On a Conjecture about Binary Strings Distribution

Abstract

It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. A lot of attacks has led to design criteria for these functions; mainly: balancedness, a high algebraic degree, a high nonlinearity, a good behavior against Fast Algebraic Attacks and also a high algebraic immunity (which is now an absolutely necessary criterion (but not sufficient) for cryptographic Boolean functions).Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many good cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true:Conjecture 0.1. Let S t,k be the following set:Then:The main contribution of the present paper is the reformulation of the problem in terms of carries which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of |S t,k | for numbers whose binary expansion is made of one block (see theorem 3.8), a proof that the conjecture is asymptotically true (see theorem 3.12) and a proof that a family of numbers (whose binary expansion has a high number of 1s and isolated 0s) reaches the bound of the conjecture (see theorem 3.17). We also conjecture that the numbers in that family are the only ones reaching the bound (see conjecture 3.20).

On the Number of Carries Occurring in an Addition Mod

“…An equivalent conjecture was originally proposed by Tu and Deng [8] in a different context. For the connection between the conjecture of Tu and Deng and the one given here, we refer the reader to [4]. Tu and Deng verified computationally the validity of their assumption for k Ä 29.…”

“…A list of the different cases proven to be true can be found in [5,Section 5]. Unfortunately a direct proof or a simple recursive one seems hard to find ( [5,Section 4]).…”

“…Up to now, different attempts ( [2][3][4][5]) were conducted and lead to partial proof of the conjecture in very specific cases. A list of the different cases proven to be true can be found in [5,Section 5].…”

“…Unfortunately a direct proof or a simple recursive one seems hard to find ( [5,Section 4]). What however came out of these works is that supposing that t has a high Hamming weight ( [2,3]) or more generally that its 0 bits are sparse ( [4,5]) greatly simplifies the study of P t;k . This condition casts a more algebraic and probabilistic structure upon it.…”

Linear Approximations of Addition Modulo 2n-1

“…Some properties of the addition modulo 2 n − 1 have been explored in [18,19]. However few results on linear approximations on the addition modulo 2 n − 1 can be found from public literature.…”