فروشگاه غیرفعال میباشد در حال حاضر تکمیل نیست خرید نفرمائید ..

تعداد اسلاید : 13 صفحه

November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-1 Chapter 27: Lattices Overview Definitions Lattices Examples November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-2 Overview Lattices used to analyze Bell-LaPadula, Biba constructions Consists of a set and a relation Relation must partially order set Partial ordering < orders some, but not all, elements of set November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-3 Sets and Relations S set, R: S  S relation If a, b  S, and (a, b)  R, write aRb Example I = { 1, 2, 3}; R is ≤ R = { (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3) } So we write 1 ≤ 2 and 3 ≤ 3 but not 3 ≤ 2 November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-4 Relation Properties Reflexive For all a  S, aRa On I, ≤ is reflexive as 1 ≤ 1, 2 ≤ 2, 3 ≤ 3 Antisymmetric For all a, b  S, aRb  bRa  a = b On I, ≤ is antisymmetric Transitive For all a, b, c  S, aRb  bRc  aRc On I, ≤ is transitive as 1 ≤ 2 and 2 ≤ 3 means 1 ≤ 3 November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-5 Bigger Example C set of complex numbers a  C  a = aR + aIi, aR, aIintegers a ≤C b if, and only if, aR ≤ bR and aI ≤ bI a ≤C b is reflexive, antisymmetric, transitive As ≤ is over integers, and aR , aI are integers November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-6 Partial Ordering Relation R orders some members of set S If all ordered, it’s total ordering Example ≤ on integers is total ordering ≤C is partial ordering on C (because neither 3+5i ≤C 4+2i nor 4+2i ≤C 3+5i holds) November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-7 Upper Bounds For a, b  S, if u in S with aRu, bRu exists, then u is upper bound Least upper if there is no t  S such that aRt, bRt, and tRu Example For 1 + 5i, 2 + 4i  C, upper bounds include 2 + 5i, 3 + 8i, and 9 + 100i Least upper bound of those is 2 + 5i November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-8 Lower Bounds For a, b  S, if l in S with lRa, lRb exists, then l is lower bound Greatest lower if there is no t  S such that tRa, tRb, and lRt Example For 1 + 5i, 2 + 4i  C, lower bounds include 0, -1 + 2i, 1 + 1i, and 1+4i Greatest lower bound of those is 1 + 4i November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-9 Lattices Set S, relation R R is reflexive, antisymmetric, transitive on elements of S For every s, t  S, there exists a greatest lower bound under R For every s, t  S, there exists a least upper bound under R November 1, 2004 Introduction to Computer Security ©2004 Matt Bishop Slide #27-10 Example S = { 0, 1, 2 }; R = ≤ is a lattice R is clearly reflexive, antisymmetric