By Idun Reiten, Sverre O. Smalø, Øyvind Solberg
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Additional info for Algebras and Modules Two
Mm0}. If we have further information about the pairs (F, Mi), then we can make use of it to make the set containing the mother smaller or to determine the mother. 5. Security and Attacks 43 instance, by studying the known information about all (F, Mi), suppose we know that twins are only possible for (F, M1). Then if twins are found in the 10 children, the mother must be/1//1. Of course, it may be technically difficult to find this character of (F, M1). In fact we can identify the keys with mothers, the algorithm with the father, and the ciphertext with the children.
For example, 10 is a primitive root of 487, b-ut not a primitive root of 4872. The following old result (, for a proof see  for example) clarifies the situation. 1 L e t p be a p r i m e , t h e n the f o l l o w i n g three assertions are equivalent: 54 Chapter 3. Primes, Primitive Roots and Sequences 1. g is a primitive root of p and gp-1 ~ 1 (mod p2); 2. g is a primitive root of p2; 3. ]or every e >_ 2, g is a primitive root of pC. Now we investigate the linear and sphere complexity of sequences with period equal to a prime power.
8N-1 x g - 1 , sOSl"'sN-1 is the first periodic s e g m e n t of s ~176 where s g = P r o o f : Note that the minimal polynomial of s cr is x N- (x N' - 1)Ph 1 gcd(x N - 1, s N (x)) gcd((x N' - 1)P*, s N ( x ) ) ' where N ' = p~l ... p~,. By assumption the minimal polynomial of s ~176 has no factor of the form ( x - 1) h. 5. 4 Primes, Primitive Roots and Binary Sequences We begin this section with two definitions. It is obvious that every odd prime p must be in one of the forms 4t 4- 1. For simplicity we call primes of the forms 4t 4- 1 with t odd o-primes, those with t even e-primes.
Algebras and Modules Two by Idun Reiten, Sverre O. Smalø, Øyvind Solberg