Matemàtiques avançades: MISSATGES CIFRATS

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lunes, 28 de noviembre de 2011

MISSATGES CIFRATS

Per a simplificar els missatges anem a utilitzar únicament lletres majúscules i espais en blanc. Per a la qual cosa, asignem números a cada u d'aquests símbols.

- A B C D E F G H I J    K L M N O P   Q R   S T  U   V W X Y Z
0 1 2   3   4   5 6  7   8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Treballem amb l'aritmètica mòdul 27.
Primer busquem un nombre senzill (a minúscula) que tinga invers en mòdul 27. Si a = 4, el seu invers és 7, perquè 4·7 és congruent a 1.
Agafem un altre número (b minúscula); és el que se vuiga, i fem la transformació:
a · x + b =  y
4 · 8 + 10 = 42
EX: Imaginem que volem enviar el missatge:
                      H O  L   A -   J   U A N
                      8 15 12   1 0 10 21 1 14
4·8+10= 42 que és congruent a 15 (mòdul 27), que equival a la lletra O
4·15+10= 70 que és congruent a 16 (mòdul 27) que equival a la lletra P
4·12+10= 58 que és congruent a 4 (mòdul 27) que equival a la lletra D
4·1+10= 14 que és congruent a 14 (mòdul 27) que equival a la lletra N
4·0+10= 10 que és congruent a 10 (mòdul 27) que equival a la lletra J
4·10+10 és congruent a 23 (mòdul 27) que equival a la lletra W
4·21+10 és congruent a 13 (mòdul 27) que equival a la lletra M
4·1+10 és congruent a 14 (mòdul 27) que equival a la lletra N
4·14+10 és congruent a 12 (mòdul 27) que equival a la lletra L
Ara qui rep el misstage (en aquest cas Juan) té que recòrrer el camí invers:
1. Convertir lletres en números
2. Restar 10 i dividir per 4 (és a dir, multiplicar per 7)
3. Passar a mòdul 27
4. Passar a lletres

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