here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

2.4. A block splitting pattern 37The γ i ′′ ’s and δ′′ i ’s are no longer pairwise independent, even between different blocks, but r(a, t) =∑d α i − γ i ′′ + δ′′ i and the following proposition is true.Proposition 2.4.3. Let k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗. Then a ∈ St,k if and only if ∑ d γ′′∑d δ′′ i .Remember that t is considered to be fixed so that the α i ’s and the β i ’s are considered to beconstants, whereas the other quantities defined in this section depend on a which ranges over allbinary strings on k bits and will be considered as random variables, whence the vocabulary weuse.2.4.2 Combining variablesIn the previous section we defined two variables for each block. However, we are only reallyinterested in the number of carries, so one should suffice, whence the following definition.Definition 2.4.4. We define ɛ i = γ i + β i − δ i , as depicted below:α 1{β 1{α i{t = 1---10---0...1---10---0...1---10---0 ,β i{α d{β d{a = ?10-0?01-1...?10-0?01-1...?10-0?01-1 .ɛ 1 ɛ i ɛ d{{Then ɛ i is “approximately” the number of carries that do not occur in the i-th block. As inthe previous subsection, we define ɛ ′ i = γ′ i + β i − δ i ′ and ɛ′′ i = γ′′ i + β i − δ i ′′ and Proposition 2.4.3is reformulated as follows.Proposition 2.4.5. Let k ≥ 2 and t ∈ ( Z/(2 k − 1)Z ) ∗. Then a ∈ St,k if and only if ∑ ∑d ɛ′′d β i = B = k − w H (t).2.4.3 One block: d = 1If t is made of only one block, we compute closed-form expressions for #C t,k,i = 2 k P (ɛ ′′ = k − i)for all i where P (ɛ ′′ = k − i) is nothing but the probability that i carries occurs while adding tand a, or equivalently the probability that k − i carries are lost.Such a t ≠ 0 (or an equivalent one) is written t = 2 k − 2 k−α (i.e. t = 1...1} {{ }0...0} {{ }) and itsα{β=k−αweight is w H (t) = α with α ≥ 1.In the following proposition, the computations are made without including the binary string0...0 in contrast with what was done in Subsection 2.4.1 because it does not complicate themtoo much.Proposition 2.4.6. The distribution of ɛ ′′ is as follows:P (ɛ ′′ = 0) = 2 −β ;i