Stream Ciphers (contd.) Debdeep Mukhopadhyay

Transcription

1 Strea Ciphers (cotd.) Debdeep Mukhopadhyay Assistat Professor Departet of Coputer Sciece ad Egieerig Idia Istitute of Techology Kharagpur IDIA Objectives iear Coplexity Berlekap Massey Algorith ow Power Ajit Pal IIT Kharagpur

2 The FSR Structure s j- s j-2 s j-+ s j- s = cs, j =, +,... j i j i A FSR is said to geerate a fiite sequece s,s,,s - whe this sequece coicides with the first output digits of the FSR for soe iitial loadig. Geeratio of a sequece If, the FSR always geerates the sequece. If <, it follows that the FSR geerates the sequece if ad oly if: s = cs, j =, +,..., j i j i ow Power Ajit Pal IIT Kharagpur 2

3 Theore If soe FSR of legth geerates the sequece s, s,..., s but ot the sequece s, s,..., s, s the ay FSR that geerates the latter sequece has legth ', satisfyig: ' + Proof Case :, the theore is trivially true. Case 2: <, let c, c,..., c ad c', c',..., c' 2 2 ' deote the coectio coefficiets of the two FSRs i questio ad assue that ' -. cs = s, j=, +,..., ' i j i j s, j = c' s = s, j = ', ' +,...,, k j k j ow Power Ajit Pal IIT Kharagpur 3

4 Proof (cotd.) Cosider, ote that { s, s,..., s } is a subset ' ' + ' + of { s, s,..., s }. cs cs = c c' s i i i i k i k k= ' = c' k i i k k= ' = c' s = s k = i cs k k ote that { s, s,..., s } is a subset of { s, s,..., s }. + ' ' + Thus we have a cotradictio. This proves the result. iear Coplexity Defie (s) as the iiu legth of all FSRs that geerate s, s,, s - Clearly, (s) Moreover, (s) ust be ootoically decreasig with icreasig. Covetio: all sequece is geerated by the FSR with = Whe s, s,, s - are all s but s =, the =+ ow Power Ajit Pal IIT Kharagpur 4

5 ea If soe FSR of legth geerates the sequece s, s,..., s but ot the sequece s, s,..., s, s the () s ax[ (), s + ()] s + Fro the ootoicity of ( s) ( s). + + Fro Theore, ( s) + ( s). Thus the lea follows. Berlekap Massey s Algorith A recursive algorith for producig oe of the FSRs of legth (s), which geerates s, s,, s - for =, 2, 3, C(D)=+C D+ +C D which has degree at ost i the ideteriate. Covetio: C(D)= for the FSR of legth = ow Power Ajit Pal IIT Kharagpur 5

6 Coectio Polyoial For a give s, let C ( D) = + C ( D) C ( D) ( ) ( ) ( S) ( s) deote the coectio polyoial of a iial legth ( s) FSR that geerates s, s,..., s Discrepacy ea is actually a equality. We have see this for the base case. Assue a iductio hypothesis for ( s). The correspodig polyoial is C ( D). ( s), ( ) ( s),..., j = sj ci sj i = d, j = d : ext discrepacy (betwee s ad the (+)st bit geerated by the iial legth FSR, which we have foud to geerate the first bits of s. ow Power Ajit Pal IIT Kharagpur 6

7 Correctig the discrepacy Case: d = FSR also geerates the first + bits of s. Thus, ( + ) + ( s) = ( s), C ( D) = C ( D) Case: d = et be the sequece legth before the last legth chage i the iial legth register, i,e ( s) < ( s) ( s) = ( s) + Provig the Iductio Hypothesis Sice a legth chage was required < ( s), c ( D) > could ot geerate s, s,..., s + ( s), ( ) ( ),..., j = s sj ci sj i = d, j = By iductio hypothesis, ( s) = ( s) = ax[ ( s), + ( s)] ( s) < ( s), ( s) = + ( s) ow Power Ajit Pal IIT Kharagpur 7

8 Recursive costructio of polyoial Clai : CD = C D + ( ) (D) D C (D) is a valid ext choice for C ( ). ote : degree of C(D)=ax[ ( s), + ( s)] =ax[ ( s), + ( s)] CD ( ) is a allowable coectio polyoial for a FSR of legth =ax[ ( s), + ( s)] Proof that C(D) geerates s + ( s) ( ) j i j i j i j i s c s = s c s ( s) ( ) [ sj + ci sj + i], j =, +,..., = =, j = ow Power Ajit Pal IIT Kharagpur 8

9 Coclusios The FSR with legth ad coectio polyoial C(D) geerates s,s,,s Sice satisfies ea with equality, the iductio is also proved. The fial Algorith ow Power Ajit Pal IIT Kharagpur 9

10 Exaple Cosider the sequece of periodicity 2: We plot the variatio of the liear coplexity with. this is obtaied by the Berlekap Massey Algorith this is called iear Profile Exaple ow Power Ajit Pal IIT Kharagpur

11 Exercise Recostruct a FSR (of the shortest legth) which geerates the sequece. s d T(D) C(D) B(D) D D 3 +D+D D 3 +D+D D 3 +D+D D 3 +D+D D+D 3 +D+D 3 + D D+D 3 8 ow Power Ajit Pal IIT Kharagpur

12 Further Readig Jaes Massey, Shift-Register Sythesis ad BCH Decodig, IEEE Trasactios o Iforatio Theory, 969 D. Stiso, Cryptography: Theory ad Practice, Chapa & Hall/CRC A. Meezes, P. Va Oorschot, Scott Vastoe, Hadbook of Applied Cryptography (Available olie) ext Days Topic Strea Ciphers (cotd.) ow Power Ajit Pal IIT Kharagpur 2