“P _i ⊕ P _j Theorem” on Standard Systems and “P _i ⊕ P _j Theorem” with Any ξ _max

Abstract

In this chapter, we will study and prove the “P _i ⊕ P _j Theorem” of Patarin (On linear systems of equations with distinct variables and small block size, Springer, 2005) on “standard systems” and the “P _i ⊕ P _j Theorem” with any ξ _max . Then, in Chap. 17, we will use these “P _i ⊕ P _j Theorems” (essentially on standard systems) to obtain tight security results on classical Feistel ciphers. “Standard systems” and ξ _max will be defined in Sect. 16.1. This chapter is essentially a generalization of what we did in Chap. 15. Moreover, since we will use the same proof technique (orange equation and then differentials on purple equations) it is recommended to read Chap. 15 before, or in parallel of this chapter.