Correlation Analysis of the Alternating Step Generator

The alternating step generator is a well-known keystream generator consisting of two stop/go clocked LFSRs, LFSR₁ and LFSR₂, whose clocks are controlled by another LFSR, LFSR₃, which is clocked regularly. A probabilistic analysis of this generator is conducted which shows that the posterior probabilites of individual bits of the first derivatives of the regularly clocked LFSR₁ and LFSR₂ sequences, when conditioned on a given segment of the first derivative of the keystream sequence, can be computed efficiently in a number of probabilistic models of interest. The expected values of these probabilities, for a random keystream sequence, are derived by an approximate theoretical analysis and are also verified by systematic computer experiments. It is pointed out that these posterior probabilities can be enhanced in a resynchronization scenario and thus used for a low-complexity fast correlation attack on the two LFSRs. More generally, it is argued that even without resynchronization these probabilities may be significantly different from one half for fast correlation attacks based on iterative decoding algorithms to be successful, although with incresead complexity. A related method for computing the posterior probabilities of individual bits of the LFSR₃ sequence, when conditioned on both the keystream sequence and the LFSR₁ and LFSR₂ sequences, is also developed. As these posterior probabilities are much more different from one half, they can be used for a low-complexity fast correlation attack on LFSR₃, provided that the initial states of LFSR₁ and LFSR₂ are previously reconstructed.     

Free Subgroups in Certain Generalized Triangle Groups of Type (2, m, 2)

A generalized triangle group is a group that can be presented in the form where p,q,r ≥ 2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product . Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p,q,r) is one of (3, 3, 2), (3, 4, 2), (3, 5, 2), or (2, m, 2) where m=3, 4, 5, 6, 10, 12, 15 , 20, 30, 60. In this paper, we show that the Tits alternative holds in the cases (p,q,r)=(2, m, 2) where m=6, 10, 12, 15, 20, 30, 60.     

On the ranks of certain semigroups of order-preserving transformations

Communicated by R. B. McFadden     

Crystalline effects in backscattering and Auger production

Measurements have been made of the variation of the backscattered intensity and of the Auger signal intensity as a function of the direction of the incident electron beam using a (111) Al surface and an incidence energy of 1.5 keV. The dynamical theory of electron diffraction and backscattering is able to account for the results quantitatively to within a factor of about two in the case of the quasi elastic and first plasma loss backscattered components. The variation measured for the Auger signal is significantly greater than for backscattered electrons of similar energy.