Error linear complexity measures for multisequences

Abstract

Complexity measures for sequences over finite fields, such as the linear complexity and the k-error linear complexity, play an important role in cryptology. Recent developments in stream ciphers point towards an interest in word-based stream ciphers, which require the study of the complexity of multisequences. We introduce various options for error linear complexity measures for multisequences. For finite multisequences as well as for periodic multisequences with prime period, we present formulas for the number of multisequences with given error linear complexity for several cases, and we present lower bounds for the expected error linear complexity.

Periodic multisequences with large error linear complexity

“…In this section we present the details of the definitions of error linear complexity measures in [16,Sect. 2] for the case of periodic multisequences.…”

“…But only recently in [16], generalizing the concept of k-error linear complexity for single sequences, Meidl et al introduced three options of defining error linear complexity measures for (finite length or periodic) multisequences. In [16] the definitions of k-error joint linear complexity, k-error F q -linear complexity, and k-error joint linear complexity were presented in detail only for the case of finite length multisequences. We present the detailed definitions for the periodic case in Sect.…”

“…2.3. The paper [16] marks the beginning of the theory of error linear complexity measures for multisequences. As suggested in that paper, an interesting task would be to find analogs of all major results on the k-error linear complexity of single sequences for the case of multisequences.…”

Linear Complexity Measures of Binary Multisequences

“…A necessary requirement for unpredictability of keystream sequence is long period, which can be attained by large linear complexity. Recent developments in stream ciphers point towards an interest in word based stream ciphers which require the study of complexity theory of multisequences i.e., of parallel streams of finitely many sequences, and of their complexity properties ( [3], [4], [11]). A cryptographically strong sequence should not only have a large linear complexity, but also changing a few terms should not cause any significant decrease of the linear complexity.…”

“…A cryptographically strong sequence should not only have a large linear complexity, but also changing a few terms should not cause any significant decrease of the linear complexity. This unfavorable property leads to the concept of k-error linear complexity [11]. Recently many authors studied various properties of k-error linear complexity of single and multisequences ( [2], [4], [5], [8], [9], [10], [11], [13], [14],16], [18] and [19]).…”