# On the inverses of Kasami and Bracken-Leander exponents

We explicitly determine the binary representation of the inverse of all Kasami exponents K_r=2^2r-2^r+1 modulo 2^n-1 for all possible values of n and r. This includes as an important special case the APN Kasami exponents with (r,n)=1. As a corollary, we determine the algebraic degree of the inverses of the Kasami functions. In particular, we show that the inverse of an APN Kasami function on F_2^n always has algebraic degree n+1/2 if n≡ 0 3. For n≢0 3 we prove that the algebraic degree is bounded from below by n/3. We consider Kasami exponents whose inverses are quadratic exponents or Kasami exponents. We also determine the binary representation of the inverse of the Bracken-Leander exponent BL_r=2^2r+2^r+1 modulo 2^n-1 where n=4r and r odd. We show that the algebraic degree of the inverse of the Bracken-Leander function is n+2/2.

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