Pantelimon Stănică

Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions

Recent research shows that the class of Rotation Symmetric Boolean Functions (RSBFs), i.e., the class of Boolean functions that are invariant under circular translation of indices, is potentially rich in functions of cryptographic significance. Here we present new results regarding the Rotation Symmetric (rots) correlation immune (CI) and bent functions. We present important data structures for efficient search strategy of rots bent and CI functions. Further, we prove the nonexistence of homogeneous rots bent functions of degree ≥ 3o n a single cycle.

Fast evaluation, weights and nonlinearity of rotation-symmetric functions

Abstract We study the nonlinearity and the weight of the rotation-symmetric ( RotS ) functions defined by Pieprzyk and Qu. We give exact results for the nonlinearity and weight of 2-degree RotS functions with the help of the semi-bent functions and we give the generating function for the weight of the 3-degree RotS function. Based on the numerical examples and our observations we state a conjecture on the nonlinearity and weight of the 3-degree RotS function.

Boolean Functions with Five Controllable Cryptographic Properties

The strict avalanche criterion (SAC) was introduced by Webster and Tavares [10] in a study of cryptographic design criteria. This is an indicator for local property. In order to improve the global analysis of cryptographically strong functions, Zhang and Zheng [11] introduced the global avalanche characteristics (GAC). The sum-of-squares indicator related to the GAC is defined as σf=∑vΔf2(v), where Δf(v)=∑x (−1)f(x)⊕f(x⊕ v). In this paper, we give a few methods to construct Boolean functions controlling five good cryptographic properties, namely balancedness, good local and GAC, high nonlinearity and high algebraic degree. We improve upon the results of Stanica [8] and Zhang and Zheng [11].

Improving the nonlinearity of certain balanced Boolean functions with good local and global avalanche characteristics

The design and evaluation of cryptographic functions requires the definition of design criteria. In [2], Preneel et al. introduced the propagation criterion of degree k (PC of degreek or PC(k)): a function satisfies thePC(k) if by complementing at most k bits the output changes with probability exactly one half. This is a generalization of the Strict Avalanche Criterion or SAC (for k = 1), introduced by Webster and Tavares in [6]. As many authors observed, the PC is a very important concept in designing cryptographic primitives used in data encryption algorithms and hash functions. However, the PC captures only local properties of the function. In order to improve the global analysis of cryptographically strong functions, Zhang and

Stream Cipher Design

The stream ciphers considered in this chapter are binary additive stream ciphers, that is, a ciphertext bitstring is produced by adding a plaintext bitstring and a binary keystream. Thus the security of the cipher depends entirely on the randomness of the keystream. It is not easy to define precisely what a random bitstring is, but certainly the essence of a random bitstring is the inability to predict any bits in the string with a better probability of success than one would have by simply making guesses. Many statistical tests have been devised such that any pseudorandom bit generator (PRBG) must pass each test in order not to have some specific statistical weakness. The theory of testing whether a given generator passes all possible statistical tests is explained. Many kinds of generators and various attacks on them are discussed. In particular, the relatively recent algebraic and linearization attacks are summarized. This is a very active current area of research, so details of the many recent results cannot be included due to lack of space.

On the prime power factorization of n

Abstract In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p1,…,pk, and for fixed integers m1,…,mk, with p i ∤m i , the numbers (ep1(n),…,epk(n)) are uniformly distributed modulo (m1,…,mk), where ep(n) is the order of the prime p in the factorization of n!. That implies one of Sanders conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence e i ∈{0,1}, i⩽f(x) , there exists n e p i (n)≡e i ( mod 2) hold for all i⩽f(x). Here, pi is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least c 1 ( log x/( log log x) 6 ) 1/9 , with some absolute constant c1, provided that only the first odd prime numbers are involved.

Chapter 7 – Block ciphers

Publisher Summary This chapter deals with symmetric encryption algorithms, focusing on block ciphers. Any block cipher can be operated in several modes. There are four modes that are more often used; electronic codebook (ECB), cipher block-chaining (CBC), output feedback (OFB), and cipher feedback (CFB). The ECB mode uses the simplest idea, that is, to have the message partitioned into several blocks of n-bit length padding if necessary, and then to apply the cipher to each block. The advantage of this mode is that one can decrypt the n-bit blocks independently, in parallel. Thus, if one error occurs in transmission or encryption, that error affects only that particular block, where the error occurred. The CBC mode encrypts the first block using the ECB method applied to the xor of the first block with a block IV. The OFB mode, which is an alternative to CBC to avoid error propagation, produces a unique stream independent from other streams produced by the same encryption key, without rekeying. It uses the output of the encryption function as a feedback, as opposed to the CFB mode, where the output of the ciphertext is used as a feedback.

PRIME DIVISORS OF LUCAS SEQUENCES AND A CONJECTURE OF SKAŁBA

In this paper, we give some heuristics suggesting that if (un)n≥0 is the Lucas sequence given by un = (an - 1)/(a - 1), where a > 1 is an integer, then ω(un) ≥ (1 + o(1))log n log log n holds for almost all positive integers n.

Cullen Numbers in Binary Recurrent Sequences

Mentioned in the excellent book of R. Guy [5], the Cullen numbers are elements of the sequence C n := n·2 n + 1 (see Section B20 of [5]). They happen to be composite (see [2] and [8] ) for all 1 ≤ n n > 0 and a similar result holds for the Woodall numbers. Here, and elsewhere throughout this paper, we use the Vinogradov symbols ≫ and ≪, as well as the Landau symbols O and o with their usual meanings, and for a real number x ≥ 1 we use log x for the natural logarithm of x.

pq-Catalan numbers and squarefree binomial coefficients

In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binom{sn}{n}. We find all