D:\sbornik\...\InfHidingSys.DVI Mathematical Problems of Computer Science 23, 2004, 20{31. On E stimates of Rate-r eliability-distor tion Function for I nfor mation H iding System¤ Ma r ia m E . H a r o u t u n ia n a n d S m b a t A . To n o ya n Institute for Informatics and Automation Problems of NAS of RA e-mails: armar@ipia.sci.am, smbatt@ipia.sci.am Abstract The model of information hiding system, introduced and studied by P. Moulin and J. A. O'Sullivan [1] is explored. The rate-reliability-distortion function for this system is investigated. Upper and lower estimates of rate-reliability-distortion function, called the random coding and the sphere packing bounds are constructed. The limit of random coding bound, when E ! 0, coincides with the information hiding capacity stated by P. Moulin and J. A. O'Sullivan. Refer ences [1 ] P . Mo u lin a n d J. A . O'S u lliva n , " In fo r m a t io n -t h e o r e t ic a n a lys is o f in fo r m a t io n h id in g " , IE E E Trans. Inform. Theory, vo l. 4 9 , n o . 3 , p p . 5 6 3 -5 9 3 , Ma r . 2 0 0 3 . [2 ] F. A . P . P e t it c o la s , R . J. A n d e r s o n , a n d M. G. K u h n , " In fo r m a t io n h id in g { A S u r ve y," P roc. IE E E (Special Issue on Identi¯cation and P rotection of M ultimedia Information), vo l. 8 7 , p p . 1 0 6 2 -1 0 7 8 , Ju ly 1 9 9 9 . [3 ] P . Mo u lin , " Th e r o le o f in fo r m a t io n t h e o r y in wa t e r m a r kin g a n d it s a p p lic a t io n t o im a g e wa t e r m a r kin g ," Signal P rocessing, vo l. 8 1 , p p . 1 1 2 1 -1 1 3 9 , 2 0 0 1 . [4 ] E . A . H a r o u t u n ia n , " U p p e r e s t im a t e o f t r a n s m is s io n r a t e fo r m e m o r yle s s c h a n n e l wit h c o u n t a b le n u m b e r o f o u t p u t s ig n a ls u n d e r g ive n e r r o r p r o b a b ilit y e xp o n e n t " , ( in R u s - s ia n ) , 3rd All-Union Conf. on Theory of Information Transmission and Coding, Uzh- gorod, P ublication house of Uzbek Academy of Sciences, Tashkent, p p . 8 3 { 8 6 , 1 9 6 7 . [5 ] M. E . H a r o u t u n ia n a n d S . A . To n o ya n , " R a n d o m c o d in g b o u n d o f in fo r m a t io n h id in g E-c a p a c it y" , P roc. of IE E E Intern. Symp. Infrom. Theory, p . 5 3 6 , U S A , Ch ic a g o , 2 0 0 4 . [6 ] S . I. Ge l'fa n d a n d M. S . P in s ke r , " Co d in g fo r c h a n n e l wit h r a n d o m p a r a m e t e r s ," P roblems of Control and Information Theory, vo l. 9 , n o . 1 , p p . 1 9 -3 1 , 1 9 8 0 . ¤The work was partially supported by INTAS Grant 00-738 and by 04.10.31 Target Program of RA. 2 0 M. E. Haroutunian and S. A. Tonoyan 2 1 [7 ] M. E . H a r o u t u n ia n , " N e w b o u n d s fo r E-c a p a c it ie s o f a r b it r a r ily va r yin g c h a n n e l a n d c h a n n e l wit h r a n d o m p a r a m e t e r " Trans. IIAP NAS R A and YSU, M athematical P rob- lems of Computer sciences, vo l. 2 2 , p . 4 4 -5 9 , 2 0 0 1 . [8 ] M. E . H a r o u t u n ia n , " B o u n d s o f E-c a p a c it y fo r m u lt ip le -a c c e s s c h a n n e l wit h r a n d o m p a r a m e t e r " , s p e c ia l b o o k is s u e d in t h e fr a m e wo r k o f r e s e a r c h p r o je c t " Ge n e r a l Th e o r y o f In fo r m a t io n Tr a n s fe r a n d Co m b in a t o r ic s " a t ZiF, B ile fe ld U n ive r s it y, Ge r m a n y, 2 0 0 4 . [9 ] N . Me r h a v, " On r a n d o m c o d in g e r r o r e xp o n e n t s o f wa t e r m a r kin g s ys t e m s " , IE E E Trans. Inform. Theory, vo l. 4 6 , n o . 2 , p p . 4 2 0 -4 3 0 , Ma r . 2 0 0 0 . [1 0 ] N . Me r h a v a n d A . S o m e kh -B a r u c h , " On t h e e r r o r e xp o n e n t a n d c a p a c it y g a m e s o f p r iva t e wa t e r m a r kin g s ys t e m s " , IE E E Trans. Inform. Theory, vo l. 4 9 , n o . 3 , p p . 5 3 7 - 5 6 2 , Ma r . 2 0 0 3 . [1 1 ] I. Cs is z ¶a r a n d J. K Äo r n e r , Information Theory: Coding theorems for discrete memoryless systems, A c a d e m ic P r e s s , N e w Y o r k, 1 9 8 1 . [1 2 ] I. Cs is z ¶a r , " Th e m e t h o d o f t yp e s " , IE E E Trans. Inform. Theory, vo l. 4 4 , n o . 6 , p p . 2 5 0 5 { 2 5 2 3 , 1 9 9 8 . îíÛ³ÉÝ»ñ óùóÝáÕ Ñ³Ù³Ï³ñ·»ñÇ Ñ³Ù³ñ ³ñ³·áõÃÛáõÝ-Ñáõë³ÉÇáõÃÛáõÝ-ß»ÕáõÙ ýáõÝÏódzÛÇ ·Ý³Ñ³ï³Ï³ÝÝ»ñÇ Ù³ëÇÝ Ø. º. гñáõÃÛáõÝÛ³Ý ¨ ê. ². îáÝáÛ³Ý ²Ù÷á÷áõÙ ²ß˳ï³ÝùáõÙ áõëáõÙݳëÇñí³Í ¿ ä. ØáõÉÇÝÇ ¨ æ. úªêáõÉÇí³ÝÇ ÏáÕÙÇó ¹Çï³ñÏí³Í ïíÛ³ÉÝ»ñ óùóÝáÕ Ñ³Ù³Ï³ñ·Ç ÁݹѳÝáõñ Ùá¹»ÉÁ: ²Û¹ ѳٳϳñ·Ç ѳٳñ Ý»ñÙáõÍí³Í ¿ ³ñ³·áõÃÛáõÝ-Ñáõë³ÉÇáõÃÛáõÝ-ß»ÕáõÙ ýáõÝÏódzÛÇ ·³Õ³÷³ñÁ, áñÇ Ñ³Ù³ñ ϳéáõóí³Í »Ý ëïáñÇÝ ¨ í»ñÇÝ ·Ý³Ñ³ï³Ï³ÝÝ»ñ, áñáÝù ѳٳå³ï³ë˳ݳµ³ñ ÏáãíáõÙ »Ý å³ï³Ñ³Ï³Ý Ïá¹³íáñÙ³Ý ¨ ëý»ñ³Ý»ñÇ ÷³Ã»Ã³íáñÙ³Ý ë³ÑÙ³ÝÝ»ñ: êïáñÇÝ ·Ý³Ñ³ï³Ï³ÝÁ ë³ÑٳݳÛÇÝ ¹»åùáõÙ, »ñµ E ! 0 , ѳÙÁÝÏÝáõÙ ¿ ѳٳϳñ·Ç áõݳÏáõÃÛ³Ý Ñ»ï: