Evaluating Algebraic Complexity of a S-box
The interpolation of Sbox is based on Lagrange algorithm.
the following three steps are used to calculate the interpolation of S-box:


represent the S-box values as polynomial system, where P and C are polynomial representation of S-box input and output.
M (N x N ) is a matrix composed of P power polynomial. N is sbox size (256 for 8-bit sbox)
solve B form the following equation: M.B=C 



# the Equation is M.B=C.
#     1  p0  p0^2  p0^3  p0^4  p0^5 .......  p0^d   
#     1  p1  p1^2  p1^3  p1^4  p1^5 .......  p1^d
#  M= 1  p2  p2^2  p2^3  p2^4  p2^5 .......  p2^d
#     1  p3  p3^2  p3^3  p3^4  p3^5 .......  p3^d   
#       .
#       .
#       .
#     1  pn  pn^2  pn^3  pn^4  pn^5 .......  pn^d 
#  size of M is dXn , d and n are equal for sbox, they are GF (2^8) (8-bit sbox)
# Calculation steps. 
#   1. convert Sbox table to polynomial of input and output, vectors (P is input  and C is output)
#   2. calculate M table  
#   3. B=M.solve_right(C)
#



The following code is implementation of interpolation representation of Rijndael S-box

f=x^8+x^4+x^3+x^1+1  # irreducible polynomial
L.&lt;z&gt;= GF(2^8,modulus=f)
R=PolynomialRing(L,'x')
N=256
M=matrix(L,N,N)
S=[0x63, 0x7C, 0x77, 0x7B, 0xF2, 0x6B, 0x6F, 0xC5, 0x30, 0x01, 0x67, 0x2B, 0xFE, 0xD7, 0xAB, 0x76, 
                             0xCA, 0x82, 0xC9, 0x7D, 0xFA, 0x59, 0x47, 0xF0, 0xAD, 0xD4, 0xA2, 0xAF, 0x9C, 0xA4, 0x72, 0xC0,
                             0xB7, 0xFD, 0x93, 0x26, 0x36, 0x3F, 0xF7, 0xCC, 0x34, 0xA5, 0xE5, 0xF1, 0x71, 0xD8, 0x31, 0x15,
                             0x04, 0xC7, 0x23, 0xC3, 0x18, 0x96, 0x05, 0x9A, 0x07, 0x12, 0x80, 0xE2, 0xEB, 0x27, 0xB2, 0x75,
                             0x09, 0x83, 0x2C, 0x1A, 0x1B, 0x6E, 0x5A, 0xA0, 0x52, 0x3B, 0xD6, 0xB3, 0x29, 0xE3, 0x2F, 0x84,
                             0x53, 0xD1, 0x00, 0xED, 0x20, 0xFC, 0xB1, 0x5B, 0x6A, 0xCB, 0xBE, 0x39, 0x4A, 0x4C, 0x58, 0xCF,
                             0xD0, 0xEF, 0xAA, 0xFB, 0x43, 0x4D, 0x33, 0x85, 0x45, 0xF9, 0x02, 0x7F, 0x50, 0x3C, 0x9F, 0xA8,
                             0x51, 0xA3, 0x40, 0x8F, 0x92, 0x9D, 0x38, 0xF5, 0xBC, 0xB6, 0xDA, 0x21, 0x10, 0xFF, 0xF3, 0xD2,
                             0xCD, 0x0C, 0x13, 0xEC, 0x5F, 0x97, 0x44, 0x17, 0xC4, 0xA7, 0x7E, 0x3D, 0x64, 0x5D, 0x19, 0x73,
                             0x60, 0x81, 0x4F, 0xDC, 0x22, 0x2A, 0x90, 0x88, 0x46, 0xEE, 0xB8, 0x14, 0xDE, 0x5E, 0x0B, 0xDB,
                             0xE0, 0x32, 0x3A, 0x0A, 0x49, 0x06, 0x24, 0x5C, 0xC2, 0xD3, 0xAC, 0x62, 0x91, 0x95, 0xE4, 0x79,
                             0xE7, 0xC8, 0x37, 0x6D, 0x8D, 0xD5, 0x4E, 0xA9, 0x6C, 0x56, 0xF4, 0xEA, 0x65, 0x7A, 0xAE, 0x08,
                             0xBA, 0x78, 0x25, 0x2E, 0x1C, 0xA6, 0xB4, 0xC6, 0xE8, 0xDD, 0x74, 0x1F, 0x4B, 0xBD, 0x8B, 0x8A,
                             0x70, 0x3E, 0xB5, 0x66, 0x48, 0x03, 0xF6, 0x0E, 0x61, 0x35, 0x57, 0xB9, 0x86, 0xC1, 0x1D, 0x9E,
                             0xE1, 0xF8, 0x98, 0x11, 0x69, 0xD9, 0x8E, 0x94, 0x9B, 0x1E, 0x87, 0xE9, 0xCE, 0x55, 0x28, 0xDF,
                             0x8C, 0xA1, 0x89, 0x0D, 0xBF, 0xE6, 0x42, 0x68, 0x41, 0x99, 0x2D, 0x0F, 0xB0, 0x54, 0xBB, 0x16] # rajS-box 

C=vector(L,N)
P=vector(L,N)

# step 1
for i in range(0,N):

    P[i]= (i) + z^1*(i&gt;&gt;1) + z^2*(i&gt;&gt;2) + z^3*(i&gt;&gt;3) + z^4*(i&gt;&gt;4) + z^5*(i&gt;&gt;5) + z^6*(i&gt;&gt;6) + z^7*(i&gt;&gt;7)
    C[i]= S[(i)] + z^1*(S[i]&gt;&gt;1) + z^2*(S[i]&gt;&gt;2) + z^3*(S[i]&gt;&gt;3) + z^4*(S[i]&gt;&gt;4) + z^5*(S[i]&gt;&gt;5) + z^6*(S[i]&gt;&gt;6) + z^7*(S[i]&gt;&gt;7)


#print P
#print C
# step 2
for j in range(0,N):
    for i in range(0,N):
        M[j,i]= P[j]^i





#step 3
B=M.solve_right(C)



print B