A substitution box or S-box is one of the basic components of symmetric key cryptography. In general, an S-box takes n input bits and transforms them into m output bits. This is called an n x m S-box and is often implemented as a lookup table.

Another representation is by vectorial Boolean function, which is defined by a Polynomial Ring in x over Finite Field in a of size 2ⁿ with primitive element g.

Note that by default bits are interpreted in little endian notation, e.g.

6₁₀ = 110₂

10₁₀ = 1010₂ = A₁₆

• Oleksandr Kazymyrov
• Maksim Storetvedt
• Anna Maria Eilertsen

1. In Sage. Go to the ''Sage'' directory and type load("./Sbox.sage").
2. Use externaly (see examples in ''Main.sage'' ).

• Create an example of a 3 x 3 substitution

    sage: S = Sbox(n=3,m=3,sbox=[1,2,3,0,7,6,5,4])
    sage: S.get_sbox()
    [1, 2, 3, 0, 7, 6, 5, 4]
    
    sage: F=S.interpolation_polynomial()
    sage: F
    'g^2*x^6 + g^3*x^5 + g^4*x^4 + g^5*x^3 + g^6*x^2 + 1'
    
    sage: S.generate_sbox(method='polynomial',G=F)
    sage: S.get_sbox() == [1, 2, 3, 0, 7, 6, 5, 4]
    True
  

• Generate APN permutation for n = 6 [1]

    sage: S=Sbox(n=6,m=6)
    sage: S.generate_sbox(method='APN6')
    sage: S.is_APN()
    True
    sage: S.MDT()
    2
    sage: S.is_bijection()
    True
  

• Apply CCZ and EA equivalence

    sage: S=Sbox(n=3,m=3)
    sage: S.generate_sbox(method='polynomial',G='x^3')
    sage: S
    [0, 1, 3, 4, 5, 6, 7, 2]
    
    sage: S.generate_sbox(method='polynomial',G='x^3',T='CCZ')
    sage: S # random
    [0, 6, 2, 3, 6, 3, 6, 4]
    sage: S.is_APN()
    True
    
    sage: S.generate_sbox(method='polynomial',G='x^3',T='EA')
    sage: S # random
    [7, 0, 7, 5, 0, 6, 6, 5]
  
    sage: S.is_APN()
    True
  

1. J. F. Dillon, APN Polynomials: An Update. Fq9, International Conference on Finite Fields and their Applications, University College Dublin. July 2009.
2. Sage