Storia della crittografia - Crittanalisi - Parametri statistici
La funzione Kappa
Presenze

Linguaκ
Dansk7.24
Deutsch7.90
English6.70
Espanol7.70
Francais7.90
Italiano7.45
Latino7.00
Nederlands7.00
Portugues7.00
??????? (Russo)5.00

William Friedman introdusse nel 1925 questo indicatore con il nome di indice di coincidenza; l'indice conteggia le coincidenze tra due testi $T, U$ di uguale lunghezza e cioè il numero di caratteri uguali alla stessa posizione in rapporto al numero totale di caratteri del singolo testo. Il Bauer definisce l'equivalente funzione Kappa come segue:

$$ K(T, U) = \sum_{i=1}^{M} \frac{δ(t_{i}, u_{i})}{M} $$

dove $T, U$ sono due testi di uguale lunghezza $M$, mentre $t_{i}, u_{i}$ sono i caratteri al posto $i$ dei due testi e la funziona delta è una semplice funzione di confronto che ritorna 1 se i due argomenti sono uguali, 0 se sono diversi; in simboli:

$$ δ(x, y) = \begin{cases} 1 \quad se \quad x = y \\ 0 \quad se \quad x \ne y \end{cases} $$

È evidente che se i due testi sono uguali si avrà $K = 1$, mentre due testi che non hanno nemmeno una lettera in comune avranno $K = 0$; visto che per definizione Kappa non può essere negativo, segue che $0 \le K \le 1$.

Si può calcolare il valore di Kappa anche per un solo testo, con il semplice accorgimento di spezzarlo in due testi di uguale lunghezza; se il numero di caratteri è dispari se ne scarterà uno, cosa statisticamente irrilevante.

La cosa sorprendente è che per testi abbastanza lunghi (almeno mille caratteri) il valore di Kappa tende a stabilizzarsi su un numero che è caratteristico per ogni lingua, vedi la tabella accanto per i valori stimati per alcune lingue. Se è ovvio che questo valore sia invariante rispetto a una cifratura monoalfabetica, che è una corrispondenza biunivoca, un po' meno ovvio è che lo sia anche rispetto ad una cifratura polialfabetica come il Vigenère, con la stessa chiave si intende.

Grazie a questa caratteristica il Kappa dà un'indicazione sulla lingua in cui è scritto un messaggio cifrato; considerato che i valori di Kappa per le varie lingue non presentano differenze enormi, si tratta solo di un'indicazione, tanto più attendibile quanto più lungo è il messaggio.

Esempio interattivo

Scrivere nell'area di testo qui sotto un brano in una delle lingue considerate in questa pagina, quindi premere il pulsante "Calcola Kappa"; vengono eliminati tutti i caratteri che non appartengono all'alfabeto scelto:


NELMEZZODELCAMMINDINOSTRAVITAMIRITROVAIPERUNASELVAOSCURACHELADIRITTAVIAERASMARRITAAHIQUANTOADIRQUALERAECOSADURAESTASELVASELVAGGIAEASPRAEFOR
TECHENELPENSIERRINNOVALAPAURATANTOEAMARACHEPOCOEPIUMORTEMAPERTRATTARDELBENCHIOVITROVAIDIRODELALTRECOSECHIOVHOSCORTEIONONSOBENRIDIRCOMIOVINT
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