further Bellissard, lochum, Scoppola and Testard [1] extended this result for any irrational number a. However, for the author's knowledge, the absence of the point spectrum of H Q for non-zero θ is not yet Brought to you by | University of Arizona Authenticated Download Date | 5/30/15 5:58 AM

We consider the Schrödinger operator with Poisson type random potential, and derive the spectrum which is deterministic almost surely. Apart from some exceptional cases, the spectrum is equal to [0, ∞) if the single-site potential is nonnegative, and is equal to R if the negative part of it does not vanish with positive probability, which is consistent with the naive observation. To prove that, we use the theory of admissible potential and the Weyl asymptotics.

SUMMARYThis article presents a differential fault analysis (DFA) technique using round addition for a generalized Feistel network (GFN) including CLEFIA and RC6. Here the term "round addition" means that the round operation executes twice using the same round key. The proposed DFA needs bypassing of an operation to count the number of rounds such as increment or decrement. To verify the feasibility of our proposal, we implement several operations, including increment and decrement, on a microcontroller and experimentally confirm the operation bypassing. The proposed round addition technique works effectively for the generalized Feistel network with a partial whitening operation after the last round. In the case of a 128-bit CLEFIA, we show a procedure to reconstruct the round keys or a secret key using one correct ciphertext and two faulty ciphertexts. Our DFA also works for DES and RC6.

SUMMARYPublic-key cryptography such as RSA cryptography and elliptic curve cryptography are used in electronic transactions. Since the security of the cryptography depends on the cryptographic key which is stored on an IC chip, security was believed to have been established as long as we use smart cards. However, since the mid-1990s, one technique after another has been developed for extracting the secret key without unsealing the IC chip. In particular, the power analysis by Kocher's group is an attack that can be carried out by using practical resources, and various applications have been proposed. In this paper, we study the power analysis of modular exponentiation, which is a primitive of RSA cryptography, and its countermeasures. Specifically, starting with the countermeasures to single-exponent multiple-data (SEMD), multiple-exponent single-data (MESD), and zero-exponent multiple-data (ZEMD) attacks of Messerges's group, we illustrate feasible attacks such as the Big Mac attack of Walter and Thompson, the template attack of Chari's group, and an attack proposed by the authors. We propose a countermeasure and present an implementation in a smart card. If the proposed method is applied to 1024-bit modular exponentiation, adequate security can be obtained in a processing time less than twice the time without countermeasures.

No abstract