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+0000190421 00000 n +0000190732 00000 n +0000192864 00000 n +0000193116 00000 n +0000193396 00000 n +0000193705 00000 n +0000193885 00000 n +0000194068 00000 n +0000194400 00000 n +0000194584 00000 n +0000194768 00000 n +0000195070 00000 n +0000195254 00000 n +0000195564 00000 n +0000195875 00000 n +0000196054 00000 n +0000196528 00000 n +0000199514 00000 n +0000199641 00000 n +0000200728 00000 n trailer << /Size 722 /Root 721 0 R /Info 719 0 R - /ID [(/fx20SdojlwYDvgeHqJybg==) (/fx20SdojlwYDvgeHqJybg==)] + /ID [(WZIrsXeWp92HH68waVuisg==) (WZIrsXeWp92HH68waVuisg==)] >> startxref -200945 +201058 %%EOF \ No newline at end of file diff --git a/git-obzornik.typ b/git-obzornik.typ index c11e15a..cd7faf0 100644 --- a/git-obzornik.typ +++ b/git-obzornik.typ @@ -85,8 +85,8 @@ $ H: B -> {0, 1, dots, 2^(n)-1}, $ ki vsakemu besedilu $b$ priredi $n$-bitno vrednost $H(b)$. Vrednosti zgoščevalne funkcije $H(b)$ pravimo _zgostitev_ vsebine $b$. -Git hrani datoteke pod imeni, ki so enaka zgostitvi vsebine. Kaj pa če imata dve različni vsebini isto zgostitev? Funkcija $H$ ni injektivna, saj je množica nizov, bistveno večja od množice zgostitev. To pomeni, da imata lahko dve različni datoteki enako zgostitev. Če se to zgodi, rečemo, da pride do _kolizije zgostitve_. V primeru kolizije zgostitve bi Git shranil le eno datoteko, za drugo pa bi predpostavil da je že shranjena. Zato je funkcija $H$ izbrana tako, da -sprememba enega samega bita v besedilu $b in B$ spremeni vrednost $H(b)$ in je porazdelitev vrednosti $H(b)$ čim bližje enakomerni porazdelitvi. To pomeni, da so vse vrednosti $H(b)$ približno enako verjetne. Na ta način zmanjšamo verjetnost kolizije( glej @sec_kolizije). Verjetnost kolizije je izjemno majhna, zato Git lahko predpostavi, da je niz $b$ enolično določen z njegovo zgostitvijo $H(b)$. +Git hrani datoteke pod imeni, ki so enaka zgostitvi vsebine. Kaj pa če imata dve različni vsebini isto zgostitev? Funkcija $H$ ni injektivna, saj je množica nizov, bistveno večja od množice zgostitev. To pomeni, da imata lahko dve različni datoteki enako zgostitev. Če se to zgodi, rečemo, da pride do _trka zgostitve_. V primeru trka zgostitve bi Git shranil le eno datoteko, za drugo pa bi predpostavil da je že shranjena. Zato je funkcija $H$ izbrana tako, da +sprememba enega samega bita v besedilu $b in B$ spremeni vrednost $H(b)$ in je porazdelitev vrednosti $H(b)$ čim bližje enakomerni porazdelitvi. To pomeni, da so vse vrednosti $H(b)$ približno enako verjetne. Na ta način zmanjšamo verjetnost trka( glej @sec_trk). Verjetnost trka je izjemno majhna, zato Git lahko predpostavi, da je niz $b$ enolično določen z njegovo zgostitvijo $H(b)$. Git uporablja $160$ bitno zgoščevalno funkcijo _SHA1_. Funkcija SHA1 je posebna implementacija zgoščevalne funkcije, ki se je uporabljala v kriptografiji#footnote[Leta 2017 so raziskovalci iz CWI Amsterdam in Google Research našli prvi praktični primer dveh različnih pdf datotek, ki imata isto SHA1 zgostitev#cite(). Opisan napad so poimenovali _SHAttered_. Git je zato z verzijo `v2.13.0` @@ -166,13 +166,13 @@ vsebina objekta določena z njegovo zgostitvijo. To pomeni, da lahko enostavno p ] -== Kolizije zgostitev in rojstnodnevni paradoks +== Trk zgostitev in rojstnodnevni paradoks Git hrani datoteke pod imeni, ki so enaka zgostitvi vsebine. Če imata dve datoteki z različno vsebino isto zgostitev, Git shrani le eno datoteko in pride do izgubil podatkov. Git se zanaša na to, da je verjetnost za to izjemno majhna. Kako bi ocenili to verjetnost? -Koliko datotek bi morali shraniti v Git, da bi z znatno verjetnostjo prišlo do kolizije? Vprašanje je povezano z rojstnodnevnim problemom. Kako velika naj bo skupina ljudi, +Koliko datotek bi morali shraniti v Git, da bi z znatno verjetnostjo prišlo do trka? Vprašanje je povezano z rojstnodnevnim problemom. Kako velika naj bo skupina ljudi, da bo vsaj $50%$ verjetnost, da imata dve osebi na isti dan rojstni dan? Velikost skupine je presenetljivo majhna(23), zato rojstnodnevnei problem imenujemo tudi rojstnodnevni paradoks. Vprašanje zastavimo matematično. Naključno izberemo $n Za vrednosti $1 << n << h$ je $1- e^(-n^2/(2h))$ tudi dobra aproksimacija za $p(n, h)$. -Da bi odgovorili kako odporna je zgoščevalna funkcija na morebitne kolizije, moramo rešiti obratno nalogo: največ koliko števil $n(p, d)$ lahko izberemo, da bo verjetnost pojava dveh enakih števil manjša od $p in [0,1]$? Natančen odgovor na to vprašanje ni tako preprost #cite(). Lahko pa uporabimo oceno (@eq_ocena) in čez palec ocenimo vrednost $n(p, h)$: +Da bi odgovorili kako odporna je zgoščevalna funkcija na morebitne trka, moramo rešiti obratno nalogo: največ koliko števil $n(p, d)$ lahko izberemo, da bo verjetnost pojava dveh enakih števil manjša od $p in [0,1]$? Natančen odgovor na to vprašanje ni tako preprost #cite(). Lahko pa uporabimo oceno (@eq_ocena) in čez palec ocenimo vrednost $n(p, h)$: $ -n^2 approx log(1-p) =>\ n(p, h) approx sqrt(2h log(1/(1-p))) approx sqrt(2h). $ Funkcija $sqrt(log(1/(1-p)))$ zelo počasi narašča, ko se $p$ približuje $1$, zato jo lahko zanemarimo. Če je zgoščevalna funkcija $160$ bitna, kot na primer SHA1, je -$n approx sqrt(2^(160)) approx 2^(80)$. Znatna verjetnost, da pride do kolizije zgostitev, bi se pojavila, ko bi shranili $2^(80)$ različnih verzij datotek v Git. -Raziskovalci, ki so razvili napad _SHAttered_, so se posebej potrudili in so potrebovali "zgolj" približno $2^(63)$ primerov, da so prišli do kolizije. +$n approx sqrt(2^(160)) approx 2^(80)$. Znatna verjetnost, da pride do trka zgostitev, bi se pojavila, ko bi shranili $2^(80)$ različnih verzij datotek v Git. +Raziskovalci, ki so razvili napad _SHAttered_, so se posebej potrudili in so potrebovali "zgolj" približno $2^(63)$ primerov, da so prišli do trka. = Zgodovinski graf sprememb