@@ -26,9 +26,9 @@ This allows us to state the result precisely, and makes it clear how each round
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values of the previous rounds.
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<p />
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It seems convenient to change the round counter <math ><mi >q</mi ></math > to be 1-based (and
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- <math ><msup ><mi >X</mi ><mrow >(< mn >0</mn >) </mrow ></msup ></math >
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mn >0</mn >< mo >)</ mo > </mrow ></msup ></math >
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is an alias for the initial value, <math ><mi >X</mi ></math >), so that the final result is
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- <math ><msup ><mi >X</mi ><mrow >(< mi >r</mi >) </mrow ></msup ></math >.
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mi >r</mi >< mo >)</ mo > </mrow ></msup ></math >.
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</p >
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</discussion >
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@@ -60,20 +60,20 @@ the <math><mi>i</mi></math>th element of <math><mi>Y</mi></math> after applying
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and the length-<math ><mi >n</mi ></math > sequence <math ><mi >X</mi ></math > into a length-<math ><mi >n</mi ></math > output
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sequence <math ><mi >Y</mi ></math >. Philox applies an <math ><mi >r</mi ></math >-round substitution-permutation network to
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the values in <math ><mi >X</mi ></math >. <del >A single round of the generation algorithm performs the following steps:</del >
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- <ins >That is, there are intermediate values <math ><msup ><mi >X</mi ><mrow >(< mn >0</mn >) </mrow ></msup ></math >,
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- <math ><msup ><mi >X</mi ><mrow >(< mn >1</mn >) </mrow ></msup ></math >, … ,
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- <math ><msup ><mi >X</mi ><mrow >(< mi >r</mi >) </mrow ></msup ></math >, where
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- <math ><msup ><mi >X</mi ><mrow >(< mn >0</mn >) </mrow ></msup ><mo >:=</mo ><mi >X</mi ></math >, and for each round
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- <math ><mi >q</mi ></math > (with <math ><mi >q</mi ><mo >=</mo ><mn >1</mn >, … , <mi >r</mi ></math >),
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- <math ><msup ><mi >X</mi ><mrow >(< mi >q</mi >) </mrow ></msup ></math > is computed from
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- <math ><msup ><mi >X</mi ><mrow >(< mi >q</mi ><mo >-</mo ><mn >1</mn >) </mrow ></msup ></math > as follows. The output sequence
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- is <math ><msup ><mi >X</mi ><mrow >(< mi >r</mi >) </mrow ></msup ></math >.</ins >
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+ <ins >That is, there are intermediate values <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mn >0</mn >< mo >)</ mo > </mrow ></msup ></math >,
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mn >1</mn >< mo >)</ mo > </mrow ></msup ></math >, … ,
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mi >r</mi >< mo >)</ mo > </mrow ></msup ></math >, where
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mn >0</mn >< mo >)</ mo > </mrow ></msup ><mo >:=</mo ><mi >X</mi ></math >, and for each round
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+ <math ><mi >q</mi ></math > (with <math ><mi >q</mi ><mo >=</mo ><mn >1</mn >,< mo > … </ mo > , <mi >r</mi ></math >),
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mi >q</mi >< mo >)</ mo > </mrow ></msup ></math > is computed from
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+ <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mi >q</mi ><mo >-</mo ><mn >1</mn >< mo >)</ mo > </mrow ></msup ></math > as follows. The output sequence
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+ is <math ><msup ><mi >X</mi ><mrow >< mo >(</ mo >< mi >r</mi >< mo >)</ mo > </mrow ></msup ></math >.</ins >
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</p >
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<ol style =" list-style-type: none" >
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<li ><p >(4.1) — <del >The output sequence <math ><mi >X</mi ><mi >'</mi ></math > of the previous round (<math ><mi >X</mi ></math >
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in case of the first round) is permuted to obtain the intermediate state <math ><mi >V</mi ></math >:</del ></p >
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<blockquote ><pre >
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- <del ><math ><msub ><mi >V</mi ><mrow ><mi >j</mi ></mrow ></msub ><mo >=</mo ><msub ><mi >X</mi >'<mrow ><msub ><mi >f</mi ><mrow ><mi >n</mi ></mrow ></msub >(< mi >j</mi >) </mrow ></msub ></math ></del >
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+ <del ><math ><msub ><mi >V</mi ><mrow ><mi >j</mi ></mrow ></msub ><mo >=</mo ><msub ><mi >X</mi >'<mrow ><msub ><mi >f</mi ><mrow ><mi >n</mi ></mrow ></msub >< mo >(</ mo >< mi >j</mi >< mo >)</ mo > </mrow ></msub ></math ></del >
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</pre ></blockquote >
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<p >
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<ins >An intermediate state <math ><msup ><mi >V</mi ><mrow >(<mi >q</mi >)</mrow ></msup ></math > is obtained
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