{"id":2310,"date":"2016-11-21T22:15:55","date_gmt":"2016-11-21T21:15:55","guid":{"rendered":"http:\/\/agora.xtec.cat\/iesm-jmzafra\/?p=2310"},"modified":"2016-11-21T23:24:38","modified_gmt":"2016-11-21T22:24:38","slug":"anem-x-matematiques-criptografia-i-aritmetica-modular","status":"publish","type":"post","link":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/eso\/eso-3\/anem-x-matematiques-criptografia-i-aritmetica-modular\/","title":{"rendered":"Anem x + matem\u00e0tiques: criptografia i aritm\u00e8tica modular"},"content":{"rendered":"

\"anemxmates\"S\u00f3c la Duna Galup i estic fent 3r d’ESO, i us vull explicar que el dissabte passat vaig estar a la tercera sessi\u00f3 d‘Anem x + matem\u00e0tiques<\/a>, unes classes gratu\u00eftes que s’imparteixen a\u00a0la Universitat Polit\u00e8cnica de Catalunya per a joves de 3r o 4t d’ESO.<\/span><\/p>\n

De fet, les classes estan pensades per alumnes de 4t, per\u00f2 com que jo tenia moltes ganes de participar-hi, finalment em van acceptar.<\/p>\n

Farem un total de nou sessions a la Facultat de Matem\u00e0tiques i estad\u00edstica de la UPC\u00a0al llarg del curs. \u00c9s una iniciativa impulsada pel Ministeri d’Educaci\u00f3 i creada per la Federaci\u00f3\u00a0Espanyola de Professors de Matem\u00e0tiques<\/a>; a Catalunya est\u00e0 a c\u00e0rrec de FEEMCAT<\/a> (Federaci\u00f3 d’Entitats per a l’Ensenyament de les Matem\u00e0tiques a Catalunya).<\/p>\n

Aquest dissabte, la classe tractava sobre criptografia. La criptografia estudia la manera de crear<\/span> missatges que nom\u00e9s puguin llegir les persones a qui van destinats per mitj\u00e0 de xifrat o codificaci\u00f3. Per tal de aconseguir un codi segur que permeti la confidencialitat del missatge, s’utilitzen les matem\u00e0tiques.<\/p>\n

Actualment, la criptografia \u00e9s present en molts aspectes de la nostra vida. Amb l’augment de la tecnologia, s’han hagut de crear codis segurs per tal que ning\u00fa pugui accedir a la nostra informaci\u00f3. Per\u00f2 la criptografia s’utilitza des de fa molts segles, i ha tingut especial import\u00e0ncia durant els per\u00edodes b\u00e8l\u00b7lics.<\/p>\n

Un dels m\u00e8todes de xifratge m\u00e9s antic \u00e9s el Xifrat de C\u00e8sar, que \u00e9s utilitzat des de l’antiga Roma. Aquest xifrat consisteix a despla\u00e7ar l’abecedari tres espais cap a la dreta, de manera que cada lletra se substitueix per la lletra que hi ha tres espais despr\u00e9s. Un exemple senzill, la paraula SECRET, en xifratge de C\u00e8sar s’escriuria VHELHW. En realitat, aquest m\u00e8tode no \u00e9s gaire segur, ja que es pot desxifrar amb facilitat el codi, com va succeir a la Segona guerra mundial amb la m\u00e0quina Enigma.<\/p>\n

Tamb\u00e9 vam treballar l’anomenada aritm\u00e8tica del rellotge, proposada pel matem\u00e0tic alemany Karl Friederich Gauss. Qualsevol persona que entengui el sistema horari sexagesimal pot afirmar que 4 hores despr\u00e9s de les 10 del mat\u00ed, s\u00f3n les dues. Estar\u00edem dient que 10 + 4 = 2. Aquesta operaci\u00f3 nom\u00e9s t\u00e9 sentit si sabem que estem parlant de nombres enters fins al 12, per tant, quan arribem al nostre m\u00e0xim (12) hem de “tornar a comen\u00e7ar<\/i>”<\/p>\n

Aquest model rep el nom d’Aritm\u00e8tica modular, i la podem fer servir per a fer sumes, restes i multiplicacions. El resultat que obtinguem ser\u00e0 en funci\u00f3 del modul n que estiguem utilitzant. Algunes operacions b\u00e0siques:<\/p>\n

8 + 9 + 2 \u2261 5 (mod 7)<\/p>\n

9 \u00b7 11 \u2261 3 (mod 12)<\/p>\n

26<\/sup> \u2261 1 (mod 7)<\/p>\n

Es pot dir que la divisi\u00f3 \u00e9s l’invers de la multiplicaci\u00f3, per\u00f2 quan treballem amb m\u00f2duls, no sempre podem tenir l’invers.<\/p>\n

Si intentem calcular inversos amb mod 7, veiem que els podem obtenir tots excepte el 0.<\/p>\n\n\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
N\u00famero<\/td>\n\n

0<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

5<\/p>\n<\/td>\n

\n

6<\/p>\n<\/td>\n<\/tr>\n

Invers<\/td>\n\n

–<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

5<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

6<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Provem a buscar l’invers amb mod 12<\/p>\n\n\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
N\u00famero<\/td>\n\n

0<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

2<\/p>\n<\/td>\n

\n

3<\/p>\n<\/td>\n

\n

4<\/p>\n<\/td>\n

\n

5<\/p>\n<\/td>\n

\n

6<\/p>\n<\/td>\n<\/tr>\n

Invers<\/td>\n\n

–<\/p>\n<\/td>\n

\n

1<\/p>\n<\/td>\n

\n

–<\/p>\n<\/td>\n

\n

–<\/p>\n<\/td>\n

\n

–<\/p>\n<\/td>\n

\n

5<\/p>\n<\/td>\n

\n

–<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Si ho f\u00e9ssim amb altres m\u00f2duls, veur\u00edem que nom\u00e9s els nombres primers tenen tots els inversos.<\/p>\n

Hem treballat aquests temes en grups per tal d’aprendre de manera cooperativa i amb l’ajuda de la professora encarregada de l’activitat. Ha estat una molt bona classe, en la que hem pogut aprendre molt tots junts.<\/p>\n

Enlla\u00e7 de les activitats treballades: Si lo escondo, \u00bflo encuentras? Aritmetica del reloj<\/em><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

a Duna Galup de 3r A ens explica la seva participaci\u00f3 en l’activitat Anem x+ Matem\u00e0tiques<\/p>\n","protected":false},"author":34,"featured_media":2336,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[141,6,29],"tags":[272,271,273,274],"class_list":["post-2310","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-departament-de-matematiques","category-eso-3","category-portada","tag-activitats-extraescolars","tag-anem-x-matematiques","tag-criptografia","tag-upc"],"post_mailing_queue_ids":[],"_links":{"self":[{"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/posts\/2310","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/users\/34"}],"replies":[{"embeddable":true,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/comments?post=2310"}],"version-history":[{"count":5,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/posts\/2310\/revisions"}],"predecessor-version":[{"id":2376,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/posts\/2310\/revisions\/2376"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/media\/2336"}],"wp:attachment":[{"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/media?parent=2310"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/categories?post=2310"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/agora.xtec.cat\/inscaterinaalbert\/wp-json\/wp\/v2\/tags?post=2310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}