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Extra resources for Robustness and Efficiency of Non-linear Side-Informed Watermarking

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These rules are to represent bijections stable under cutf so the following equations must hold: • g o /® = (g o /)® for g : Ai,C, A2 —> A 3 and f : Fi —> F 2 ,C,F 3 , and where T\ contains the sequence A,B. • /® og = (fog)® for g : Ax —-> A 2 ,C,A 3 and f : Fi,C,r 2 —> F 3 , and where one o/Fi,F 2 contains the sequence A,B. f = /®. om for f :Fi,A,£,F 2 — F 3 =% •gof® = {gof)® for g : Ai,C, A2 —> A 3 and f : Tx —> F 2 ,C,r 3 , and where one o/F 2 ,F 3 contains the sequence A,B. • f® og = (fog)® for g : Ai —> A 2 ,C,A 3 oncf / : Fi,C,F 2 —> F 3 , and where F 3 contains the sequence A, 2?.

We can also obtain some slightly stronger results by taking advantage of the ordering. We omit most of the proofs, which may be easily obtained from the results above, using monotonicity and continuity of composition. 1 Let C be an ordered ccc and let (T, 6,6,7) be a computational comonad over C with a computational pairing. • //"split o merge > id then curry o uncurry > id. • //split o merge < id then cuTfy o uncurry < id. • If merge0 split > id then unoTrry ocuTry > id. ry < id. Next we introduce a simple generalization of the notion of cartesian closed ordered category, obtained by relaxing the requirement that currying and uncurrying form an isomorphism.

Then: curfy(g 6 ( / x id)) = = < = = = < = = cilrry(# o T(f x e) o Tsplit 06) curry(# o T(f x e) o T split oS o merge) curry(# o T(f x e) o merge 0(6 x 6)) curry(# o merge o(Tf x Te) o (6 x 6)) curry(# o merge o((Tf o 6) x (Te o 6))) curry(# o mergeo((T/ o S) x id)) curry(# o merge) o (Tf o S) cuTry(^) o Tf o 6 curfy(g)bf. • To show that uncITrry is a lax natural transformation, suppose that / : A - A Ai a n d h . A> _+c uncurry(/i) 6 ( / x id) = > = = = > = = uncurry(/i) o split o T ( / x e) o Tsplit o£ uncurry(/i) o (Tf x Te) o split oT split o£ uncurry(Zi) o (Tf x Te) o (8 x 8) o split uncurry(/i) o ((Tf o 8) x (Te o 8)) o split uncuny(h) o ((Tf o 8) x \d) o sptit uncurry(/i o T / o 8) o split uncurry(/i 6 / ) o split uncurry(/i 6 / ) .

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Robustness and Efficiency of Non-linear Side-Informed Watermarking by Silvestre, Hurley, Furon

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