By Dan Gusfield

Typically a space of research in desktop technology, string algorithms have, in recent times, develop into an more and more vital a part of biology, rather genetics. This quantity is a complete examine laptop algorithms for string processing. as well as natural machine technology, Gusfield provides huge discussions on organic difficulties which are solid as string difficulties and on equipment built to unravel them. this article emphasizes the elemental rules and methods relevant to cutting-edge functions. New methods to this complicated fabric simplify tools that during the past were for the expert by myself. With over four hundred workouts to augment the fabric and improve extra issues, the booklet is acceptable as a textual content for graduate or complicated undergraduate scholars in machine technology, computational biology, or bio-informatics.

**Read Online or Download Algorithms on Strings, Trees and Sequences - Computer Science and Computational Biology PDF**

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**Extra info for Algorithms on Strings, Trees and Sequences - Computer Science and Computational Biology**

**Example text**

Es gibt einfache, praktische Verfahren, um große Primzahlen zu erzeugen. Man baut große Primzahlen iterativ aus kleinen Primzahlen zusammen. 17 Seien p1 , . . , pr prim und p = 1 + (p−1)/pi ai =1 (mod p), r ei i=1 pi . ap−1 =1 i Aus (mod p) f¨ ur i = 1, 2, . . , r folgt, daß p prim ist. (p−1)/pi Beweis. ai Nach CRT folgt prim. = 1 (mod p), ap−1 = 1 (mod p) impliziert pei i ord(ai ). i r ei i=1 pi |ord(ai ), und somit ϕ(p) = p − 1. Damit ist p ✷ F¨ ur prime p, p − 1 = r ei i=1 pi gilt andererseits f¨ ur zuf¨allige a ∈ Z∗p Ws a(p−1)/pi = 1 (mod p) = 1 − 1 pi .

Wir zeigen, daß ordpe (y) = ϕ(pe ) bereits ordpe+1 (y) = ϕ(pe+1 ) ¨ impliziert. Zur Ubung beweist man die Behauptung: z = 1 mod pe , z = 1 mod pe+1 =⇒ e z p = 1 mod pe+1 , z p = 1 mod pe+2 e Sei ordpe (y) = ϕ(pe ) und z = y ϕ(p )/p , wobei ϕ(pp ) = ϕ(pe−1 ). Dann gilt e z = 1 mod pe−1 , aber z = 1 mod pe . Diese Behauptung liefert y ϕ(p ) = 1 (mod pe+1 ) und folglich ordpe+1 (y) = ϕ(pe+1 ). KAPITEL 4. RSA UND STRUKTUR VON Z∗N 38 2. 1) (mod 2r ) : j = 1, 2, . . 2) folgt jetzt aus −1 = 5j (mod 2r ) f¨ ur j = 1, 2, .

R . Beweis. 4 schließt man: λ(N ) = kgV {λ(pei i ) : i = 1, 2, . . 8. ✷ Man w¨ ahlt den RSA-Modul N = P1 ·P2 so, daß ggT(P1 − 1, P2 − 1) = 2. Dann gilt λ(N ) = 21 ϕ(N ). 3 ∈ {0, 1, . . , ϕ(N ) − 1} ∗ Pseudoprimzahlen und Carmichael-Zahlen Zum RSA-Schema ben¨ otigt man große zuf¨allige Primzahlen. Weil die Primzahlen ≤ N etwa die Dichte log1N haben, gen¨ ugt ein effektiver Primheitstest. 3. ∗ PSEUDOPRIMZAHLEN UND CARMICHAEL-ZAHLEN 39 F¨ ur jede Primzahl p gilt die Fermat-Identit¨ at ap−1 = 1 mod p f¨ ur alle ∗ a ∈ Zp .