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Home Page Hebern's machines Home Page Hebern 1 rotor Home Page Cryptanalysis, Home Page
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IntroductionAlan G. Konheim published in one of his books concerning cryptography a method that allows you to find the plain text associated with a cryptogram encrypted by the Hebern machine 1 rotor where only the wiring of the rotor is unknown. The methodIf T corresponds to the encryption of the plain letter S at the key A (or 0 in numerical notation), then, the expression (T – i) corresponds to the plain letter for the ith line (key i) for each key i from 0 (A) to 25 (Z). If we follow the law of large numbers, the following formula applies:
Chi[t,s] = SUM_FROM_i=0_TO_i=25 (PI[t-i] * NL[s][i] / N[i])
With:
Chi[t,s] = probability of the plain letter 't' is ciphered by
's' when the key (i) is zero.
NL[s][i] = Number of letter 's'-i at the "i" key.
N[i] = Number of letters ciphered at the "i" key.
PI[x] = Probability of the plain letter "x".
Using Latex: \( X[t,s] = \sum_{t=0}^{25}\pi(t-i)( \frac{N_{s-i}(i)}{N(i)}) \sim 0.06875 \) If the solution is correct, Chi[t,s] has a value close to 0.06875. Note: Konheim uses the term column when personally I use the term line (to be consistent with my other web pages). An exampleWe take the example taken by Konheim. Here is the cryptogram, it is 650 characters long: C:\H1_TOOLS> python groupe.py < MSGS\konheim.cry RBLFO GZBKG CWUQF YDKIB KRFOL SFJMY XVLHD WFEYU IWHGU YBQSR HHBOL KAGIX XKRIT JWDHR QCWRK DAEPC DYFRE KDXWW UOVRA RQXIH BRINT JRRRY HDFCW SMKIC NNIDG OXDHN IWLBY TELCQ BMOXP WMSDS ZEGLA EFFQP YLTBM KRWOO KDQJO DLWBK UAKOA PEEOB EHTVO OCNKR RPVLL LIQBG YIJZY ZQKRG SGMRS DCBIE LYVZA GPETY WBGPE IJUIC GVWCZ QTBCH ZFWVO ZHDIB SHMMN HTJDU FABTS SZFLH HVJDR QEMBK PHELV HFYJD KXRIR MJMLC NVWGT SFVIS GPFRG OJRWI STJZZ XUBPW LHWET JNCBJ WDLMP XLHTZ JNMAX MMKEE MXLEG ZLDXW COSLY EHXJN CHJSC WQUEM QEJLI NZZCX ZWOJI QGVCN ICDWR DRMIB DOXYR VAJHC IXGWU ZOHTR WGQSA XQNOZ SJZZR LNHUN EKPOO AQXIH LAVYC CIPHB OBQHW BPDMC NKWCF YJRGE KCLYF VOLJS LXOHF EMKIZ FYJFV LXUIY FGRFI QILTK QQHKC PIKLA VCXPZ ADUWP ISUTL XPYUX USFRD MMABH XDZXX BUJJG AQFYD KOEJU KTGJP TFPZJ YBEOT OMKPL YVRST RNBAQ Show program syntax:
C:\H1_TOOLS> python konheim.py
konheim: search of probability of a plain letter
is ciphered by a specific letter at key 0
Syntax: konheim.py language cryptogram plain_letter
We launch the attack by looking for the encrypted letter in
position A
corresponds to the letter E. The highest value found (0.0844)
corresponds to the letter X.
C:\H1_TOOLS> python konheim.py english MSGS\konheim.cry E Plain letters probability: [0.0834, 0.0144, 0.0273, 0.0414, 0.126] ... cryptogram : RBLFOGZBKGCWUQFYDKIBKRFOLSFJMYXVLHDWFEYUIWHGUYBQSR ... Cryptogram Length : 650 Plain letter: E E -> a 0.0525 E -> b 0.0533 E -> c 0.0496 E -> d 0.0411 E -> e 0.0297 E -> f 0.0421 E -> g 0.0448 E -> h 0.0495 E -> i 0.0459 E -> j 0.0385 E -> k 0.0335 E -> l 0.0268 E -> m 0.0244 E -> n 0.0325 E -> o 0.0357 E -> p 0.0324 E -> q 0.0281 E -> r 0.0348 E -> s 0.0268 E -> t 0.0386 E -> u 0.0290 E -> v 0.0265 E -> w 0.0270 E -> x 0.0844 E -> y 0.0343 E -> z 0.0369 The program uses the probabilities of the letters present in the file STATS\english.proba if the language used is "english". C:\H1_TOOLS> more STATS\english.proba A,0.0834 G,0.0192 L,0.0424 Q,0.0009 V,0.0106 B,0.0144 H,0.0611 M,0.0253 R,0.0568 W,0.0234 C,0.0273 I,0.0671 N,0.0680 S,0.0611 X,0.0020 D,0.0414 J,0.0023 O,0.0770 T,0.0937 Y,0.0204 E,0.1260 K,0.0087 P,0.0166 U,0.0285 Z,0.0006 F,0.0203 Then, we search for the value of the other letters of the alphabet starting by the most frequent letters (T,A,O,N,S): ... Plain letter: T T -> a 0.0525 T -> b 0.0310 T -> c 0.0409 T -> d 0.0397 T -> e 0.0385 T -> f 0.0371 T -> g 0.0443 T -> h 0.0255 T -> i 0.0388 T -> j 0.0278 T -> k 0.0418 T -> l 0.0453 T -> m 0.0225 T -> n 0.0528 T -> o 0.0292 T -> p 0.0304 T -> q 0.0329 T -> r 0.0584 T -> s 0.0290 T -> t 0.0454 T -> u 0.0384 T -> v 0.0293 T -> w 0.0435 T -> x 0.0543 T -> y 0.0323 T -> z 0.0367 ... Plain letter: A A -> a 0.0463 A -> b 0.0366 A -> c 0.0382 A -> d 0.0247 A -> e 0.0348 A -> f 0.0599 A -> g 0.0444 A -> h 0.0394 A -> i 0.0540 A -> j 0.0335 A -> k 0.0448 A -> l 0.0292 A -> m 0.0272 A -> n 0.0303 A -> o 0.0352 A -> p 0.0232 A -> q 0.0421 A -> r 0.0321 A -> s 0.0333 A -> t 0.0402 A -> u 0.0344 A -> v 0.0329 A -> w 0.0384 A -> x 0.0660 A -> y 0.0344 A -> z 0.0430 … Plain letter: O O -> a 0.0411 O -> b 0.0439 O -> c 0.0481 O -> d 0.0494 O -> e 0.0418 O -> f 0.0347 O -> g 0.0544 O -> h 0.0412 O -> i 0.0393 O -> j 0.0297 O -> k 0.0448 O -> l 0.0253 O -> m 0.0425 O -> n 0.0285 O -> o 0.0361 O -> p 0.0406 O -> q 0.0380 O -> r 0.0273 O -> s 0.0320 O -> t 0.0495 O -> u 0.0330 O -> v 0.0244 O -> w 0.0299 O -> x 0.0454 O -> y 0.0283 O -> z 0.0493 ... Plain letter: N N -> a 0.0350 N -> b 0.0504 N -> c 0.0458 N -> d 0.0371 N -> e 0.0437 N -> f 0.0453 N -> g 0.0452 N -> h 0.0325 N -> i 0.0230 N -> j 0.0237 N -> k 0.0491 N -> l 0.0355 N -> m 0.0383 N -> n 0.0362 N -> o 0.0243 N -> p 0.0388 N -> q 0.0408 N -> r 0.0343 N -> s 0.0210 N -> t 0.0526 N -> u 0.0319 N -> v 0.0247 N -> w 0.0278 N -> x 0.0572 N -> y 0.0219 N -> z 0.0824 ... Plain letter: S S -> a 0.0492 S -> b 0.0410 S -> c 0.0408 S -> d 0.0801 S -> e 0.0471 S -> f 0.0233 S -> g 0.0550 S -> h 0.0312 S -> i 0.0424 S -> j 0.0459 S -> k 0.0402 S -> l 0.0296 S -> m 0.0357 S -> n 0.0288 S -> o 0.0334 S -> p 0.0314 S -> q 0.0334 S -> r 0.0342 S -> s 0.0324 S -> t 0.0470 S -> u 0.0200 S -> v 0.0241 S -> w 0.0452 S -> x 0.0429 S -> y 0.0306 S -> z 0.0336 In partial conclusion, we obtain, for the effective key A, the letters numbers corresponding to the most frequent plain text letters: E->x T->r (but the letters A and Z give plausible values) A->x (Konheim finds the letter F) O->z (Konheim finds the letter P) N->z S->D Note: Slightly different results are obtained from those of Konheim. I think it comes from the statistics used. We do the same for the other letters. Reference
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