<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">Independ\303\252ncia Linear</Font></Text-field>

Exerc\303\255cios de \303\201lgebra Linear e Geometria 5 maart 2012

Li\303\247\303\243o 3 - Exerc\303\255cio 1

Solu\303\247\303\265es

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JSFH

De vectoren v1,....vn uit de vectorruimte V zijn lineair onafhankelijk als uit a1*v1+...+an*vn=0 volgt dat alle co\303\253ffici\303\253nten a1 tot en met an gelijk zijn aan nul (zie definitie pagina 45).

Onderzoek of volgende vectoren lineair onafhankelijk zijn.

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We schrijven de voorwaarde neer opdat deze drie vectoren v1,v2,v3 lineair onafhankelijk zijn. Dan zetten we dit om in een stelsel met onbekenden a1, a2 en a3.

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Wanneer we de voorwaarde voor lineaire onafhankelijkheid uitschrijven dan bekomen we een homogeen stelsel, i.e. de constante termen zijn alle gelijk 0. Strikt gesproken komt met dat stelsel dus volgende matrix As overeen.

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Omdat elementaire rij-operaties deze laatste nulkolom niet zullen wijzigen, laten we deze weg voor de eenvoud. Bij de interpretatie van het stelsel, op het einde van de berekeningen, zullen we er weer concreet rekening mee houden.

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Los dit stelsel nu zelf op via het reductie-algoritme. Wat betekent het als dit stelsel een oplossing heeft die verschillend is van de nuloplossing? Trek hieruit je conclusies omtrent de lineaire onafhankelijkheid.

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Enkel nuloplossing--> lineair onafhankelijk!

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Ga voor elk van de volgende verzamelingen van vectoren na of ze lineair onafhankelijk zijn:

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(-1,-2,1) is oplossing verschillend van nuloplossing -> niet lineair onafhankelijk!

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

niet lineair onafhankelijk (vier vectoren in R^2 kunnen in het algemeen niet lineair onafhankelijk zijn, zie Lemma 3.10)

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

(2,-1,1) is oplossing verschillend van nuloplossing -> niet lineair onafhankelijk

Os exerc\303\255cios a seguir cobrem "Compartilhamento secreto" (consulte as p\303\241ginas 48-51 do curso). Aqui, os dados s\303\243o divididos em diferentes partes, sendo que os dados originais s\303\263 podem ser reconstru\303\255dos se houver um certo n\303\272mero de partes.

<Text-field style="Heading 1" layout="Heading 1">Receita secreta</Text-field>

Um gigante dos refrigerantes quer ter certeza de que o m\303\255nimo de pessoas poss\303\255vel conhe\303\247a a receita secreta completa. \303\211 por isso que cada funcion\303\241rio recebe uma equa\303\247\303\243o em que a quantidade necess\303\241ria (em centilitros) de cada ingrediente (a, b, c e d) \303\251 inclu\303\255da para uma bebida de 1 litro:(Os funcion\303\241rios n\303\243o sabem que a soma \303\251 exatamente 1 litro!)

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Tentamos para quatro pessoas aleat\303\263rias:

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Agora, tentamos sistematicamente todos os quatro:

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Portanto, vemos que cada grupo de quatro pessoas \303\251 capaz de encontrar o c\303\263digo juntos.

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<Text-field style="Heading 1" layout="Heading 1">Bankkluis</Text-field>

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No exerc\303\255cio a seguir, mostramos como compartilhar um segredo usando polin\303\264mios (esquema de Adi Shamir, consulte a p\303\241gina 50 do curso).

Quest\303\243o:

50 pessoas trabalham em um banco. Pelo menos quatro pessoas devem usar sua pr\303\263pria chave digital para abrir o cofre.

Existe um polin\303\264mio de terceiro grau f (x), onde o termo constante (ou portanto o valor no ponto x = 0) \303\251 nosso c\303\263digo secreto. Cada pessoa recebe uma chave, este \303\251 um casal

(LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJpRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1YkdGJDYlLUYsNiVRInhGJ0YvRjItRiM2JC1GLDYlUSJpRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRkVGRUZF).

Quando quatro pessoas colocam suas informa\303\247\303\265es juntas, elas podem calcular o termo m\303\272ltiplo c\303\272bico e, portanto, o termo constante procurado. A f\303\263rmula para o polin\303\264mio pode ser encontrada na p\303\241gina 50 do curso (consulte tamb\303\251m o exerc\303\255cio "Interpola\303\247\303\243o" da li\303\247\303\243o anterior).

Abaixo voc\303\252 encontrar\303\241 os detalhes deste problema.

Damos as chaves de consecutivamente

Alice: (x1,y1)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSMxMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiRRJTM5MzZGJ0Y0RjRGNEY0

Bob: (x2,y2)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSMyNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiRRJjE5MjExRidGNEY0RjRGNA==

Carol: (x3,y3)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSM0MkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIixGJ0Y0LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR1EldHJ1ZUYnLyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYxNiRRJjgxMDA2RidGNEY0RjRGNA==

David: (x4,y4)=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSI1RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFElMjAxMUYnRjRGNEY0RjQ=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEjeDNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzQyRidGOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnRk0tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGWC8lJmRlcHRoR0Zobi8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRlcvRk5GWC1GLDYlUSN5M0YnRi9GMkY1LUZQNiRRJjgxMDA2RidGOUZTLyUrZXhlY3V0YWJsZUdGPUY5

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

Agora encontre a senha correta.

Agora use a f\303\263rmula na p\303\241gina 50 (parte inferior) para encontrar f:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRI1AxRidGL0YyLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GVi1GLDYlUSNQMkYnRi9GMkZSLUYsNiVRI1AzRidGL0YyRlItRiw2JVEjUDRGJ0YvRjItRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJ0ZNLyUrZXhlY3V0YWJsZUdGPUY5

Agora simplifique este f.

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

Verificamos se n\303\243o cometemos erros. Fazemos isso inserindo o valor x em f para cada pessoa (Alice, Bob, Carol e David) e vemos se ele corresponde ao valor y fornecido. Use o comando "subs" para isso.

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpc2ltcGxpZnlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJmRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI7RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRj1GQA==

Se seu polin\303\264mio estiver correto, agora voc\303\252 pode fornecer o termo constante e, portanto, a solu\303\247\303\243o.

Resposta: 1836 \303\251 o c\303\263digo que voc\303\252 est\303\241 procurando.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

Embora a forma da informa\303\247\303\243o distribu\303\255da pare\303\247a diferente em Adi Shamir, aqui tamb\303\251m cada pessoa recebe uma equa\303\247\303\243o linear em v\303\241rias inc\303\263gnitas. (Veja tamb\303\251m a explica\303\247\303\243o na p\303\241gina 51). Descubra por si mesmo qual equa\303\247\303\243o linear Alice e Bob conseguiram aqui.

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Jan, sua bicicleta e os amigos do computador</Text-field>

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Jan \303\251 um novo aluno em Ghent e j\303\241 fez alguns amigos com os alunos de ci\303\252ncia da computa\303\247\303\243o. Ele tem uma fechadura de combina\303\247\303\243o para proteger sua bicicleta. Como \303\240s vezes fica um pouco mais tarde e Jan esquece seu c\303\263digo, ele conta com seus amigos do computador. Cada um deles recebeu os dados [x, f (x)] para um certo valor x, de modo que, quando tr\303\252s de seus amigos estiverem juntos, eles possam descobrir o c\303\263digo num\303\251rico e voltar para casa juntos. Aqui f (x) \303\251 um termo m\303\272ltiplo de segundo grau, do qual o termo constante \303\251 o c\303\263digo pesquisado.

Qual \303\251 o c\303\263digo de bloqueio de John? (Pe\303\247a a dois outros alunos aleat\303\263rios o seu par de n\303\272meros e seja o primeiro a encontrar o c\303\263digo num\303\251rico!)

Solu\303\247\303\243o:

Por exemplo, suponha que voc\303\252 tenha recebido as seguintes informa\303\247\303\265es junto com outras duas:

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Primeiro, verificamos novamente para ter certeza:

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Portanto, 5312 \303\251 o c\303\263digo que voc\303\252 est\303\241 procurando.

<Text-field style="Heading 1" layout="Heading 1">Configurar compartilhamento secreto</Text-field>

Existem v\303\241rios m\303\251todos de solu\303\247\303\243o para esses tipos de exerc\303\255cios. A maneira como isso \303\251 resolvido no curso de teoria (p. 48ss.) Por meio de sistemas de equa\303\247\303\265es lineares, vamos demonstrar aqui. Claro que tamb\303\251m existem outras solu\303\247\303\265es. O desafio \303\251 encontrar uma solu\303\247\303\243o que seja o mais simples poss\303\255vel (muitas vezes isso significa na dimens\303\243o mais baixa poss\303\255vel).

1) Crie um sistema de compartilhamento secreto para quatro pessoas A, B, C e D, onde A e

B s\303\243o chefes e C e D s\303\243o subordinados. O sistema deve ser tal que 2 pessoas

podem encontrar o c\303\263digo secreto juntos, desde que pelo menos um dos

essa pessoa \303\251 um chefe. Por exemplo, codifique o n\303\272mero 19.

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Codificamos nossas informa\303\247\303\265es: tomamos o c\303\263digo 19 como o valor z de um ponto (xp, yp, zp) no espa\303\247o tridimensional. Para xp e yp, voc\303\252 pode escolher n\303\272meros arbitr\303\241rios (de prefer\303\252ncia n\303\243o muito grandes).

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Tal ponto \303\251 determinado por 3 planos bem escolhidos, enquanto 2 planos determinam no m\303\241ximo uma linha reta. Um plano \303\251 determinado por uma equa\303\247\303\243o. Damos a cada chefe duas equa\303\247\303\265es independentes (= faces) e o subordinado obt\303\251m apenas 1 equa\303\247\303\243o (= face).

Agora preenchemos as partes homog\303\252neas das equa\303\247\303\265es (ou sem o termo constante). Podemos fazer isso arbitrariamente, desde que nossos planos n\303\243o sejam paralelos ou percorram a mesma linha em 3 ou mais. Voc\303\252 pode verificar isso atrav\303\251s do comando solve ou geom\303\251trica. O termo constante ser\303\241 fixado uma vez que determinamos os coeficientes. (afinal, o ponto (xp, yp, 19) deve estar em nosso plano).

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Agora damos o chefe A, a equa\303\247\303\243o EqA1 e EqA2, para B, EqB1 e EqB2 ...

O c\303\263digo agora pode ser encontrado resolvendo o sistema que determina as equa\303\247\303\265es de, por exemplo, chefe A e subordinado D. Voc\303\252 pode fazer isso com o comando "resolver". Verifique se sua solu\303\247\303\243o est\303\241 funcionando corretamente.

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JSFH

Agora verificamos se cada tr\303\255ade de equa\303\247\303\265es poss\303\255vel leva a uma solu\303\247\303\243o:

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Plotamos os planos e verificamos se os planos desejados se cruzam em exatamente 1 ponto. Consulte a se\303\247\303\243o de ajuda de "implicitplot3d" e "pointplot3d" para obter mais informa\303\247\303\265es. Crie todas as faces, com uma cor separada. Depois, voc\303\252 deixa as \303\241reas desejadas aparecerem com "display". Tamb\303\251m exibiremos nosso ponto (xp, yp, zp) sob o nome "solu\303\247\303\243o".

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Pode ser que o m\303\251todo anterior tenha sido aplicado corretamente, mas surgem problemas, a saber, que os tr\303\252s planos se cruzam em uma linha reta em vez de um ponto. Isso ocorre porque nossos avi\303\265es originalmente escolhidos aparentemente n\303\243o eram t\303\243o aleat\303\263rios entre si. Podemos evitar isso usando Adi Shamir da seguinte maneira:

Em Adi Shamir, voc\303\252 trabalha com polin\303\264mios, cujos coeficientes s\303\243o as inc\303\263gnitas e onde o termo constante \303\251 o c\303\263digo (portanto, 19 aqui). Assim como acima, precisamos de tr\303\252s inc\303\263gnitas, a \303\272ltima das quais \303\251 19, ent\303\243o estamos trabalhando com um polin\303\264mio de segundo grau:

f(x)=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. Se voc\303\252 der a uma pessoa um peda\303\247o de informa\303\247\303\243o na forma de (x1, y1), ent\303\243o isso nada mais \303\251 do que passar a equa\303\247\303\243o y1= 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

Visto que um polin\303\264mio de grau n \303\251 completamente determinado por seus valores em n + 1 pontos, neste caso \303\251 suficiente fornecer 3 pontos (valores xey) para determinar o polin\303\264mio de segundo grau completamente.

Primeiro configuramos a fun\303\247\303\243o f que conter\303\241 nosso c\303\263digo 19. Certifique-se de que o termo constante de f seja 19.

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

Agora geramos seis pares de pontos que satisfazem f.

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

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZicy1JI21pR0YkNiVRJnZnbEExRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEqJmNvbG9uZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZTLUYsNiVRImFGJ0YvRjItRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRlJGVC1JJW1zdXBHRiQ2JS1GLDYlUSN4MUYnRi9GMi1GIzYlLUkjbW5HRiQ2JFEiMkYnRjkvJStleGVjdXRhYmxlR0Y9RjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5Gam9GTy1GLDYlUSJiRidGL0YyRlhGaG5GT0Zmb0ZPLUYsNiVRImNGJ0YvRjItRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GTy1GLDYlUSJmRidGL0YyLUkobWZlbmNlZEdGJDYkLUYjNiVGaG5GYW9GOUY5LUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGUkZNLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRlMvJSZkZXB0aEdGZnEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRidGTy1GLDYlUSZ2Z2xBMkYnRi9GMkY1Rk9GVUZYLUZmbjYlLUYsNiVRI3gyRidGL0YyLUYjNiRGXW9GOUZjb0Zmb0ZPRlxwRlhGY3JGZm9GT0ZfcEZicEZPRmVwLUZpcDYkLUYjNiRGY3JGOUY5Rl1xRmFxRk8tRiw2JVEmdmdsQjFGJ0YvRjJGNUZPRlVGWC1GZm42JS1GLDYlUSN4M0YnRi9GMkZmckZjb0Zmb0ZPRlxwRlhGYXNGT0Zmb0ZPRl9wRmJwRk9GZXAtRmlwNiQtRiM2JEZhc0Y5RjlGXXFGYXFGTy1GLDYlUSZ2Z2xCMkYnRi9GMkY1Rk9GVUZYLUZmbjYlLUYsNiVRI3g0RidGL0YyRmZyRmNvRmZvRk9GXHBGWEZddEZPRmZvRk9GX3BGYnBGT0ZlcC1GaXA2JC1GIzYkRl10RjlGOUZdcUZhcUZPLUYsNiVRJXZnbENGJ0YvRjJGNUZPRlVGWC1GZm42JS1GLDYlUSN4NUYnRi9GMkZmckZjb0Zmb0ZPRlxwRlhGaXRGT0Zmb0ZPRl9wRmJwRk9GZXAtRmlwNiQtRiM2JEZpdEY5RjlGXXFGYXFGTy1GLDYlUSV2Z2xERidGL0YyRjVGVUZYLUZmbjYlLUYsNiVRI3g2RidGL0YyRmZyRmNvRmZvRk9GXHBGWEZldUZPRmZvRk9GX3BGYnBGT0ZlcC1GaXA2JC1GIzYkRmV1RjlGOUZdcUZhcS1GYnE2JkZkcUZncUZpcS9GXHJRJWF1dG9GJ0ZhcS1GLDYjUSFGJ0Zhb0Y5

O chefe A recebe a equa\303\247\303\243o A1 e a equa\303\247\303\243o A2, o chefe B obt\303\251m ... A solu\303\247\303\243o de um sistema que consiste em tr\303\252s dessas equa\303\247\303\265es (ou a interse\303\247\303\243o de tr\303\252s desses planos) \303\251 ent\303\243o (a, b, c), onde sabemos que c \303\251 o nosso c\303\263digo. A diferen\303\247a com o m\303\251todo anterior \303\251 que agora temos certeza de que nossos planos se cruzam em exatamente um ponto. Verifique com o solve se a nossa solu\303\247\303\243o est\303\241 correta. Trace alguns dos planos para verificar a solu\303\247\303\243o graficamente.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZALUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGaG4vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRkM2LVEifkYnRkBGRkZIRkpGTEZORlBGUi9GVUZbby9GWEZbby1GLDYlUSZmaWdBMUYnRi9GMi1GQzYtUSomY29sb25lcTtGJ0ZARkZGSEZKRkxGTkZQRlJGVEZXLUYsNiVRL2ltcGxpY2l0cGxvdDNkRidGL0YyLUY2NiQtRiM2OS1GLDYlUSZ2Z2xBMUYnRi9GMi1GQzYtUSIsRidGQEZGL0ZJRjFGSkZMRk5GUEZSRmRvL0ZYUSwwLjMzMzMzMzNlbUYnLUYsNiVRImFGJ0YvRjItRkM2LVEiPUYnRkBGRkZIRkpGTEZORlBGUkZURlctRkM2LVEqJnVtaW51czA7RidGQEZGRkhGSkZMRk5GUEZSL0ZVUSwwLjIyMjIyMjJlbUYnL0ZYRmZxLUkjbW5HRiQ2JFEiNUYnRkAtRkM2LVEjLi5GJ0ZARkZGSEZKRkxGTkZQRlJGZXFGZW9GaHFGZnAtRiw2JVEiYkYnRi9GMkZfcUZicS1GaXE2JFEiNEYnRkBGXHItRmlxNiRRIjdGJ0ZARmZwLUYsNiVRImNGJ0YvRjJGX3EtRmlxNiRRIzEwRidGQEZcci1GaXE2JFEjMzBGJ0ZARj1GQEZARkJGWUZhby1GLDYlUSZmaWdBMkYnRi9GMkZpb0ZccC1GNjYkLUYjNjgtRiw2JVEmdmdsQTJGJ0YvRjJGZnBGXHFGX3FGYnFGaHFGXHJGaHFGZnBGX3JGX3FGYnFGYnJGXHJGZXJGZnBGaHJGX3FGW3NGXHJGXnNGQEZARkJGWUZhb0Zhby1GLDYlUSVmaWdDRidGL0YyRmlvRlxwLUY2NiQtRiM2OC1GLDYlUSV2Z2xDRidGL0YyRmZwRlxxRl9xRmJxRmhxRlxyRmhxRmZwRl9yRl9xRmJxRmJyRlxyRmVyRmZwRmhyRl9xRltzRlxyRl5zRkBGQEZCRlktRlo2JkZmbkZpbkZcby9GX29RJWF1dG9GJy1GLDYjUSFGJ0Y9RkA=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Nota: por exemplo, podemos simplificar o exerc\303\255cio dando a C e D a mesma equa\303\247\303\243o neste caso. Verifique se isso n\303\243o altera a solu\303\247\303\243o para o nosso problema. Na verdade, tamb\303\251m poder\303\255amos ver todo o problema no espa\303\247o bidimensional. Pense em uma poss\303\255vel solu\303\247\303\243o para voc\303\252.

Resolva tamb\303\251m os seguintes exerc\303\255cios da \303\272ltima maneira.

2) Fa\303\247a um sistema para quatro pessoas, sendo tr\303\252s secret\303\241rias A, B e C e uma

diretor D. Pe\303\247a ao diretor, junto com duas secret\303\241rias,

para encontrar o c\303\263digo. C\303\263digo 43.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

o diretor recebe duas compara\303\247\303\265es, as secret\303\241rias 1.

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Considere 4 pessoas A, B, C e D. Cada uma delas tem um certo n\303\255vel de acesso: A est\303\241 ligado

n\303\255vel 3, B no n\303\255vel 2 e C e D no n\303\255vel grau 1. Codifique o n\303\272mero 67

de modo que um grupo de pessoas possa encontrar este n\303\272mero se e somente se a soma de seus

n\303\255veis \303\251 de pelo menos 5.

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trabalhamos com um termo m\303\272ltiplo de quarto grau, e todos t\303\252m tanta compara\303\247\303\243o quanto seu n\303\255vel.

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Desta vez, configuramos nossas equa\303\247\303\265es um pouco mais sistematicamente.

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Considere novamente quatro pessoas A, B, C e D com quatro n\303\255veis de acesso. Suponha que A

est\303\241 no n\303\255vel 3, B e C no n\303\255vel 2 e D no n\303\255vel 1. Tenha um grupo de pessoas no

o total deve ser de pelo menos n\303\255vel 4 para encontrar o c\303\263digo e codific\303\241-lo

n\303\272mero 163.

JSFH

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Voc\303\252 v\303\252 claramente a conex\303\243o entre o m\303\251todo de Adi Shamir e aquele com sistemas de equa\303\247\303\265es lineares? Isso \303\251 importante.

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JSFH