# Problems on ANF equation?

I tried to solve SBox as given in this link

I am trying to understand this paper SBox. In this on page no.4 the correct equation of given SBox is mentioned. I am not getting how to design ANF equation as given in paper.

• @kelalaka I given paper refernce which I am trying to implement. – mahima bhatnagar Oct 5 '20 at 11:32
• @kelalaka How can I ask directly to them....??? – mahima bhatnagar Oct 5 '20 at 11:36
• @kelalaka no email-id information is given. If you get answer then please tell. – mahima bhatnagar Oct 5 '20 at 11:40
• Here the original article in the IEEE and research gate. The rest is google. – kelalaka Oct 5 '20 at 11:55
• For the love of God, please typeset, do not post text as images. – hola Feb 17 at 13:07

Using the S-box package of SageMath used

S = SBox(1, 10, 4, 12, 6, 15, 3, 9, 2, 13, 11, 7, 5, 0, 8, 14);

f0 = S.component_function(1)
f1 = S.component_function(2)
f2 = S.component_function(4)
f3 = S.component_function(8)

print ( "y0 = ", f0.algebraic_normal_form())
print ( "y1 = ", f1.algebraic_normal_form())
print ( "y2 = ", f2.algebraic_normal_form())
print ( "y3 = ", f3.algebraic_normal_form())


Return a Boolean function corresponding to the component function $$b\cdot S(x)$$.

and the output \begin{align} y_0 &= x_0 x_1 + x_0 + x_1 + x_2 + x_3 + 1\\ y_1 &= x_0 x_1 + x_0 x_2 + x_0 + x_2 + x_3\\ y_2 &= x_0 x_3 + x_1 x_2 x_3 + x_1 x_3 + x_1 + x_2\\ y_3 &= x_0 x_2 x_3 + x_0 + x_1 x_3\\ \end{align}

The blow the result of the article

\begin{align} s_0 &= 1 + a + b + ba + c + d \\ s_1 &= a + ba + c + ca + d\\ s_2 &= b + c + da + db + dcb\\ s_3 &= a + bd + dca\\ \end{align}

And the result form Conchild code \begin{align} y_0 &= 1 + x_0 + x_1 + x_0*x_1 + x_2 + x_3\\ y_1 &= x_0 + x_0*x_1 + x_2 + x_0*x_2 + x_3\\ y_2 &= x_1 + x_2 + x_0*x_3 + x_1*x_3 + x_1*x_2*x_3\\ y_3 &= x_0 + x_1*x_3 + x_0*x_2*x_3\\ \end{align}

Martin R. Albrecht provides another method with SageMath

from sage.crypto.sbox import SBox

S = SBox (1, 10, 4, 12, 6, 15, 3, 9, 2, 13, 11, 7, 5, 0, 8, 14)
P.<y0 ,y1 , y2 ,y3 ,x0 , x1 ,x2 ,x3 > = PolynomialRing ( GF (2) , order ='lex')
X = [x0 ,x1 ,x2 , x3 ]
Y = [y0 ,y1 ,y2 , y3 ]
S. polynomials (X=X , Y=Y , degree =3 , groebner = True )


The result is compatible with the article but the $$y_i$$'s reversed.

[y0 + x0*x1*x3 + x0*x2 + x3,
y1 + x0*x1*x2 + x0*x2 + x0*x3 + x1 + x2,
y2 + x0 + x1*x3 + x1 + x2*x3 + x3,
y3 + x0 + x1 + x2*x3 + x2 + x3 + 1]

• I've posted this answer as two show that there is a different result with sageMath and the article. If there is a real answer, they can use these equations, freely. – kelalaka Oct 5 '20 at 11:52
• then can you suggest what should I do now because I checked this with other SBox table also of different algorithm it is happening with that also. Is Mobius Transformation method is not applicable to all SBox ?? – mahima bhatnagar Oct 5 '20 at 12:11
• I've not applied Mobius Transformation but the results should be identical. Why don't you post your calculation code instead of giving only the result? One cannot reproduce them by looking at the input and the result. – kelalaka Oct 5 '20 at 12:13
• @mahimabhatnagar, please post your detailed calculations/code given that kelalaka has spent so much effort trying to help you – kodlu Oct 5 '20 at 19:38
• @kelalaka in Boolean functions, component typically means linear combination of outputs (in contrast to coordinates - single output bits). E.g. component_function(3) return XOR of two least significant bits, not the third bit. – Fractalice Feb 19 at 14:27