# Collected mathematical papers, volume 2 by Sylvester J.J.

By Sylvester J.J.

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27. Wie lautet der Satz von Euler? 28. Erl¨autern Sie die Aufgabenstellung der Kryptographie! 29. Erl¨autern Sie die Verschiebe-Chiffre! 30. Erl¨autern Sie die Tausch-Chiffre! ¨ ¨ 31. Was muss der Schlussel einer Tausch-Chiffre erfullen? 32. Erl¨autern Sie das Prinzip eines Public-Key-Kryptosystems! 33. Erl¨autern Sie das Kryptosystem nach Hellman, Merkle, Diffie! 34. Erl¨autern Sie das RSA-System! 35. Erl¨autern Sie die digitale Unterschrift mit Hilfe des RSA-Systems! 36. Erl¨autern Sie das ElGamal-System!

Eigenwerte und Eigenvektoren einer quadratischen Matrix berechnen konnen. ¨ 16. Entscheiden konnen, ob eine Matrix zu einer Diagonalmatrix a¨ hnlich ist. ¨ Ubungsaufgaben 1. Bestimmen Sie in der erweiterten Koeffizientenmatrix a b c −2 c ab 8 b c a 0 die Koeffizienten a, b und ¨ ¨ c so, dass das zugehorige Gleichungssystem genau die Losung x1 = 1, x2 = −1 und x3 = 2 besitzt! L¨osung: ¨ a, b, c ergeben sich als Losung des Gleichungssystems mit der Koeffizientenmatrix 1 −1 2 −1 2 1 2 b −1 . ¨ Das Gleichungssystem erhalten Sie, indem Sie die gegebene Losung in die Gleichungen ein¨ setzen und nach a, b und c sortieren.

Berechnen Sie D1 = 1 −2 7 0 3 2 5 −1 4 , D2 = 1 0 1 −1 2 12 0 2 −3 0 −1 2 5 1 2 1 und D3 = 1 −2 0 1 5 1 1 2 5 −2 1 4 0 3 0 1 . L¨osung: D1 kann mit der Regel von Sarrus oder mit dem Entwicklungssatz berechnet werden. Es ist D1 = 1 · 3 · 4 + (−2) · 2 · 5 + 7 · 0 · (−1) − 7 · 3 · 5 − 1 · 2 · (−1) − (−2) · 0 · 4 = 12 − 20 − 105 + 2 = −111. 5 Lineare Gleichungssysteme und der Gauß-Algorithmus 59 D2 ist (Entwicklung nach der ersten Spalte) 12 D2 = 0 2 0 1 2 −3 5 2 −3 5 −1 2 + 12 0 1 + 12 0 1 = 102. 2 1 2 2 1 0 −1 2 D3 ist (Entwicklung nach der vierten Zeile) 1 D3 = 3 · 0 1 5 1 2 5 1 = 3 · 2 = 6.