Implementation and Improvement of the Partial Sum Attack on 6-Round AES

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S n denote the partial sum S n = θ(ξ1) + … + θ(ξ n ), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S N = [lim n→∞𝔼θ(ξ n )]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ N ). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ N ) = o(𝔼N).

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Asymptotic properties for partial sum processes of a Gaussian random field

Statistics & Probability Letters ◽

10.1016/j.spl.2006.05.006 ◽

2007 ◽

Vol 77 (1) ◽

pp. 9-18

Author(s):

Hee-Jin Moon ◽

Yong-Kab Choi

Keyword(s):

Random Field ◽

Asymptotic Properties ◽

Gaussian Random Field ◽

Partial Sum

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A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

Journal of Applied Probability ◽

10.2307/3212841 ◽

1976 ◽

Vol 13 (2) ◽

pp. 361-364 ◽

Cited By ~ 2

Author(s):

M. E. Solari ◽

J. E. A. Dunnage

Keyword(s):

Infinite Sequence ◽

Random Variables ◽

Finite Sequence ◽

Partial Sums ◽

Dependent Random Variables ◽

Finite Segment ◽

Exchangeable Random Variables ◽

Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

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On Absolute Summability for any Positive Order

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007902 ◽

1933 ◽

Vol 3 (3) ◽

pp. 173-178 ◽

Cited By ~ 1

Author(s):

C. E. Winn

Keyword(s):

Absolute Summability ◽

Positive Order ◽

Partial Sum ◽

Image Position

Absolute summability according to Cesàro's method has been defined by Fekete for positive integral orders, as follows:—Denoting the rth partial sum of a series Σun by and its rth mean, namely2, by we can regard as the sum of the series.

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