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2.4. A block splitting pattern 352.4 A block splitting pattern2.4.1 General situationIn this section we often compute P t,k = #S t,k /2 k rather than #S t,k . Therefore, we use the wordsproportion or probability in place of cardinality. Moreover, in this subsection, we often computecardinalities considering all the binary strings on k bits, i.e. including 1...1 and 0...0. Recallthat the modular integer 0 is considered to act as the binary string 1...1. Hence, the binarystring 0...0 should be discarded when doing final computation of P t,k . Such a choice ensuresthat the random variables we construct are truly independent.We split t ≠ 0 (once correctly rotated, i.e. we multiply it by a power of 2 so that its binaryexpansion on k bits begins with a 1 and ends with a 0) in blocks of the form [1 ∗ 0 ∗ ] (i.e. as many1’s as possible followed by as many 0’s as possible) and write it down as follows.Definition 2.4.1.α 1{β 1{α i{t = 1---10---0... 1---10---0... 1---10---0t 1 t iwith d the number of blocks, α i and β i the numbers of 1’s and 0’s of the i-th block t i . We defineB = ∑ di=1 β i = k − w H (t).We define corresponding values for a (a modular integer to be added to t) as follows.Definition 2.4.2.α 1{β 1{α i{β i{α d{t = 1---10---0...1---10---0...1---10---0 ,β i{α dt d{a = ?10-0?01-1...?10-0?01-1...?10-0?01-1γ 1 δ 1 γ i δ i γ d{{{i.e. γ i is the number of 0’s in front of the end of the 1’s subblock of t i and δ i is the number of 1’sin front of the end of the 0’s subblock of t i .{{β d{β d{{δ d,One should be aware that γ i ’s and δ i ’s depend on a and will be considered as variables.We first “approximate” r(a, t) by ∑ di=0 α i − γ i + δ i ignoring the two following facts:• if a carry goes out of the i − 1-st block (we say that it overflows) and δ i = β i , the 1’ssubblock produces α i carries, whatever value γ i takes;• and if no carry goes out of the i − 1-st block (we say that it is inert), the 0’s subblockproduces no carries, whatever value β i takes.When computing that “approximation” of the number of carries produced by the i-th block, wedo as if a carry always goes out of the i − 1-st block and no carry goes out of the 0’s subblock.So this is actually the number of carries produced by the i − 1-st block only in that situation.Then, r(a, t) > w H (t) becomes “approximately” ∑ di=1 γ i < ∑ di=1 δ i and the distributions forγ i and δ i , considered as random variables, are given in Table 2.1.Indeed, for 0 ≤ c i < α i ,P (γ i = c i ) = 2 −ci−1 ,because we have to set c i bits to 0 and one bit in front of them to 1 leaving the other bits free,andP (γ i = α i ) = 2 −αi