Discrete mathematics for cryptography, code and pseudo random number generator

2014 Fiscal Year Final Research Report

• Project/Area Number: 23740070
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Ochanomizu University
• Principal Investigator: HAGITA Mariko お茶の水女子大学, 大学院人間文化創成科学研究科, 准教授 (70338218)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 離散数学 / 暗号 / 符号 / 擬似乱数 / 誤り訂正符号系列 / グラフ彩色 / m系列 / ド・ブライン系列

Outline of Final Research Achievements

We define an (N,k,d) error-correcting sequence over X as a periodic sequence {a_i}_{i=0,1,\ldots} (a_i \in X) with period N, such that its sub k-tuples {(a_i, a_{i+1}, \ldots, a_{i+k-1})|i=0,1,\ldots, N-1} (multiset) are all distinct for 0 \leq i \leq N-1, and that they form an error-correcting code with minimum distance d:= \min_{0 \leq s<t \leq N-1}{\sum_{i=0,1,2,...,k-1}\delta(a_{i+s},a_{i+t})}, where \delta(x,y)=1 for x \neq y and =0 for x=y. If d \geq 2e+1, then one can correct up to e errors in a k-tuple, so the sequence is said to be e-error correcting.
An m-sequence over GF(q) of period {q to the n} -1 is a ({q to the n}-1,n,1) error-correcting sequence. We considered when an m-sequence will be an error-correcting sequencewith minimum distance d=3 or d=5 and we gave some new constructions of error-correcting sequences.

Free Research Field

離散数学