#A42 INTEGERS 16 (2016) THE SUM OF BINOMIAL COEFFICIENTS AND INTEGER FACTORIZATION

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1 #A4 INTEGERS 1 (01) THE SUM OF BINOMIAL COEFFICIENTS AND INTEGER FACTORIZATION Ygu Deg Key Laboatoy of Mathematcs Mechazato, NCMIS, Academy of Mathematcs ad Systems Scece, Chese Academy of Sceces, Bejg, P R Cha degy@amssacc Yab Pa Key Laboatoy of Mathematcs Mechazato, NCMIS, Academy of Mathematcs ad Systems Scece, Chese Academy of Sceces, Bejg, PR Cha ayab@amssacc Receved: 10//14, Revsed: 1/18/1, Acceted: //1, Publshed: /10/1 Abstact The combatoal sum of bomal coe cets ale X (a) := (mod ) a has bee studed wdely combatoal umbe theoy, esecally whe a = 1 ad a = 1 I ths ae, we coect t wth tege factozato fo the fst tme Moe ecsely, gve a comoste, we ove that fo ay a come to thee exsts a modulus such that the combatoal sum has a otval geatest commo dvso wth Deote by FAC(, a) the least We eset some elemetay ue bouds fo t ad beleve that some bouds ca be moved futhe sce FAC(, a) s usually much smalle the exemets We also oosed a algothm based o the combatoal sum to facto teges Ufotuately, t does ot wo as well as the exstg mode factozato methods Howeve, ou method yelds some teestg heomea ad some ew deas to facto teges, whch maes t wothwhle to study futhe 1 Itoducto Let,, ad a be teges wth > 0 ad > 0 Cosde the sum of bomal coe cets ale X (a) := a, (mod ) whee s the bomal coe cet wth the coveto = 0 fo < 0 o > The combatoal sum has bee studed wdely combatoal umbe

2 INTEGERS: 1 (01) theoy ad may of ts oetes have bee exloed Fo examle, Wesma [18] oved that fo ay me ad ay ostve tege, ale j ( 1) 0 (mod 1 ( 1) Futhemoe, Su [1] showed that fo > 1 ad a 1 (mod ), ale (a) 0 (mod b 1 '( ) c ) Othe esults about the combatoal sum ca be foud [1, 14, 1, 1] Howeve, we have to ot out that the exact value of the combatoal sum seems had to obta fo geeal, eve whe a = 1 o a = 1 Fo examle, Su [9, 10, 11] studed the values of the combatoal sum fo a = 1, whe =, 4,,, 8, 9, 10, 1, 1 Amog them, the values ae exlctly gve just fo =, 4,, wheeas the othe values ae mlctly gve by some Lucas sequeces Note that the fome wos ae combatoal atue, whch am to obta cogueces o combatoal dettes I ths ae, we coect the combatoal sum wth tege factozato fo the fst tme It s well-ow that the tege factozato oblem s oe of the most famous comutatoal oblems, as wtte by Gauss (Dsqustoes Athmetcae, 1801, at 9): The oblem of dstgushg me umbes fom comoste umbes, ad of esolvg the latte to the me factos s ow to be oe of the most motat ad useful athmetc It has egaged the dusty ad wsdom of acet ad mode geometes to such a extet that t would be suefluous to dscuss the oblem at legth Futhe, the dgty of the scece tself seems to eque that evey ossble meas be exloed fo the soluto of a oblem so elegat ad so celebated I 004, Agawal, Kayal ad Saxea [1] gave a detemstc olyomal-tme algothm to test the malty of a umbe, whch solved the oblem of dstgushg me umbes fom comoste umbes theoy Howeve, the oblem of esolvg comoste umbes to the me factos seems fa fom beg solved, sce the best ow algothm to facto teges, the umbe feld seve method [], taes subexoetal tme I ths ae, we oose a ew algothm to facto teges based o the combatoal sum The ey obsevato s that fo ay comoste ad ay tege a come to ale, thee always exsts a modulus less tha such that the combatoal sum (a) has a otval geatest commo dvso (gcd) wth By comutg the geatest commo dvso, a otval dvso of ca be obtaed easly 1 )

3 INTEGERS: 1 (01) Note that the combatoal sum ca be comuted e cetly fo fxed a ad as [1] It emas to show how to fd the modulus fo some fxed a A atual way s to chec evey fom 1 to by decdg whethe the coesodg combatoal sum has a otval geatest commo dvso wth o ot It s obvous that the tme comlexty of ths ocedue deeds o the sze of We deote by FAC(, a) the least such that the sum has a otval geatest commo dvso wth, ad call t the factozato umbe of wth esect to a Fo ay eve comoste, whch s of couse easy to be factoed, we fd that the factozato umbe s at most whe a = ±1, whch meas that the eve comoste ca also be easly factoed wth ou algothm Fo the RSA modulus, whch s cosdeed had to be factoed, we eset some elemetay ue bouds fo the factozato umbe Howeve, ou bouds seem athe ough sce the exemets show that FAC(, a) s usually much smalle tha the bouds Hece, we beleve that the bouds ca be moved futhe theoy Due to lac of good mathematcal tools to deal wth the combatoal sum, we do ot ow how to estmate the factozato umbe FAC(, a) as well as ossble whe a s fxed Futhemoe, we do ot ow how to estmate m a FAC(, a) whee a us ove some secfc set ethe We cojectue that both of the two questos ae vey d cult ad oose them as oe oblems We also mlemeted ou algothm to facto teges Ufotuately, t dd ot wo as well as the exstg mode factozato methods, such as the umbe feld seve method Howeve, ou method yelds some teestg heomea ad some ew deas to facto teges, whch maes t wothwhle to study futhe The ae s ogazed as follows We gve the defto of the factozato umbe of a comoste Secto We gve some oetes fo the factozato umbe of a eve comoste Secto I Secto 4, we ove some elemetay ue bouds fo the factozato umbe of a RSA modulus We lst some exemetal esults Secto Fally, a shot cocluso ad some oe oblems ae gve Secto The Factozato Numbe of a Comoste 1 The Combatoal Sum ad ts Poosto Defto 1 Let,, ad a be teges wth > 0 ad > 0 We defe the combatoal sum of bomal coe cets as ale X (a) := a 0aleale (mod )

4 INTEGERS: 1 (01) 4 Fo smlcty, we defe ale ale := ale The followg lemma s useful to comute (1) (a) whe s small Lemma Let C be a mtve -th oot of uty The fo 0 ale ale 1, we have ale (a) = 1 X 1 ( j + a) ( j ) Poof Sce (X + a) = P =0 j=0 a X, we have " X 1 X 1 X ( j + a) ( j ) = j=0 The lemma follows = j=0 X s=0 = s # js a s ( j ) s s=0 X 1 a s (s )j X 0alesale s (mod ) j=0 s a s Itege Factozato ad the Factozato Numbe of a Comoste Gve a comoste umbe > 1, tege factozato efes to the questo of fdg a otval dvso d of, e, d ad 1 < d < To facto a comoste, we obseve the followg: Poosto Fo ay ostve comoste, thee exsts a tege j wth 1 < j < 1 such that 1 < gcd(, j ) < Poof To ove the oosto, we cosde the followg two cases Case 1 has a squae dvso Assume has a me dvso such that wth > 1, e, but +1 - Notce that ( 1)( ) ( + 1) =! Sce - ( ) fo 1 ale ale 1, we have 1 Hece 1 < gcd(, ) < Case s squae-fee Let ad q be two me dvsos of wth < q As the fst case, t ca be cocluded that - but q Hece 1 < gcd(, ) <

5 INTEGERS: 1 (01) Note that evey j s a coe cet of the olyomal (X + 1) By Poosto, a atual way to obta a otval dvso of s exadg (X + 1), ad the comutg each geatest commo dvso of ts coe cets ad Howeve, ths wll tae exoetal tme sce thee ae a total of coe cets To educe the tme comlexty, we tu to emloy aothe olyomal X 1 (X + a) a X mod (X 1, ), =0 as cosdeed [1], whch yelds the defto of the factozato-fedly umbe of Defto 4 Let be a ostve comoste, a be a tege come to, ad be a ostve tege Cosde the olyomal wth a detemate X: X 1 (X + a) a X mod (X 1, ), =0 whee a s ae teges wth 0 ale a ale 1 fo 0 ale ale 1 We call a factozatofedly umbe of wth esect to a f thee exsts a wth 0 ale ale 1 such that gcd(, a ) s a otval dvso of By Defto 4, t s easy to coclude that fo 0 ale ale ale a (a) (mod ) 1, we have Next we show that the factozato-fedly umbe of wth esect to a abtay a come to must exst Poosto Fo ay ostve comoste ad a tege a come to, 1 s a factozato-fedly umbe of wth esect to a Poof Sce we have (X + a) = = X =0 a X, X (X + a) a + X + a X mod (X 1 1, ) The esult follows by Poosto

6 INTEGERS: 1 (01) By Poosto, we ca always facto by comutg the coesodg 1 coe cets Ufotuately, t wll tae exoetal tme too Geeally seag, fo ay factozato-fedly umbe, t wll tae e O ( log ) tme (see [17]) to obta all the a s Defto 4 Theefoe, we ae teested the least factozato-fedly umbe of Defto Fo ay ostve comoste ad a tege a come to, the least factozato-fedly umbe of wth esect to a s called the factozato umbe of wth esect to a ad s deoted by FAC(, a) By Poosto, we mmedately have Poosto 7 Fo ay ostve comoste ad a tege a come to, we have FAC(, a) ale 1 To fd the exact value of FAC(, a), a atual way s to chec evey fom 1 to 1, whch yelds a ew algothm to facto a comoste Iut: a comoste Outut: a otval dvso d of 1 Choose some a [, 1], f 1 < d = gcd(a, ) <, outut d Othewse, fo fom 1 to 1 do 4: exad (X + a) mod (X 1, ) as P 1 =0 a x, 4 comute each geatest commo dvso d of a ad If 1 < d <, outut d We ema that the tme comlexty of the algothm deeds o the sze of FAC(, a) The Factozato Numbe of a Eve Comoste Numbe I ths secto, we show that the factozato umbe of a eve comoste umbe s usually vey small whe a = ±1 1 The Factozato Numbe of a Eve Comoste Numbe whe a = 1 Let Note that fo 0 ale ale X 1 (X + 1) a X mod (X 1, ) =0 1, we have ale a (mod )

7 INTEGERS: 1 (01) 7 The statemets the followg oosto ae well-ow The fst two ae easy to chec, ad we omt the oof The thd statemet s well-ow, but fo comleteess we ovde a oof Poosto 1 Let be ay ostve tege The we have: ale (1) fo = 1, = 0 ; ale () fo =, 0 () fo =, Poof of () Set = The we ow ad Thus, we have whch mles ale 0 = 1 1 ale = 1 ale 0 ale 1 ale = 1 = 1 = 1 = 1 ; + cos + cos + cos, ( ) ( + ), 1 By Lemma, let be a mtve -d oot of uty = 1 +, = = 1 + = e, 1 + = 1, (1 + ) = e, (1 + ) = e, X (1 + j ) = 1 + e + e = 1 j=0 Smlaly, we have ale 1 ale = 1 = 1 X (1 + j ) j = 1 j=0 X (1 + j ) j = 1 j=0 + cos + cos = e + cos ( ) ( + ),

8 INTEGERS: 1 (01) 8 Theoem Let be a ostve comoste umbe We have: (1) f s eve ad s ot a owe of, the FAC(, 1) = 1; () f s odd, the FAC(, 1) ; () f s a owe of, the FAC(, 1) = Poof By Poosto 1, t s easy to ove (1) ad () It emas to ove () Suose s a owe of, ale e, = m, m If m s eve, the 1 (mod ) ad + s eve Thus a = 1 ( + ) (mod ), whch mles gcd(, a ) = If m s odd, we have gcd(, a 1 ) = smlaly The followg oosto ca be used to educe the comutato of geatest commo dvsos Poosto Let, be two ostve teges ad be a tege The we have ale ale = Poof We have ale = = X 0aleale (mod ) X 0ale ale (mod ) ale = The Factozato Numbe of a Eve Comoste Numbe whe a = 1 Let Note that fo 0 ale ale ale a X 1 (X 1) a X mod (X 1, ) =0 1, we have ( 1) = X 0aleale (mod ) ( 1) (mod )

9 INTEGERS: 1 (01) 9 Poosto 4 Let be ay ostve tege The we have: ale (1) fo = 1, ( 1) = 0; 0 1 ale ale () fo =, ( 1) = ( 1) 0 1 ad ( 1) = ( 1) ; () fo =, ale 0 ale 1 ale ( 1) = 1 cos ( 1) = 1 cos ( 1) = 1 cos, ( 4) ( 8) Poof The oof s smla to the oe of Poosto 1 ad we omt the detals Theoem Let be a ostve comoste umbe We have: (1) f s eve ad s ot a owe of, the FAC(, 1) = ; () f s odd, the FAC(, 1) ; () f s a owe of, the FAC(, 1) = Poof By Poosto 4, t s easy to ove (1) ad () It emas to ove () Suose s a owe of, e, = m, m The 4 = m 1 ad 8 = m 1 4 ale It s easy ale to coclude that ethe 4 o 8 must be eve, thus ethe ( 1) o ( 1) s 1 m 1 1 Hece we have ethe gcd(, a 1 ) = o gcd(, a ) = Smla to Poosto, we have Poosto Let, be two ostve teges, ad be a tege The we have ale ale ( 1) = ( 1) ( 1),

10 INTEGERS: 1 (01) 10 4 The Factozato Numbe of a RSA Modulus I ths secto, we cosde the factozato umbe of a RSA modulus A RSA modulus s a oduct of two dstct odd mes, that s, = q whee < q ae two dstct odd mes The RSA modulus = q whee < q ae two dstct bg odd mes wth the same umbe of bts s cosdeed had to be factoed To aalyze the tme comlexty of ou algothm fo a RSA modulus, we eset some ue bouds fo FAC(, a) Fst, we toduce some useful esults 41 Some Useful Results Theoem 41 (Lucas Theoem) Fo ay me, suose a = a 0 + a a, b = b 0 + b b, whee 0 ale a, b < fo = 0, 1,, The we have a Y a (mod ) b =0 See [] fo a oof of Lucas Theoem b Lemma 4 Fo ay ostve tege > 1, let m be a tege satsfyg 0 < m < ad gcd(, m) = 1 The we have Poof Notce that = m s a tege ad 1 m 1 we have m! m!( m)! = m m ( 1)! (m 1)!( m)! = m s also a tege We have m 1 m 1 1 m 1, whch yelds m 1 m 1 Sce gcd(, m) = 1, Lemma 4 Suose = q, whee < q ae two dstct odd mes The we have (1) fo 0 < < q, q () fo 0 < j <, ; qj but q - qj Poof Sce q - fo 0 < < q, we have q by Lucas Theoem Smlaly, we have qj fo 0 < j < Fo 0 < j <, we have qj j (mod q) by Lucas Theoem Sce < q, we have q - j Hece q - qj fo 0 < j <

11 INTEGERS: 1 (01) 11 4 The Factozato Numbe of a RSA Modulus Now we eset some ue bouds fo FAC(, 1), whee s a RSA modulus Fst we show that Theoem 44 Fo a RSA modulus = q wth < q, s a factozato-fedly umbe of wth esect to 1, whch yelds FAC(, 1) ale < Poof It s su cet to ove that s a factozato-fedly umbe of wth esect to 1 Wte q = a +, whee a > 0 ad 0 < < We wll ove the theoem by showg that ale! gcd, = Set I = { + s s Z, 0 ale + s ale } Obvously, 0, ad (0 < < q) ae ot I but q s I If qj s I fo some j wth 0 < j <, the qj j (mod ), whch yelds j 1 (mod ), that s, j = 1 Hece q I ad the othe elemets I ae come to By Lemma 4, we have ale By Lemma 4, we have = X 0aletale t (mod ) q but q - Togethe wth Theoem, we have t q (mod ) q Hece gcd, ale! = Coollay 4 Fo a tege = q whee q > s a me, we have FAC(, 1) = I fact, we ca do a lttle bette Theoem 4 Let = q be a RSA modulus wth < q <, ad c be a ostve tege such that c ale q 1 The c s a factozato-fedly umbe of wth esect to 1, whch yelds FAC(, 1) ale c Poof Wte q = +, whee 0 < < Sce s eve, we have c +, whch yelds c + 1 < c ad + c < c It s easy to show that gcd( c, c) = 1 sce ( c) + c = s a me Hece the set { c (mod c) = 1,,, c + 1} has exactly c + 1 elemets Smlaly, t ca be show that gcd( c, + c) = 1 ad the set {( + c)j (mod c) j = 1,,, c} has c elemets Thus thee must exst a elemet a the fst set but ot the secod set Let a c 1 (mod c) wth 1 ale 1 ale c + 1 ad 0 < a < c Set I = {a + ( c)s s Z, 0 ale a + ( c)s ale } Obvously, 0 / I Sce = q c( + c) (mod c), we have / I by the choce of a Fo

12 INTEGERS: 1 (01) 1 0 < < q, f I, the a (mod c) We have c c 1 (mod c), whch mles 1 (mod c) Hece = c 1 + ( c)s, s 0 Sce c 1 + c = c 1 c 1 + = q, we have = c 1, 0, whch meas that fo 0 < < q oly 0 I Fo 0 < j <, f qj I, the qj a (mod c) We have ( +c)j a (mod c) The equato has a uque soluto j 0 wth 0 ale j 0 < c We ow j 0 c + 1 by the choce of a Sce j 0 + c >, we have j = j 0, whch meas that fo 0 < j < oly qj 0 I Thus, we get ale + (mod ) a 0 qj 0 By Lemma 4, we have Sce q ale, we have Hece, we have c ale q - a 0 c, whch mles ale a ale gcd, a c c! = Theefoe c s a factozato-fedly umbe of wth esect to 1 Coollay 47 Let = q be a RSA modulus wth < q < Wte q = + wth 0 < < If < " fo some 0 < " < 1 ad 1 ", the FAC(, 1) ale b 1 " +" c t Poof Put c = b 1 " c ad the the esult follows by Theoem 4 Theoem 48 Let = q be a RSA modulus wth < q < Wte q = + wth 0 < < If 1 (mod 4) ad < <, the := + 4 s a factozato-fedly umbe of wth esect to 1, whch yelds FAC(, 1) ale + 4 Poof Sce 1 (mod 4), t ca be cocluded that oe of ad q s coguet to 1 modulo 4, wheeas the othe s coguet to 1 modulo 4 Hece we have (mod 4), whch yelds that s a ostve tege ad < Let a Z satsfy 0 ale a < ad a ( 1) (mod ) We have a > 0 sce > 1 Set I = {a + s s Z, 0 ale a + s ale } By a smla aalyss as the oof of Theoem 4, we have ( 1) I, q( ) I, ad the othe elemets I ae come to Thus ale + a ( 1) q( ) (mod )

13 INTEGERS: 1 (01) 1 Smlaly, we have ale gcd, a =, whch mles that s a factozato-fedly umbe of wth esect to 1 Coollay 49 Wth otato as Theoem 48, f we futhe suose < " wth 0 < " ale, the we have FAC(, 1) ale ( ") Rema 410 Comag the bouds Coollay 47 ad Coollay 49, t s easy to see that +" = " whe " =, +" < " whe " > +", ad > " whe " < Moeove, fom the oofs of Theoems 44, 4 ad 48 we ow that thee s oly oe bomal coe cet left whe the combatoal sum s educed modulo (o modulo q) Theefoe, all the bouds fo FAC(, 1) above hold also fo FAC(, a) wth a Z come to Fally, we have to ot out that the ue bouds fo FAC(, 1) above ae athe ough, sce ou exemets show that FAC(, 1) s usually much smalle tha See Secto fo moe detals 4 The Factozato Numbe of a RSA Modulus wth Tw Pmes We ext eset two teestg esults, whch ovde atal evdece of ou cojectue that a umbe that ca be easly factoed by othe methods ca also be easly factoed by ou method Theoem 411 Let = q be a RSA modulus If q = +, e, ad q ae tw mes, the we have FAC(, 1) ale Poof Sce FAC(1, 1) = by Coollay 4, we ca assume It s easy to coclude that, thee s a ostve tege such that = 1 ad q = + 1 Hece t ca be show that ale X X + qj (mod ) 0<<q 4 (mod ) qj 0<j< j (mod ) By Lemma 4, we ow fo each j wth 0 < j < Moeove, fo each wth 0 < < q ad 4 (mod ), we have 4 ale ale q = 1, whch mles q = = 0 (mod ) by Lucas theoem Thus ale

14 INTEGERS: 1 (01) 14 By Lemma 4, we have q fo each wth 0 < < q smlaly By Lucas theoem, we have qj j (mod q) fo each j wth 0 < j < Hece ale ale (mod q) Notce that t ca be cocluded fom [9] that ale = 1 ( ), fo a eve, ale = 1 ( ), fo a odd By Eule s cteo ad the Quadatc Recocty Law of the Legede symbol, we ow +1 = q 1 = ( 1) (q 1)/ q 1 = ( 1) = ( 1) (mod q) q Thus, both cases, we have ale ale whch mles q - Fally, we get 1 Theefoe, we have ale q - ale gcd, (mod q), = It s teestg that we ca move the boud Theoem 411 fo some s Theoem 41 Let = q be a RSA modulus Suose q = +, e, ad q ae tw mes If > ad ±1 (mod ), the FAC(, 1) ale Poof We fst cosde the case whe 1 (mod ), q (mod ) It s easy to coclude that ale ale q (mod ), 4 4 ale ale (mod q) 4

15 INTEGERS: 1 (01) 1 Defe two sequeces {u } 0, {v } 0 as follows: u 0 = 0, u 1 = 1, u +1 = u + u 1 fo 1, v 0 =, v 1 = 1, v +1 = v + v 1 fo 1 Sce the Legede symbol = = 1 = 1, we have v 1 (mod ), u 1 0 (mod ) ad u 1 (mod ) by a well-ow esult of the Fboacc sequece (see [4]) It follows that v q = v + 4 (mod ) Moeove, by a esult of Su [9], we have ale q Hece we have 4 ale q v q = 4 q sce q 8 (mod ), whch mles ale 4 O the othe had, sce the Legede symbol = 1, smlaly we have v q+1 (mod q), u q+1 0 (mod q) ad u q 1 (mod q), ad t follows that By a esult of Su [9], we have v = v q 4 (mod q) q ale Hece q -, whch mles ale v = ale q - 4 Theefoe, Fo the case ale gcd, 4 = 1(mod ), the oof s smla Notce that fo the case (mod ), the exemets show that the boud ca ot be elaced by

16 INTEGERS: 1 (01) 1 Exemetal Results We have doe umeous exemets usg NTL lbay [8] These exemets show the emaable fact that FAC(, a) s, eve FAC(, 1) s, ae usually much smalle tha, ad they gow vey slowly as ceases I Table 1 we lst some values of FAC(, 1) s fo = q whee ad q have thee dgts = q F (1) = q F (1) = q F (1) 1040=101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =101* =9* =9* =9*719 47=7* =7* =7* =79*7 8=79* =79*8 4097=9*91 419=9* =9* =07* =07* =07* =701*887 71=709* =77* =8* =87* =89* =9* =9* =991*997 9 Table 1 Fo a fxed, d eet a s wll geeally lead to dstct FAC(, a) s Usually, thee exsts some a such that the coesodg FAC(, a) s emaably less tha FAC(, 1), whch dcates that we ca obta that combatoal sum much moe qucly whe choosg such a We also lst some exemetal esults Table = q FAC(, 1) a FAC(, a) 91011=14* =1487* =1871* =0* =001* =7* =789* =0071* =9* =1911* =700* =10171* Table

17 INTEGERS: 1 (01) 17 Cocluso ad Oe Poblems It s well-ow that tege factozato s a vey motat comutatoal oblem Howeve, thee has bee o substatal ogess o solvg ths oblem sce the veto of the geeal umbe feld seve method 199 [,, ] We oose a ew method to facto teges based o combatoal sums of bomal coe cets ths ae As we ow, t s the fst tme to coect the combatoal sum wth tege factozato We beleve that ou method yelds ew ad motat dea, whch maes t wothwhle to study futhe Of couse, thee ae stll some oe oblems left Oe s to obta a tghte ue boud of FAC(, a) fo some fxed a sce the exemets show that FAC(, a) s usually much smalle tha ou bouds ths ae The othe s to gve a bette theoetc estmate fo m a FAC(, a) whe a us ove some secfc set Acowledgmets We tha the aoymous efeees fo the may valuable suggestos o how to move the esetato of ths ae The wo of ths ae was suoted by the NNSF of Cha (Gats Nos , 17490), ad the Natoal Cete fo Mathematcs ad Itedsclay Sceces, CAS Refeeces [1] M Agawal, N Kayal ad N Saxea, Pmes s P, A of Math () 10 (004), [] P J Cameo, Combatocs: Tocs, Techques, Algothms, Cambdge Uvesty Pess, Cambdge, 1994 [] H Cohe, A Couse Comutatoal Algebac Numbe Theoy, Gaduate Texts Mathematcs, vol 18, Sge, Bel, 199 [4] R Cadall ad C Pomeace, Pme Numbes, A Comutatoal Pesectve, Secod edto, Sge, New Yo, 00 [] A K Lesta, Itege Factog, Towads a Quate-cetuy of Publc Key Cytogahy, Des Codes Cytog 19 (000), [] A K Lesta ad H W Lesta, J(Eds), The Develomet of the Numbe Feld Seve, Lectue Notes Mathematcs, vol 14, Sge, Bel, 199 [7] R L Rvest, A Sham ad L Adlema, A Method fo Obtag Dgtal Sgatues ad Publc-ey Cytosystems, Comm ACM 1(1978), 10 1 [8] V Shou, NTL: A Lbay fo Dog Numbe Theoy, Avalable at htt://wwwshouet/tl/ [9] Z-H Su, Combatoal Sum P =0, (mod m) ad ts Alcatos Numbe Theoy (I), Najg Daxue Xuebao Shuxue Baa Ka 9 (199), 7 40 [10] Z-H Su, Combatoal Sum P =0, (mod m) ad ts Alcatos Numbe Theoy (II), Najg Daxue Xuebao Shuxue Baa Ka 10(199),

18 INTEGERS: 1 (01) 18 [11] Z-H Su, Combatoal Sum P =0, (mod m) ad ts Alcatos Numbe Theoy (III), Najg Daxue Xuebao Shuxue Baa Ka 1(199), [1] Z-W Su, O the Sum P (mod m) 18(00), 1 1 ad Related Cogueces, Isael J Math [1] Z-W Su, Polyomal Exteso of Flec s Coguece, Acta Ath 1(00), [14] Z-W Su, O Sums of Bomal Coe cets ad the Alcatos, Dscete Math 08(008), [1] Z-W Su ad D Davs, Combatoal Cogueces Modulo Pme Powes, Tas Ame Math Soc 9(007), [1] Z-W Su ad R Tauaso, Cogueces fo Sums of Bomal Coe cets, J Numbe Theoy 1(007), 87 9 [17] J vo zu Gathe ad J Gehad, Mode Comute Algeba, Cambdge Uvesty Pess, Cambdge, 1999 [18] C S Wesma, O -adc D eetablty, J Numbe Theoy 9(1977), 79 8