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+ <doc-data|<doc-title|bkshuf: A Shuf-Like Utility with Pre-Image Resistance
+ and Relative Order Preservation for Random Sampling of Long
+ Lists>|<doc-author|<author-data|<author-name|Steven Baltakatei
+ Sandoval>>>|<doc-date|2023-02-14>>
+
+ <section|Summary>
+
+ <shell|bkshuf> is a <shell|shuf>-like utility designed to output randomly
+ sized groups of lines with a group size distribution centered around some
+ characteristic value.
+
+ <section|Objective>
+
+ The author desires to create a <shell|shuf>-like utility named
+ <shell|bkshuf> to mix line lists in order to produce output line lists with
+ the following somewhat conflicting properties:
+
+ <\description>
+ <item*|Pre-image resistance (PIR)>An output line's position should not
+ contain information about its input line position.
+
+ <item*|Relative order preservation (ROP)>Two neighboring lines in the
+ input stream should have a high probability of remaining neighbors in the
+ output stream.
+ </description>
+
+ The objective is to improve the value of a short random scan of a small
+ fraction of a potentially large input list; output that demonstrates ROP as
+ well as some degree of PIR may achieve this objective. In contrast, the
+ <shell|shuf> utility provides PIR but no ROP: a line's neighbor in the
+ output of <shell|shuf> is equally likely to be any other random line from
+ the input.\
+
+ In other words, output produced by <shell|bkshuf> should group together
+ sequential segments of the input lines in order to partially preserve
+ relationships that may exist between sequential files. For example, this
+ could be done by jumping to a random position in the input lines, consuming
+ (i.e. reading, outputting, and marking a line not to be read again) some
+ amount of sequential lines, then repeating the process until every line is
+ consumed. The amount of sequential lines to read between jumps affects how
+ well the above desired properties are satisfied.
+
+ The objective of <shell|bkshuf> is not to completely prevent the
+ possibility of reassembling the input given the output. Additionally, a
+ valuable property desired of <shell|bkshuf> is output which demonstrates
+ sufficiently high PIR compared to ROP such that only a short (compared to
+ the logarithm of the input list size) sequential scan of the output list
+ from a random starting position is required before a jump to a new group is
+ is encountered; this would permit the overal contents of very large input
+ line lists to be sampled.
+
+ <section|Design>
+
+ <subsection|Definitions>
+
+ <\eqnarray*>
+ <tformat|<table|<row|<cell|l>|<cell|:>|<cell|<text|number of
+ lines>>>|<row|<cell|l<rsub|<text|in>>>|<cell|:>|<cell|<text|input line
+ count>>>|<row|<cell|l<rsub|<text|out>>>|<cell|:>|<cell|<text|output line
+ count>>>|<row|<cell|c>|<cell|:>|<cell|<text|target group
+ count>>>|<row|<cell|s>|<cell|:>|<cell|<text|target group
+ size>>>|<row|<cell|p<rsub|<text|seq>><rsub|<text|>>>|<cell|:>|<cell|<text|probability
+ to include next sequential line>>>|<row|<cell|s<around*|(|l<rsub|<text|in>,0>|)>>|<cell|:>|<cell|<text|<math|target
+ group size parameter>>>>|<row|<cell|l<rsub|<text|in>,0>>|<cell|:>|<cell|<text|input
+ line count parameter>>>>>
+ </eqnarray*>
+
+ <subsection|Process>
+
+ <\enumerate-numeric>
+ <item>Acquire and count input lines (via <shell|/dev/stdin> or positional
+ arguments).
+
+ <item>Calculate line count <math|l<rsub|<text|in>>> .
+
+ <item>Calculate target group size <math|s>.
+
+ <item>Select random unconsumed input line and consume it to
+ output.<label|jump-to-random>
+
+ <item>Consume the next sequential line with probability
+ <math|p<rsub|<text|seq>>>. Otherwise if some input lines remain
+ unconsumed, go to step <reference|jump-to-random>. Otherwise, exit.
+ </enumerate-numeric>
+
+ <subsection|Parameter analysis>
+
+ <subsubsection|Target group size calculation>
+
+ The simultaneous presence of ROP and PIR properties in the output depends
+ upon the amount of sequential lines that are read before <shell|bkshuf>
+ jumps to a new random position in the input list. This amount is the
+ <em|target group size>, <math|s>; it is the \Ptarget\Q since <math|s>
+ represents the average of a distribution of group sizes that may be
+ selected, not a single group size. In this analysis, the total number of
+ lines in the input list is <math|l<rsub|<text|in>>>. For small input line
+ counts, (e.g. <math|l<rsub|<text|in>>\<cong\>10>) the target group size
+ should be nearly one (e.g. <math|s\<cong\>1>) since group sizes any larger
+ than this would have almost no PIR (e.g. a group size of <math|s=8> for
+ <math|l<rsub|<text|in>>=10> would be 80% identical to the input). For
+ modest line input counts (e.g. <math|l<rsub|<text|in>>\<cong\>100>), the
+ target group size may be allowed to be larger, such as a tenth of the input
+ line count (e.g. <math|s\<cong\>10>); this would provide some PIR
+ (approximately <math|10!> permutations between the approximately
+ <math|<frac|l<rsub|<text|in>>|s>\<cong\><frac|100|10>\<cong\>10> groups)
+ while each line in groups around size <math|10> would have a low
+ probability of not being next to its neighbor (<math|8> of the 10 lines
+ would retain the same two neighbors while the two ends would retain one
+ each). For very large input line counts (e.g.
+ <math|l<rsub|<text|in>>\<cong\>1<separating-space|0.2em>000<separating-space|0.2em>000>),
+ however, breaking up and randomizing the input into ten groups of
+ <math|100<separating-space|0.2em>000> offers very little PIR; the benefit
+ of the very high ROP is also lost since sequential scanning of tens of
+ thousands of lines is required before a random jump to a new group may be
+ encountered; therefore, the target group size should be a much smaller
+ fraction of <math|l<rsub|<text|in>>>, (e.g. <math|s\<cong\>20>) while still
+ increasing. The relationship between a desireable target group size
+ <math|s> and the input line count <math|l<rsub|<text|in>>> is non-linear.
+ The author believes a reasonable approach is to scale the group size to the
+ logarithm of input line count.
+
+ <hgroup|Figure <reference|fig ex-plot-s>> shows an example plot of
+ <math|s<around*|(|l<rsub|<text|in>>|)>> that is tuned to achieve a target
+ group size of <math|s<around*|(|l<rsub|<text|in>>=10<rsup|6>|)>=25> for an
+ input list length of <math|l<rsub|<text|in>>=10<rsup|6>> lines.\
+
+ <\big-figure|<with|gr-mode|<tuple|edit|point>|gr-frame|<tuple|scale|1.18922cm|<tuple|0.299593gw|0.120812gh>>|gr-geometry|<tuple|geometry|1par|0.6par>|gr-grid|<tuple|cartesian|<point|0|0>|1>|gr-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-aspect|<tuple|<tuple|axes|none>|<tuple|1|none>|<tuple|10|none>>|gr-edit-grid|<tuple|cartesian|<point|0|0>|1>|gr-edit-grid-old|<tuple|cartesian|<point|0|0>|1>|gr-grid-aspect-props|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|10|pastel
+ blue>>|gr-grid-aspect|<tuple|<tuple|axes|#808080>|<tuple|1|#c0c0c0>|<tuple|10|pastel
+ blue>>|magnify|1.18920711463847|gr-auto-crop|false|<graphics||<math-at|5|<point|-0.221848749616356|1.0>>|<math-at|10|<point|-0.397940008672038|2.0>>|<math-at|15|<point|-0.397940008672038|3.0>>|<math-at|20|<point|-0.397940008672038|4.0>>|<math-at|25|<point|-0.397940008672038|5.0>>|<math-at|30|<point|-0.397940008672038|6.0>>|<math-at|s|<point|0.0719460474170896|6.34360008343183>>|<point|0|0.2>|<point|6|5>|<math-at|10<rsup|1>|<point|1.0|-0.4>>|<math-at|10<rsup|2>|<point|2.0|-0.4>>|<math-at|10<rsup|3>|<point|3.0|-0.4>>|<math-at|10<rsup|4>|<point|4.0|-0.4>>|<math-at|10<rsup|5>|<point|5.0|-0.4>>|<math-at|10<rsup|6>|<point|6.0|-0.4>>|<math-at|1|<point|1.0|-0.8>>|<math-at|2|<point|2.0|-0.8>>|<math-at|3|<point|3.0|-0.8>>|<math-at|4|<point|4.0|-0.8>>|<math-at|5|<point|5.0|-0.8>>|<math-at|6|<point|6.0|-0.8>>|<math-at|x|<point|7.0|-0.8>>|<math-at|l<rsub|<text|in>>|<point|7.0|-0.4>>|<point|5.0|3.5>|<point|4.0|2.3>|<point|3.0|1.4>|<point|2.0|0.7>|<point|1.0|0.3>>>>
+ <label|fig ex-plot-s>A plot of a possible function that relates target
+ group size <math|s> and input lines <math|l<rsub|<text|in>>> that provide
+ some ROP and PIR. The function is tuned to achieve
+ <math|s<around*|(|l<rsub|<text|in>>=10<rsup|6>|)>=25>.\
+ </big-figure>
+
+ The following is a set of equations that are used to derive a definition
+ for <math|s<around*|(|l<rsub|<text|in>>|)>> that satisfies the plot in
+ <hgroup|Figure <reference|fig ex-plot-s>>.\
+
+ <\eqnarray*>
+ <tformat|<table|<row|<cell|x>|<cell|=>|<cell|<text|log>
+ <around*|(|l<rsub|<text|in>>|)>=<frac|ln
+ <around*|(|l<rsub|<text|in>>|)>|<text|ln>
+ <around*|(|10|)>><eq-number><label|eq
+ rel-x-lin>>>|<row|<cell|10<rsup|x>>|<cell|=>|<cell|l<rsub|<text|in>>>>|<row|<cell|x<rsub|0>>|<cell|=>|<cell|<text|log>
+ <around*|(|l<rsub|<text|in>,0>|)>=<frac|ln
+ <around*|(|l<rsub|<text|in>,0>|)>|<text|ln>
+ <around*|(|10|)>><eq-number><label|eq
+ rel-x0-lin0>>>|<row|<cell|10<rsup|x<rsub|0>>>|<cell|=>|<cell|l<rsub|<text|in>,0>>>|<row|<cell|>|<cell|>|<cell|>>|<row|<cell|s<around*|(|x|)>>|<cell|=>|<cell|<around*|(|k**x|)><rsup|2>+1<eq-number><label|eq
+ gsize-model>>>|<row|<cell|s<around*|(|x=6|)>=25>|<cell|=>|<cell|k<rsup|2>\<cdot\><around*|(|6|)><rsup|2>+1>>|<row|<cell|25>|<cell|=>|<cell|k<rsup|2>\<cdot\><around*|(|36|)>+1>>|<row|<cell|s<around*|(|x<rsub|0>|)>>|<cell|=>|<cell|<around*|(|k*x<rsub|0>|)><rsup|2>+1<eq-number><label|eq
+ gsize-param-rel>>>|<row|<cell|<frac|s<around*|(|x<rsub|0>|)>-1|x<rsub|0><rsup|2>>>|<cell|=>|<cell|k<rsup|2>>>|<row|<cell|k>|<cell|=>|<cell|<sqrt|<frac|s<around*|(|x<rsub|0>|)>-1|x<rsub|0><rsup|2>>>>>|<row|<cell|k>|<cell|=>|<cell|<frac|<sqrt|s<around*|(|x<rsub|0>|)>-1>|x<rsub|0>>>>|<row|<cell|k<rsup|2>>|<cell|=>|<cell|<frac|s<around*|(|x<rsub|0>|)>-1|x<rsub|0><rsup|2>><eq-number><label|eq
+ gsize-const-ksq>>>|<row|<cell|s<around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|<around*|(|<frac|s<around*|(|x<rsub|0>|)>-1|x<rsub|0><rsup|2>>|)>\<cdot\><around*|(|<frac|<text|ln><around*|(|l<rsub|<text|in>>|)>|<text|ln>
+ <around*|(|10|)>>|)><rsup|2>+1>>|<row|<cell|>|<cell|=>|<cell|<around*|(|<frac|<text|ln>
+ <around*|(|10|)>|ln <around*|(|l<rsub|<text|in>,0>|)>>|)><rsup|2>\<cdot\><around*|(|s<around*|(|x<rsub|0>|)>-1|)>\<cdot\><around*|(|<frac|<text|ln><around*|(|l<rsub|<text|in>>|)>|<text|ln>
+ <around*|(|10|)>>|)><rsup|2>+1>>|<row|<cell|s<around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|<around*|(|s<around*|(|l<rsub|<text|in>,0>|)>-1|)>\<cdot\><around*|(|<frac|<text|ln><around*|(|l<rsub|<text|in>>|)>|ln
+ <around*|(|l<rsub|<text|in>,0>|)>>|)><rsup|2>+1>>|<row|<cell|s<around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|<around*|(|<frac|s<around*|(|l<rsub|<text|in>,0>|)>-1|<around*|[|ln
+ <around*|(|l<rsub|<text|in>,0>|)>|]><rsup|2>>|)>\<cdot\><around*|[|<text|ln><around*|(|l<rsub|<text|in>>|)>|]><rsup|2>+1<eq-number><label|eq
+ gsize-lin>>>>>
+ </eqnarray*>
+
+ Equation <reference|eq gsize-model> defines a quadratic equation for the
+ linear range <math|s> and the logarithmic domain <math|x>. <math|x> is
+ defined in terms of <math|l<rsub|<text|in>>> via a domain transformation
+ defined by <hgroup|Equation <reference|eq rel-x-lin>>. The result is
+ <hgroup|Equation <reference|eq gsize-lin>> which defines
+ <math|s<around*|(|l<rsub|<text|in>>|)>> as a function of
+ <math|l<rsub|<text|in>>> and parameters
+ <math|s<around*|(|l<rsub|<text|in>,0>|)>> and <math|l<rsub|<text|in>,0>>.
+ The parameters define how quickly or slowly the quadratic equation grows.
+ In other words, if a user wishes for a <math|1<separating-space|0.2em>000<separating-space|0.2em>000>
+ line input to be split into groups each containing, on average, <math|25>
+ lines, then they should plug in <math|l<rsub|<text|in>,0>=1<separating-space|0.2em>000<separating-space|0.2em>000>
+ and <math|s<around*|(|l<rsub|<text|in>,0>|)>=25> into <hgroup|Equation
+ <reference|eq gsize-lin>> as is done in <hgroup|Equation <reference|eq
+ gsize-ex-1>>. This equation can then be used to calculate target group
+ sizes <math|s> as a function of other input line counts
+ <math|l<rsub|<text|in>>> besides <math|l<rsub|<text|in>>=1<separating-space|0.2em>000<separating-space|0.2em>000>.
+ For example, plugging <math|l<rsub|<text|in>>=500> into <hgroup|Equation
+ <reference|eq gsize-ex-1>> yields <hgroup|Equation <reference|eq
+ gsize-ex-1-lin500>> which specifies a target group size of
+ <math|5.85629\<cong\>6>.
+
+ <\eqnarray*>
+ <tformat|<table|<row|<cell|s<around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|<around*|(|<frac|s<around*|(|l<rsub|<text|in>,0>|)>-1|<around*|[|ln
+ <around*|(|l<rsub|<text|in>,0>|)>|]><rsup|2>>|)>\<cdot\><around*|[|<text|ln><around*|(|l<rsub|<text|in>>|)>|]><rsup|2>+1>>|<row|<cell|s<around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|<around*|(|<frac|25-1|<around*|[|<text|ln
+ ><around*|(|1<separating-space|0.2em>000<separating-space|0.2em>000|)>|]><rsup|2>>|)>\<cdot\><around*|[|<text|ln><around*|(|l<rsub|<text|in>>|)>|]><rsup|2>+1<eq-number><label|eq
+ gsize-ex-1>>>|<row|<cell|>|<cell|>|<cell|>>|<row|<cell|s<around*|(|l<rsub|<text|in>>=500|)>>|<cell|=>|<cell|<around*|(|<frac|25-1|<around*|[|<text|ln
+ ><around*|(|1<separating-space|0.2em>000<separating-space|0.2em>000|)>|]><rsup|2>>|)>\<cdot\><around*|[|<text|ln><around*|(|500|)>|]><rsup|2>+1<eq-number><label|eq
+ gsize-ex-1-lin500>>>|<row|<cell|s<around*|(|l<rsub|<text|in>>=500|)>>|<cell|=>|<cell|5.85629>>>>
+ </eqnarray*>
+
+ \;
+
+ <subsubsection|Jump from expected value>
+
+ A method <shell|bkshuf> may employ to decide when read the next sequential
+ unconsumed input line is to simply do so with probability
+ <math|p<rsub|<text|seq>>> such that the expected value of the average group
+ size trends towards <math|s>.
+
+ <\eqnarray*>
+ <tformat|<table|<row|<cell|p<rsub|<text|seq>>>|<cell|=>|<cell|<around*|(|1-p<rsub|<text|jump>>|)>>>|<row|<cell|p<rsub|<text|jump>>>|<cell|=>|<cell|1-p<rsub|<text|seq>>>>|<row|<cell|s>|<cell|=>|<cell|<frac|1|p<rsub|<text|jump>>>=<frac|1|1-p<rsub|<text|seq>>><eq-number>>>|<row|<cell|s>|<cell|=>|<cell|<frac|1|1-p<rsub|<text|seq>>>>>|<row|<cell|1-p<rsub|<text|seq>>>|<cell|=>|<cell|<frac|1|s>>>|<row|<cell|p<rsub|<text|seq>>-1>|<cell|=>|<cell|<frac|-1|s>>>|<row|<cell|p<rsub|<text|seq>>>|<cell|=>|<cell|1-<frac|1|s<around*|(|l<rsub|<text|in>>|)>><eq-number><inactive|<label|eq
+ pseq-from-s-lin>>>>|<row|<cell|p<rsub|<text|jump>>>|<cell|=>|<cell|<frac|1|s<around*|(|l<rsub|<text|in>>|)>><eq-number><inactive|<label|eq
+ pjump-from-s-lin>>>>|<row|<cell|>|<cell|>|<cell|>>|<row|<cell|p<rsub|<text|seq>><around*|(|l<rsub|<text|in>>|)>>|<cell|=>|<cell|1-<around*|[|<around*|(|<frac|s<around*|(|l<rsub|<text|in>,0>|)>-1|<around*|[|ln
+ <around*|(|l<rsub|<text|in>,0>|)>|]><rsup|2>>|)>\<cdot\><around*|[|<text|ln><around*|(|l<rsub|<text|in>>|)>|]><rsup|2>+1|]><rsup|-1><eq-number><inactive|<label|eq
+ pseq-from-s-lin-exp>>>>>>
+ </eqnarray*>
+
+ <subsubsection|Jump from random variate of inverse gaussian distribution>
+
+ Another method <shell|bkshuf> may employ to decide when to read the next
+ sequential unconsumed input line is to use an inverse gaussian
+ distribution. This may be done by generating from the distribution a float
+ sampled from the inverse gaussian with range 0 to infinity with mean
+ <math|\<mu\>> whenever a new random position in the input list is selected;
+ the float is rounded to the nearest integer.<\footnote>
+ See <name|Michael, John R.> "Generating Random Variates Using
+ Transformations with Multiple Roots". 1976.
+ <hlink|https://doi.org/10.2307/2683801|https://doi.org/10.2307/2683801> .
+ </footnote> Then, after consuming an input line, this integer is
+ decremented by one and another sequential line is consumed provided the
+ integer does not become less than or equal to zero. The inverse gaussian
+ distribution requires specifying the mean <math|\<mu\>> and the shape
+ parameter <math|\<lambda\>>; a higher <math|\<lambda\>> results in a
+ greater spread. An upper bound may also be specified since the distribution
+ has none except for that imposed by its programming implementation.
+
+ The result of using an inverse gaussian distribution is an output with
+ potentially much more regular group sizes than using the previously
+ mentioned expected value method. However, the implementation of the inverse
+ gaussian sampling operation described by (Michael, 1976) requires several
+ exponent calculations and a square root calculation in addition to various
+ multiplication and division operations. If sufficient processing power is
+ available, this may not necessarily be an issue.
+
+ <subsubsection|Output structure>
+
+ Regardless of whether group sizes are determined by the expected value
+ method or using variates of an inverse gaussian distribution, mimicking the
+ <shell|shuf> property of all input lines being present in the output,
+ albeit rearranged, results in a side effect: the first output lines are
+ more likely to contain groups with uninterrupted sequence runs (high ROP)
+ while groups in the last output lines are almost certain to contain
+ sequence jumps within a group (less ROP). The reason for this is that
+ <shell|bkshuf>, when it encounters an input line that has already been
+ consumed, will skip to the next available input line. The decision could be
+ made to skip to a new random line, but, it is simpler to simply read the
+ next available input line. The author's original intention of sampling only
+ a short initial portion of the output is compatible with the behavior that
+ ROP is preserved mostly at the beginning of the output.
+
+ <section|Version History>
+
+ <\big-table|<tabular|<tformat|<table|<row|<cell|Version
+ No.>|<cell|Date>|<cell|Path>|<cell|Description>>|<row|<cell|<verbatim|0.0.1>>|<cell|2023-02-14>|<cell|<verbatim|unitproc/bkshuf>>|<cell|Initial
+ draft implemented in <name|Bash>.>>>>>>
+ A table listing versions of <shell|bkshuf>.
+ </big-table>
+
+ <\description>
+ <item*|v0.0.1>Initial implementation in <shell|bash> <verbatim|5.1.16>
+ with <shell|bc> <verbatim|1.07.1> and <name|GNU Coreutils>
+ <verbatim|8.32> and tested on <name|Pop!_OS> <verbatim|22.04 LTS>. Saved
+ to the author's <name|BK-2020-03> repository<\footnote>
+ See commit <verbatim|22772fe> at <hlink|https://gitlab.com/baltakatei/baltakatei-exdev|https://gitlab.com/baltakatei/baltakatei-exdev>
+ .
+ </footnote>.
+ </description>
+</body>
+
+<initial|<\collection>
+</collection>>
+
+<\references>
+ <\collection>
+ <associate|auto-1|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-10|<tuple|3.3.3|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-11|<tuple|3.3.4|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-12|<tuple|4|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-13|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-14|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-2|<tuple|2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-3|<tuple|3|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-4|<tuple|3.1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-5|<tuple|3.2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-6|<tuple|3.3|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-7|<tuple|3.3.1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-8|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|auto-9|<tuple|3.3.2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-const-ksq|<tuple|5|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-ex-1|<tuple|7|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-ex-1-lin500|<tuple|8|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-lin|<tuple|6|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-model|<tuple|3|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq gsize-param-rel|<tuple|4|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq rel-x-lin|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|eq rel-x0-lin0|<tuple|2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|fig ex-plot-s|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|footnote-1|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|footnote-2|<tuple|2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|footnr-1|<tuple|1|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|footnr-2|<tuple|2|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ <associate|jump-to-random|<tuple|4|?|../../../../../wr/20230213..bkshuf_draft/src/doc.tm>>
+ </collection>
+</references>
+
+<\auxiliary>
+ <\collection>
+ <\associate|figure>
+ <tuple|normal|<\surround|<hidden-binding|<tuple>|1>|>
+ A plot of a possible function that relates target group size
+ <with|mode|<quote|math>|s> and input lines
+ <with|mode|<quote|math>|l<rsub|<with|mode|<quote|text>|in>>> that
+ provide some ROP and PIR. The function is tuned to achieve
+ <with|mode|<quote|math>|s<around*|(|l<rsub|<with|mode|<quote|text>|in>>=10<rsup|6>|)>=25>.
+ </surround>|<pageref|auto-8>>
+ </associate>
+ <\associate|table>
+ <tuple|normal|<\surround|<hidden-binding|<tuple>|1>|>
+ \;
+ </surround>|<pageref|auto-13>>
+ </associate>
+ <\associate|toc>
+ <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Summary>
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-1><vspace|0.5fn>
+
+ <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>Objective>
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-2><vspace|0.5fn>
+
+ <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Design>
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-3><vspace|0.5fn>
+
+ <with|par-left|<quote|1tab>|3.1<space|2spc>Definitions
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-4>>
+
+ <with|par-left|<quote|1tab>|3.2<space|2spc>Process
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-5>>
+
+ <with|par-left|<quote|1tab>|3.3<space|2spc>Parameter analysis
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-6>>
+
+ <with|par-left|<quote|2tab>|3.3.1<space|2spc>Target group size
+ calculation <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-7>>
+
+ <with|par-left|<quote|2tab>|3.3.2<space|2spc>Jump from expected value
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-9>>
+
+ <with|par-left|<quote|2tab>|3.3.3<space|2spc>Jump from random variate
+ of inverse gaussian distribution <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-10>>
+
+ <with|par-left|<quote|2tab>|3.3.4<space|2spc>Output structure
+ <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-11>>
+
+ <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>Version
+ History> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
+ <no-break><pageref|auto-12><vspace|0.5fn>
+ </associate>
+ </collection>
+</auxiliary>
\ No newline at end of file
projects / BK-2020-03.git / commitdiff