REBEL: Reconfigurable Block Encryption Logic Page: 3
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2.2.3 Symmetry over Input Variables:
Any gate g can be represented as a boolean function of input variables. Let the set of n input variables to
the gate g is I(g) = {x1, x2, ..., xn}. The support set of g, Sup(g) C I(g), which is the set of all input
variables g's value depends on, i.e., if x E Sup(g) then 6g/6xi 0. A set of gates G is symmetric if for
each g E G, if any pair of input variables in Sup(g) are swapped in the expression for g leading to a gate g',
then g' E G. This property exhibits G's bias towards certain input variables. For example if the set G has
more gates with variable xi than any other variables then the set becomes biased towards xi and any change
in xi is reflected at the output with higher probability than a change in any other input variables.
Definition 2.2. A set of gates G is said to be symmetric in input variables ifV g E G the gate g' obtained
by swapping any two input variables from the support set of g is also present in G.
For g E Gb, the balance property dictates that there be 2n-1 minterms. A swap of xi and xj in the
expression for g does not alter the number of 1's (minterms). It merely, recodes the minterm locations.
Hence the gate g' derived from g through a swap of two variables will also have 2n-1 minterms, and hence
will be balanced, implying g' E Gb. Hence, the set Gb is symmetric over input variables. Now on, we will
use the term symmetric set G to mean a set symmetric in input variables.
2.2.4 Closed over variable-complement:
Definition 2.3. We define a set of gates G as closed over input variable-complement if V g E G the gate
g' obtained by complementing all literal instances of an input variable xi is also present in G.
Let g E Gb and let g' be obtained by complementing the variable xi. Since g is balanced the number
of 1's (minterms) is 2n-1. Each affected minterm gets relocated to a different row in the truth table through
complementing of a literal xi or xi. However, the number of minterms still remains at 2n-1, and hence g' is
also balanced. Thus Gb is closed over input variable-complement.
2.2.5 input-collision probability:
Definition 2.4. For any gate g, the input-collision probability pgoll is defined as the probability that any pair
of inputs x, x' ER In s.t. x x', produce the same output, i.e., pColl = P[g(x) = g(') I g]. Note that the
collision probability is averaged over all the input pairs.
For every g E Gb there are equal number (2n-1) of 0's and l's. For the collision to happen, the input x
has 2n out of 2n choices and the input zx has 2n-1 - 1 out of 2n 1 choices. Hence g 2l-1-1
2.2.6 gate-collision probability:
Definition 2.5. For any given input x E In gate-collision probability is defined as the probability that any
pair of gates g, g' ER G collide i.e., pGoll(x) = P[g(x) = g'(x) zx]. Note that the collision probability is
averaged over all the gate pairs.
The set Gb is closed over complement. Hence for every g E Gb the complement gate g is also present
in Gb. Thus every row of the truth table for the set is also balanced and p ol (x) - .
2.2.7 set-collision probability:
Definition 2.6. For any set of gates G, the set-collision probability p (x, x') is defined as the probability
that any gate g ER G collides for any given pair of inputs x, x' E In s.t. x x', i.e, pCOll(x, ')
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Gomathisankaran, Mahadevan & Tyagi, Akhilesh. REBEL: Reconfigurable Block Encryption Logic, paper, September 1, 2006; (https://digital.library.unt.edu/ark:/67531/metadc96662/m1/3/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Engineering.