ICS141: Discrete Mathematics for Computer Science I

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1 ICS4: Discrete Mthemtics for Computer Sciece I Dept. Iformtio & Computer Sci., J Stelovsky sed o slides y Dr. Bek d Dr. Still Origils y Dr. M. P. Frk d Dr. J.L. Gross Provided y McGrw-Hill 3-

2 Quiz. gcd(84,96). lcm(84,96) 3. gcd(84,96) x lcm(84,96) Hits Wht's the prime fctoriztio of 84? Wht's the prime fctoriztio of 96? Try the primes, 3 d 7 3-

3 Lecture 8 Chpter 3. The Fudmetls 3.6 Itegers d Algorithms 3-3

4 Review: Gretest Commo Divisor The gretest commo divisor gcd(,) of itegers, (ot oth ) is the lrgest iteger d tht is divisor oth of d of. d gcd(,) mx(d: d d ) d d e Z, (e e ) (d e) If the prime fctoriztios re writte s d p p p, the the GCD is give y: p p p gcd(, ) p mi(, ) p mi(, ) p mi(, ). Exmple: d gcd(84,96)

5 Review: Lest Commo Multiple lcm(,) of positive itegers,, is the smllest positive iteger tht is multiple oth of d of. E.g. lcm(6,) 3 m lcm(,) mi(m: m m) m m Z: ( ) (m ) If the prime fctoriztios re writte s d p p p, the the LCM is give y: p p p lcm(, ) p mx(, ) p mx(, ) p mx(, ). Exmple: d lcm(84,96)

6 GCD d LCM Theorem: Let d e positive itegers. The gcd(,) lcm(,) Exmple ( 3 7 ) ( ) mi(,5) 3 mi(,) 7 mi(,) mx(,5) 3 mx(,) 7 mx(,) gcd(,) lcm(,) 3-6

7 Itegers d Algorithms Topics: Bse- represettios of itegers. Especilly: iry, hexdeciml, octl. Algorithms for computer rithmetic: Biry dditio d multiplictio. Euclide lgorithm for fidig GCD s. 3-7

8 Bse- Numer Systems Ordirily, we write se- represettios of umers, usig digits -9. But, is t specil! Ay se > will work. For y positive itegers d, there is uique sequece k k- of digits i < such tht: k k i k i i k k k k Nottio: ( k k- ) The se- expsio of 3-8

9 Prticulr Bses of Iterest Bse (deciml): digits:,,,3,4,5,6,7,8,9. Bse (iry): digits:,. ( Bits iry digits. ) Bse 8 (octl): 8 digits:,,,3,4,5,6,7. Bse 6 (hexdeciml): Used oly ecuse we hve figers 6 digits:,,,3,4,5,6,7,8,9,a,b,c,d,e,f Used iterlly i ll moder computers Octl digits correspod to groups of 3 its Hex digits give groups of 4 its 3-9

10 Exmples Exmple : Deciml expsio of the iteger with iry expsio ()? () (35) Exmple : Deciml expsio of the iteger with hexdeciml expsio (AEB) 6? (AEB) (7567) 3-

11 Covertig to Bse (A lgorithm, iformlly stted.) To covert y iteger to y se > : To fid the vlue of the rightmost (lowestorder) digit, simply compute mod. Now, replce with the quotiet. Repet ove two steps to fid susequet digits, util is goe ( ). Exercise: Write this out i pseudocode 3-

12 3- Covertig to Bse q q q q q ) ( ) ( ) ( ) (

13 Exmples Exmple 3: Fid the se 8, i.e. octl, expsio of (345) Therefore, (345) (37) 8 3-3

14 Biry Hexdeciml Hexdeciml expsio of ( ) 3 E B C ( ) (3EBC) 6 Biry expsio of (A8D) 6 (A) 6 (), (8) 6 (), (D) 6 () (A8D) 6 ( ) 3-4

15 Additio of Biry Numers Crry: Crry itsum / s itidex itsum mod itsum crry 3-5

16 Additio of Biry Numers procedure dd(, : iry represettios of o-egtive itegers, ) crry : for itidex : to egi {go through its} itsum : itidex itidex crry {-it sum} crry : itsum / s itidex : itsum crry ed s : crry {high it of sum} {low it of sum} retur s s : iry represettio of iteger s 3-6

17 Multiplictio of Biry Numers ( - - ) ( ) ( ) ( - - ) x shift it to the left, i.e. pped extr -it shift it to the left, i.e. pped extr -its 3-7

18 Multiplictio of Biry Numers ( - - ) ( ) ( ) ( - - ) procedure multiply(, : iry represettios of positive itegers,) product : for i : to if i the retur product i extr -its ppeded fter the digits of i times product : dd(, product) 3-8

19 Divisio with Remider procedure div-mod( Z,d Z ) {quotiet & remider of /d} q : r : while r d egi r : r d q : q ed if < d r > the egi { is egtive} r : d r q : (q ) ed {q div d (quotiet), r mod d (remider)} 3-9

20 Euclid s Algorithm for GCD Fidig GCDs y comprig prime fctoriztios c e difficult whe the prime fctors re ot kow! Euclid discovered: Let q r, where,, q, d r re itegers. The gcd(,) gcd(,r) (i.e. gcd(,) gcd(, ( mod ))) Exmple: gcd(36, 4) gcd(4, ) Sort, so tht >, d the (give > ) ( mod ) <, so prolem is simplified. 3-

21 Euclid s Algorithm Exmple gcd(37, 64) gcd(64, 37 mod 64) 37 mod / gcd(64, 44) gcd(44, 64 mod 44) 64 mod / gcd(44, 3) gcd(3, 44 mod 3) gcd(3, ) gcd(, 3 mod ) gcd(, 8) gcd(8, mod 8) gcd(8, 4) gcd(4, 8 mod 4) gcd(4, ) 4 3-

22 Euclid s Algorithm Pseudocode procedure gcd(, : positive itegers) x : y : while y egi r : x mod y; x : y; y : r; ed retur x {x gcd(, )} 3-

23 Proof Tht Euclid s Algorithm Works Theorem : gcd(,) gcd(,c) if c mod. Proof: First, c mod implies t: t c. Let g gcd(,), d gʹ gcd(,c). Sice g d g (thus g t) we kow g ( t), i.e. g c. Sice g g c, it follows tht g gcd(,c) gʹ. Now, sice gʹ (thus gʹ t) d gʹ c, we kow gʹ (tc), i.e., gʹ. Sice gʹ gʹ, it follows tht gʹ gcd(,) g. Sice we hve show tht oth g gʹ d gʹ g, it must e the cse tht g gʹ. 3-3

24 Two s Complemet I iry, egtive umers c e coveietly represeted usig two s complemet ottio. I this scheme, strig of its c represet y iteger i such tht i <. The leftmost it is used to represet the sig (:positive, :egtive iteger) The egtio of y -it two s complemet umer is give y. The itwise logicl complemet of the -it strig. 3-4

25 Two s Complemet Exmple i < ( i < ) vlue 3-it ptter To oti the results for 4, cosider, the I the iry represettio of, replce ech y, d ech y, This is the oe s complemet of. Add (i.e. ) to the result from the previous step. This is the two s complemet of. Exmple 3: -> 3-5

26 Sutrctio of Biry Numers procedure sutrct(, : iry two s complemet reps. of itegers, ) retur dd(, dd(,)) { ( ) } Note tht this fils if either of the dds cuses crry ito or out of the positio, sice, d ( ) is t represetle! We cll this overflow. 3-6

27 Modulr Expoetitio Prolem: Give lrge itegers (se), (expoet), d m (modulus), efficietly compute mod m. Note tht itself my e completely ifesile to compute d store directly. E.g. if is,-it umer, the itself will hve fr more digits th there re toms i the uiverse! Yet, this is type of clcultio tht is commoly required i moder cryptogrphic lgorithms! 3-7

28 3-8 Algorithm Cocept ) ( ) ( ) ( k k k k k k k k Note tht: We c compute to vrious powers of y repeted squrig. The multiply them ito the prtil product, or ot, depedig o whether the correspodig i it is. Crucilly, we c do the mod m opertios s we go log, ecuse of the vrious idetity lws of modulr rithmetic. All the umers sty smll. The iry expsio of

29 Modulr Expoetitio procedure modulrexpoetitio(: iteger, ( k ), m: positive itegers) x : {ccumultes the result} i : mod m { i mod m; i iitilly} for i : to k egi {go thru ll k its of } if i the x : (x i) mod m i : (i i) mod m ed retur x {x equls mod m} i i ( i ) ( i ) 3-9

Laws of Integral Indices

Laws of Integral Indices A Lws of Itegrl Idices A. Positive Itegrl Idices I, is clled the se, is clled the idex lso clled the expoet. mes times.... Exmple Simplify 5 6 c Solutio 8 5 6 c 6 Exmple Simplify Solutio The results i