Method for determining affine equivalence of two Boolean functions of arbitrary variables
A technology of Boolean functions and determination methods, which is applied in the fields of digital integrated circuits and cryptography, can solve the problems of unresolved affine equivalence determination of Boolean functions and the selection of representative elements, not given, etc., so as to reduce workload and enrich The effect of the function
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Embodiment 1
[0036] Embodiment 1 Determine whether two Boolean functions are affine equivalent
[0037] according to figure 1 , taking n=3 as an example, the set of Boolean functions containing 4 1s is set to F 4 , for F 4 For any two Boolean functions f and g in , the truth table is shown in Table 1, and the corresponding variable value matrix is shown in formula (1). And abbreviate f and g as: f=(0,1,1,0,1,1,0,0), g=(1,1,0,0,0,0,1,1).
[0038] Table 1: Truth tables for f and g
[0039] X 1
[0040]
[0041] Through the rules of affine transformation, the matrix is subjected to several column elementary transformations, and the rank(A f )=3,rank(A g )=2, so f and g are not affine equivalent.
[0042] Conversely, if two functions are affine equivalent, then the ranks of the variable value matrices corresponding to the two functions are equal. For example: suppose the truth table corresponding to f is shown in Table 1, and under the action of the affine transformatio...
Embodiment 2
[0049] Embodiment 2 Select the representative element of the same equivalence class
[0050] For steps 4 and 5, follow the figure 1 In the process of , taking the case of n=3 as an example, when m=4, select the representative element of each affine equivalence class in F4.
[0051] Let the set of Boolean functions similar to f be denoted as M f ,M f The representative element r in f The selection method is as follows:
[0052] Put the matrix A corresponding to f f Carry out a series of elementary column transformations, put A f Matrixize to R f , such as formula (4), then R f The corresponding Boolean function is the representative element of this kind of affine-equivalent Boolean function.
[0053]
[0054] Then M f The representative element r f =(1,1,1,0,1,0,0,0,0).
[0055] The representative element corresponding to g can be obtained by a similar method R g , such as formula (5);
[0056]
[0057] m g The corresponding representative element r g =(1,1...
Embodiment 3
[0059] Embodiment 3 Application in combinational logic circuits
[0060] Taking the Boolean function of 4 variables as an example, when m=7, the representative element of the Boolean function is determined by the above method such as image 3 shown, according to Figure 4 A circuit Ci can be designed for each representative element ri, where r1=(1,1,1,1,1,1,1,0,0,0,0,0,0,0,0 ,0), r2=(1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0), r3=(1,1,1,1 ,1,0,0,0,1,0,0,0,1,0,0,0). Here take r2 as an example to design circuit C such as Figure 7 , after simplification according to the truth table of r2, the CNF expression of r2 is r2=x1’x2’+x1’x2x3’+x1x2’x3’x4’.
[0061] The circuit realization of the affine equivalent Boolean function f=(1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0) can be achieved by The linear combination of the input gate circuits is obtained, where f=x1'x2'+x1x2'x3'+x1'x2x3'x4'. Obtain the affine transformation (A, b) formula (6) according to f=r2(AX+b) as follows:
[0062]
[0063]...
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