here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

1.2. Families of Boolean functions with good cryptographic properties 153. of optimal algebraic immunity n/2 if Conjecture 1.2.2 is verified,4. of nonlinearity at least∑t−12 n−1 − 2 n/(2i+1 )−1 − 2 (m−1)/2 ;i=0and their second family is1. 1-resilient,2. of optimal algebraic degree n − 2,3. of algebraic immunity at least n/2 − 1 if Conjecture 1.2.2 is verified,4. of nonlinearity at least{( ∑t−1)2 n−1 − 2 n/2−1 − 3i=1 )−1 2n/(2i+1 − 2 (m−1)/2if m = 1 ,2 n−1 − 2 n/2−1 − 3 ∑ t−1i=1 2n/(2i+1 )−1 − 2 (m+1)/2 − 6 if m ≥ 2 .Unfortunately, Carlet [36] observed that the functions introduced by Tu and Deng are weakagainst fast algebraic attacks and unsuccessfully tried to repair their weakness. It was subsequentlyshown by Wang and Johansson [277] that this family can not be easily repaired.Nonetheless, more recent developments have shown that the construction of Tu and Dengand the associated conjecture are not of purely æsthetic interest, but are interesting tools in acryptographic context.1.2.4 The Tang–Carlet–Tang familyIn 2011, inspired by the previous work of Tu and Deng [264], Tang, Carlet and Tang [259]constructed an infinite family of Boolean functions with many good cryptographic properties.The main idea of their construction is to change the division in the construction of Tu and Dengby a multiplication. The associated combinatorial conjecture is then modified as follows.Conjecture 1.2.7 (Tang–Carlet–Tang conjecture). For all k ≥ 2 and all t ∈ ( Z/(2 k − 1)Z ) ∗,{# (a, b) ∈ ( Z/(2 k − 1)Z ) }2| a − b = t; wH (a) + w H (b) ≤ k − 1 ≤ 2 k−1 .They verified it experimentally for k ≤ 29, as well as the following generalized property fork ≤ 15 where u ∈ Z/(2 k − 1)Z is such that gcd(u, 2 k − 1) = 1 and ɛ = ±1.Conjecture 1.2.8 (Tang–Carlet–Tang conjecture). Let k ≥ 2 be an integer, t ∈ ( Z/(2 k − 1)Z ) ∗,u ∈ Z/(2 k − 1)Z such that gcd(u, 2 k − 1) = 1 and ɛ ∈ {−1, 1}. Then{# (a, b) ∈ ( Z/(2 k − 1)Z ) }2| ua + ɛb = t; wH (a) + w H (b) ≤ k − 1 ≤ 2 k−1 .This generalized conjecture includes the original conjecture proposed by Tu and Deng (Conjecture1.2.2) for u = 1 and ɛ = +1.The construction of their functions is as follows.