Statistically optimized bit permutations in interated block substitution systems

Claim 3

The method of claim 1, wherein m is even.

Claim 4

The method of claim 2, wherein in (d), selecting the permutation comprises: (d)(1) selecting a quasi-sequencing {0, a 1 , a 2 , . . . , a c-1 } which is a sequence of elements {a i } of a group of c elements {0, a 1 , a 2 , . . . ,a c-1 }, where the following occurs: the group element a i such that a i =a i -1 occurs once and for each other group element: either (i) a i occurs twice and a i -1 does not occur, or, (ii) a i and a i -1 each occurs once, or (iii) a i -1 occurs twice and a i does not occur; where the group of c elements is arranged such that the sequence of partial sums b 0 =0, b 1 ,=a 1 , b 2 =a 1 +a 2 , . . . , b c-1 are all different; (d)(2) locating all pairs of elements a i and a j in the quasi-sequencing where a i =a j ; (d)(3) adding a constant c to a i , for all such pairs, replacing a i with a i +c and modifying the quasi-sequencing to {0,a 1 ,a 2 , . . . ,a c-1 } where a i =a i +c; (d)(4) constructing a sequencing of size 2c, where the sequencing is {0,a 1 ,a 2 , . . . ,a c-1 ,c, 2c-a c-1 , . . . 2c-a 2 ,2c-a 1 }; (d)(5) repeating step (d)(3); (d)(6) constructing a sequencing of size 4c, where {0,a 1 , . . . ,a c-1 ,c,2c-a c-1 , . . . ,2c-a 1 ,2c, 4c- 2c-a 1 ! . . . ,4c-a 1 }; (d)(7) repeating step (d)(3) k times to obtain a sequencing of size m=c2 k ; and (d)(8) providing the permutation corresponding to the sequencing.

Claim 5

The method of claim 1, wherein in (d), selecting the permutation comprises: (d)(1) selecting a quasi-sequencing {0, a 1 , a 2 , . . . , a c-1 } which is a sequence of elements {a i } of a group of c elements {0,a 1 ,a 2 , . . . ,a c-1 } where the following occurs: the group element a i such that a i =a i -1 occurs once and for each other group element: either (i) a i occurs twice and a i -1 does not occur, or, (ii) a i and a i -1 each occurs once, or (iii) a i -1 occurs twice and a i , does not occur; where the group of c elements is arranged such that the sequence of partial sums b 0 =0, b 1 ,=a 1 , b 2 =a 1 +a 2 , . . . , b c-1 , are all different; (d)(2) selecting one or more but not all pairs of elements a i and a j in the quasi-sequencing where a i =a j ; (d)(3) adding a constant c to a j for all such selected pairs, replacing a j with a j +c and modifying the quasi-sequencing to {0,a 1 ,a 2 , . . . ,a c-1 } where a j =a j +c; (d)(4) constructing a quasi sequencing of size 2c, where the quasi sequencing is {0,a 1 ,a 2 , . . . ,a c-1 ,c, 2c-a c-1 , . . . 2c-a 2 ,2c-a 1 }; (d)(5) repeating step (d)(3); (d)(6) constructing a quasi sequencing of size 4c, where {0,a 1 , . . . ,a c-1 ,c,2c-a c-1 , . . . ,2c-a 1 ,2c, 4c- 2c-a 1 ! . . . ,4c-a 1 }; (d)(7) repeating step (d)(3) k times to obtain a quasi sequencing of size m=c2 k ; and (d)(8) providing the permutation corresponding to the quasi sequencing.

Claim 6

The method of claim 1 wherein m is odd.

Claim 7

The method of claim 5 wherein in (d) selecting the quasi-quick trickle permutation comprises: (d)(1) selecting a quasi-sequencing {0, a 1 , a 2 , . . . , a c-1 } which is a sequence of elements {a i } of a group of m elements {0, a 1 , a 2 , . . . ,a m-1 }, where the following occurs: the group element a i such that a i =a i -1 occurs once and for any other group element, a i one of the following occurs: either (i) a i occurs twice and a i -1 does not occur, or (ii) a i and a i -1 each occurs once, or (iii) a i -1 occurs twice and a i does not occur; where the group of elements are arranged in such an order that the sequence of partial sums b 0 =0, b 1 ,=a 1 , b 2 =a 1 +a 2 , b 3 =a1+a 2 +a 3 , . . . , b m-1 are all different, wherein the sequence of distinct elements b i is called a terrace, and wherein the quasi-quick trickle permutation is one in which a totality of spacings from a location of an element i to a location of an element i+1 constitute a quasi-sequencing.