Cryptography - Sage

ZYXWVUT T T TSRQPONNMLKJIIH H HGFEEDC CBA A A A1 2 3 4 5 6 7 8 9 10 11 12Table 2.1: Transposition and substitution axes for ACATINTHEHATList notation for permutationsThe map π(j) = i j can be denoted by [i 1 , . . . , i n ]. This is the way, in effect, that we havedescribed a key for a substitution cipher — we list the sequence of characters in the imageof A, B, C, etc. Although these permutations act on the set of the characters A, . . . , Z ratherthan the integers 1, . . . , n, the principle is identical.Cycle notation and orbit structureGiven a permutation π in S n there exists a unique orbit decomposition:{1, . . . , n} =t⋃{π j (i k ) : j ∈ Z},k=1where union can be taken over disjoint sets, i.e. i k is not equal to π j (i l ) for any j unlessk = l. The sets {π j (i k ) : j ∈ Z} are called the orbits of π, and the cycle lengths of π arethe sizes d 1 , . . . , d t of the orbits.Asociated to any orbit decomposition we can express an element π asπ = ( i 1 , π(i 1 ), . . . , π d 1−1 (i 1 ) ) · · · (it , π(i t ), . . . , π dt−1 (i t ) )14 Chapter 2. Classical Cryptography