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Table 1: Comparison of Over all Performance.
Parameters Bitwise XOR Addition Slash Nmix
Nonlinearity of output bit y
i
0 2
2i
2
2i+1
− 2
i+1
2
2i+1
− 2
i+2
Bias for best linear approx. of output bit y
i
0.5 0.25 2
−(i+1)
2
−i
Bias for best linear approx. of y
i
⊕ y
i+1
0.5 0.25 0.25 0.0625
Provide differential resistance No Yes No Yes
Algebraic degree of output bit y
i
1 i+ 1 2 2
Number of gates to implement n-bit function n XOR 2n+ 1 XOR 3n− 3 XOR 5n− 7 XOR
(froward transformation) 0 AND 2n− 4 AND n− 1 AND 3n− 5 AND
0 OR n− 2 OR 0 OR 0 OR
Time Complexity O(1) O(n) O(n) O(n)
So it is observed that the probability of a particu-
lar output difference ∆Y occurs given a particular in-
put difference ∆X is function of all x
i
’s and x
′
i
’s for
a fixed K. In other words Nmix has the property that
a given XOR difference does not necessarily yield a
fixed output difference. Therefore the proposed func-
tion provides differential resistance.
4.5 Comparison
In this section, we discuss cryptographic properties,
hardware and time complexity of our proposed func-
tion as well as exiting modular addition and Slash.
Table 1 shows a comparison of the proposed function
with the similar existing key mixing functions. From
table 1, it is observed that XOR provides no nonlin-
earity, modulo Slash has maximum nonlinearity and
Nmix has nonlinearity greater than modulo addition
but less compared to Slash. The bias for best lin-
ear approximation of y
i
decreases exponentially with
bit position in case of Nmix and Slash but remains
constant both in modulo addition and bitwise XOR.
Also the proposed function has lowest value for bias
of best linear approximation of y
i
⊕ y
i+1
compared
to other three functions. Similar to modulo addition,
Nmix provides differential resistance but bitwise XOR
and Slash does not provide any differential resistance.
Addition has higher algebraic degree compared to
both Slash and Nmix. To implement a n-bit key mix-
ing layer bitwise XOR requires minimum number of
logic gates. Proposed function needs 5n − 7 XOR
gates and 3n−5 AND gates to implement a n-bit mix-
ing layer. Bitwise XOR requires minimum computa-
tion time, whereas other three nonlinear mixing func-
tions have O(n) time complexity. Therefore Nmix is
an ideal key mixing function for cryptographic prim-
itives.
5 CONCLUSIONS
In this work, a highly nonlinear, balanced and re-
versible Boolean function Nmix has been proposed
which can be used as nonlinear key mixing func-
tion in block ciphers. It has been shown that the
proposed function provides higher nonlinearity com-
pared to modular addition. Therefore Nmix gives bet-
ter resistance against linear cryptanalysis. Also Nmix
provides resistance against differential attack.
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