Finding Primitive Roots Pseudo-Deterministically

Transcription

1 Electronc Colloquum on Computatonal Complexty, Report No 207 (205) Fndng Prmtve Roots Pseudo-Determnstcally Ofer Grossman December 22, 205 Abstract Pseudo-determnstc algorthms are randomzed search algorthms whch output unque solutons (e, wth hgh probablty they output the same soluton on each executon) We present a pseudo-determnstc algorthm that, gven a prme p, fnds a prmtve root modulo p n tme exp(o( p log )) Ths mproves upon the prevous best known provable determnstc (and pseudo-determnstc) algorthm whch runs n exponental tme p 4 +o() Our algorthm matches the problem s best known runnng tme for Las Vegas algorthms whch may output d erent prmtve roots n d erent executons When the factorzaton of p s known, as may be the case when generatng prmes wth p n factored form for use n certan applcatons, we present a pseudo-determnstc polynomal tme algorthm for the case that each prme factor of p s ether of sze at most log c (p) or at least p /c for some constant c>0 Ths s a sgnfcant mprovement over a result of Gat and Goldwasser [5], whch descrbed a polynomal tme pseudo-determnstc algorthm when the factorzaton of p was of the form kq for prme q and k = poly() We remark that the Generalzed Remann Hypothess (GRH) mples that the smallest prmtve root g satsfes g apple O(log 6 (p)) Therefore, assumng GRH, gven the factorzaton of p, the smallest prmtve root can be found and verfed determnstcally by brute force n polynomal tme Introducton Pseudo-determnstc algorthms are randomzed search algorthms whch, wth hgh probablty, output the same soluton on each executon Formally, A s a pseudo-determnstc algorthm for a bnary relaton R f there exsts some functon s such that when executed on nput x, the algorthm A outputs s(x) wth hgh probablty, and (x, s(x)) 2 R In other words, when we execute A on nput x, we get the same output s(x) for almost all random seeds Standard randomzed search algorthms, on the other hand, may output a d erent y satsfyng (x, y) 2 R on each executon wth nput x In [5], Gat and Goldwasser ask whether there exsts a pseudo-determnstc algorthm that fnds a prmtve root mod p faster than the best known determnstc algorthm, whch runs n tme p 4 +o() We answer ths queston n the a rmatve: Theorem There exsts a pseudo-determnstc algorthm for Prmtve-Root that runs n expected tme L p (/2) = exp(o( p log )) ogrossma@mtedu Department of Mathematcs, MIT ISSN

2 We note that ths matches the tme bound for the best known Las Vegas algorthms for Prmtve-Root Ths problem may have cryptographc applcatons, as protocols based on the D e-hellman problem [4] rely on prmtve roots to establsh keys It may be desrable for two partes to ndependently generate the same key, or prmtve root, for F p In ths stuaton, pseudo-determnstc algorthms are helpful whle standard randomzed algorthms wll not su ce A closely related problem to Prmtve-Root s Prmtve-Root-Gven-Factorzaton Ths problem asks for a prmtve root mod p, gven both p and the factorzaton of p Prmtve-Root-Gven-Factorzaton may be relevant to applcatons snce the factorzaton of p s often known For example, protocols may requre e cent ways to verfy that an element s a prmtve root, n whch case the factorzaton of p wll be known For such applcatons, t s possble to e cently generate prmes p wth p n factored form [] Assumng the generalzed Remann Hypothess (GRH), Shoup proved n [7] that the smallest non-resdue mod p s of sze O(log 6 (p)), whch mples a brute force polynomal tme algorthm for Prmtve-Root-Gven-Factorzaton Wthout the GRH assumpton, the best determnstc algorthm remans the p 4 +o() algorthm from [2] In [5], polynomal tme pseudo-determnstc algorthms are presented for Prmtve-Root- Gven-Factorzaton when the nput prme satsfes p =kq, wth q prme and k = poly() We mprove upon ths result by fndng polynomal tme pseudo-determnstc algorthms for prmes satsfyng p = Q k = qe, where for some constant c each of the q s ether at most of sze log c (p) or at least of sze p /c (our dependence on c s exponental) It remans open to fnd a polynomal tme pseudo-determnstc algorthm for Prmtve-Root-Gven-Factorzaton for general prmes 2 Prelmnares In ths secton we establsh some lemmas we wll later use All lemmas n ths secton assume p s a prme, a, b 6 0 mod p, and ord refers to the order n F p (the multplcatve group of F p ) Lemma 2 Suppose a, b 2 F p If ord(a) and ord(b) are relatvely prme, then ord(ab) = ord(a)ord(b) Proof Frst, we note that (ab) ord(a)ord(b) = Therefore, ord(ab) ord(a)ord(b) Suppose ord(ab) < ord(a)ord(b) Let q be a prme dvdng ord(a)ord(b) ord(ab) We know that (ab) ord(a)ord(b)/q = However, q dvdes ether ord(a) or ord(b) Suppose wthout loss of generalty that q ord(a) Then =a ord(a)ord(b)/q b ord(b) (ord(a)/q) = a ord(a)ord(b)/q Therefore, ord(a) (ord(a)/q)ord(b) However, because ord(a) and ord(b) are relatvely prme, ths mples ord(a) (ord(a)/q), whch s mpossble Defnton 22 (qth resdue) Let q p be a prme We call an element a whch s a qth power (e, there exsts some b such that a = b q )aqth resdue Otherwse, we call a a qth non-resdue Lemma 23 Suppose q e s the largest power of q dvdng p dvsble by q e Then a qth non-resdue has order 2

3 Proof Suppose g s a prmtve root mod p An element a = g k satsfes ord(a) = p gcd(p,k) If a s a qth non-resdue, then we know k s not dvsble by q Therefore, q - gcd(p that ord(a) sdvsblebyq e, where q e s the largest power of q dvdng p,k) It follows The followng lemma wll show that to fnd a prmtve root modulo p, t s enough f for each prme q dvdng p wefndaq th non-resdue Lemma 24 Let p Then the product s a prmtve root = Q m = qe Suppose that for each, the element a s a q th non-resdue my a (p = )/qe )/qe = Proof We can wrte a = g k for some prmtve root g, and k not dvsble by q Then a (p g k (p )/q e must have order exactly q e, snce qe s the smallest number N such that Nk (p )/q e s dvsble by p, whch s the order of g Therefore, the element a (p )/qe has order exactly q e It follows that the orders of each of the a (p )/qe are relatvely prme, and so by Lemma 2,! my ord a (p )/qe my = The order of a (p s a prmtve root )/qe = = ord a (p )/qe s q e, so the product of the orders s Q m = qe = p Hence Q m = a(p )/qe Lemma 25 Gven p and q p, there exsts a pseudo-determnstc algorthm that fnds a qth non-resdue n tme q poly() Proof See Theorem 3 n [5] Lemma 26 Gven the factorzaton p ord(a) n poly() tme = Q m = qe and an element a 2 F p, we can compute Proof See page 329 n [8] The followng theorem from [3] gves a bound on smooth numbers (we say that n s m-smooth f all prme factors of n are at most m) Theorem 27 (Canfeld-Erdös-Pomerance) Let (x, y) denote the number of y-smooth postve ntegers bounded by x Let u = log x log y Suppose that u<( ) log x log log x for some > 0 Then holds unformly as u and x approach x u+o(u) (x, y) =u 3

4 3 Algorthm and Analyss In ths secton, we present and analyze our algorthm The dea for the algorthm s as follows Frst we factor p Now, for each prme factor q of p, we fnd a qth non-resdue We then use Lemma 24, to construct a prmtve root To fnd a qth non-resdue, we frst check f q s large or small (compared to exp( p log )) If q s small, we run the algorthm from Lemma 25 If q s large, we check the elements {, 2,,p } (n order) untl we fnd one whch s a qth non-resdue Lemma 3 guarantees that for large q, we wll encounter a qth non-resdue wthn the frst exp( p log ) elements: Lemma 3 For all su cently large p, for all q exp( p log ) dvdng p, there exsts a postve s apple exp( p log ) whch s a qth non-resdue Proof Our strategy wll be to assume Lemma 3 s false and then to wrte an nequalty comparng the number of exp( p log )-smooth numbers wth the number of qth resdues We wll then reduce ths nequalty to a contradcton We frst calculate (p, exp( p log )) We use the Canfeld-Erdös-Pomerance theorem p (Theorem 27), and see that u = p log = p log Therefore, p (p, exp(p log )) = p p log p +o p p p log log () For the sake of contradcton, assume that every element s apple exp( p log ) saqth resdue Snce the product of two elements whch are qth resdues s also a qth resdue, every exp( p log )-smooth number s a qth resdue We therefore know that (p, exp( p log )) s bounded above by the number of qth resdues, whch s p/q apple p/ exp( p log ) Combnng ths wth () yelds p (p/ exp(p log )) p p log p +o p p p log log Takng the log of both sdes gves p p p p log p + o p log log log Multplyng both sdes by And ths mples log apple p log p results n p log + p o p p log log log apple ( + o()) log 2 The above nequalty s clearly false, completng the proof p p log 4

5 Now that we have proven Lemma 3, we are ready to analyze the algorthm (Fgure ) Prmtve-Root(p) Factor p = Q m = qe 2 for each q : 3 f q > exp( p log ) 4 Compute the order of, 2,, untl an element a wth q e ord(a ) s found 5 f q apple exp( p log ) 6 Fnd a q th non-resdue a usng Lemma 25 7 return Q m = )/qe a(p Fgure : A pseudo-determnstc algorthm fndng a prmtve root modulo a gven prme p Correctness of the algorthm follows mmedately from Lemma 24 We wll now analyze the tme complexty of the algorthm: Lemma 32 The algorthm n Fgure runs n tme L p (/2) = exp(o( p log )) Proof By Lenstra and Pomerance s factorng algorthm [6], lne takes tme L p (/2) For each q > exp( p log ), by Lemma 3, n lne 4 we have to fnd the order of at most L p (/2) elements By Lemma 26, fndng the order each requres poly() tme, so lne 4 takes a total of L p (/2) poly() =L p (/2) tme For q apple exp( p log ), lne 6 takes at most exp( p log ) poly() =L p (/2) tme by Lemma 25 Snce there are at most prmes dvdng p, the loop n lne 2 takes a total of L p (/2) = L p (/2) tme Calculatng the product n lne 7 takes poly() tme Therefore, the algorthm as a whole termnates n expected tme L p (/2) We now show that the algorthm s pseudo-determnstc Note that the only randomzed steps of the algorthm are lne and lne 6 In lne, we use an algorthm that wth hgh probablty outputs the factorzaton of p, whch s always the same In lne 6, we use an algorthm whch s pseudo-determnstc by Lemma 25 Ths mples our man theorem: Theorem 33 There exsts a pseudo-determnstc algorthm for Prmtve-Root that runs n expected tme L p (/2) 4 Fndng a Prmtve Root Gven Factorzaton A related problem to Prmtve-Root s Prmtve-Root-Gven-Factorzaton: Defnton 4 The Prmtve-Root-Gven-Factorzaton problem s the problem of fndng a prmtve root mod p when both p and the factorzaton of p are gven as nput 5

6 For Prmtve-Root-Gven-Factorzaton, the best known Las-Vegas algorthm runs n polynomal tme The best prevously known pseudo-determnstc algorthm runs n tme p 4 +o() The algorthm from secton 3 mproves ths to L p (/2) In [5], Gat and Goldwasser pose as a problem to fnd a polynomal tme pseudo-determnstc algorthm for Prmtve-Root-Gven-Factorzaton The authors present a polynomal tme algorthm for the case p =kq, where q s prme and k s of sze poly() We mprove upon ths result wth a polynomal tme algorthm for all p where each prme factor of p s of sze ether at most log c (p) or at least p /c, for some constant c> Our algorthm runs n tme log c (p) poly() We descrbe our algorthm n Fgure 2 Prmtve-Root-Gven-Factorzaton(p, p for each q : 2 f q > exp( p log ) = Q m = qe ) 3 Compute the order of, 2,, untl an element a wth q e ord(a ) s found 4 f q apple exp( p log ) 5 Fnd a q th non-resdue a usng Lemma 25 6 return Q m = )/qe a(p Fgure 2: A pseudo-determnstc algorthm fndng a prmtve root modulo a prme p, gven both p and the factorzaton of p Correctness of the algorthm follows mmedately from Lemma 24 We now prove that f there s some constant c such that all q satsfy ether q < log c p or q >p /c, then the algorthm termnates n tme at most log c (p) poly() Frst, note that for large enough p, f q < log c p then q < exp( p log ) Also, f q >p /c then q > exp( p log ) To prove that lne 3 takes polynomal tme, we argue that for all fxed " > 0, for large enough p, f q >p /c then there exsts an a<log c+" (p) that s a q th non-resdue We do ths wth a smlar strategy to our proof of Lemma 3 We know that there are at most p q elements whch are q th resdues Suppose for the sake of contradcton that all a<log c+" (p) are q th resdues Ths mples that there are at least (p, log c+" (p)) elements whch are q th resdues Therefore, we have the nequalty q p q (p, log c+" (p)) By the Canfeld-Erdös-Pomerance theorem (Theorem 27), (p, log c+" (p)) = pu u+o(u), where u = Pluggng ths n and takng the log of both sdes yelds log log c+" p log + o log log c+" log (p) log log c+" (p) Smplfyng gves log q apple + o 6 log

7 But we know that q p /c Therefore, log q c c apple + o Further smplfyng now gves c apple (c + ") log + o Pluggng ths n and smplfyng yelds log log log However, the rght sde approaches c+", whereas the left sde s c Therefore, we have reached a contradcton, and so wthn the frst log c+" (p) elements that we test n lne 3, we wll encounter a q th non-resdue Therefore, lne 3 of the algorthm requres calculatng the order of up to log c+" (p) elements, each of whch takes poly() tme by Lemma 26 Lne 5 takes up to log c (p) poly() tme by Lemma 25 Snce there are at most prmes dvdng p, the loop n lne s of length up to It follows that our algorthm termnates and outputs a prmtve root n expected tme log c (p) poly() Note that on every executon of the algorthm, we output the same prmtve root, snce the only randomzed step of the algorthm s lne 5 whch s pseudo-determnstc by Lemma 25 Ths completes the proof of the followng theorem: Theorem 42 For any constant c>, there exsts a pseudo-determnstc algorthm for Prmtve- Root-Gven-Factorzaton that runs n polynomal tme for all p where each prme factor q of p satsfes ether q<log c (p) or q>p /c 5 Dscusson It would be nterestng to fnd a polynomal tme pseudo-determnstc algorthm for Prmtve- Root-Gven-Factorzaton for general prmes The slowest step n Las Vegas algorthms for Prmtve-Root s factorng p It would be nterestng to fnd an algorthm whch can verfy an element s a prmtve root wthout usng the factorzaton of p Acknowledgments I would lke to thank Shaf Goldwasser for ntroducng me to the prmtve root problem, for helpful dscussons, and for advce and encouragement on the paper I would also lke to thank Andrew Sutherland for helpful dscussons References [] Erc Bach How to generate factored random numbers SIAM Journal on Computng, 7(2):79 93, 988 [2] DA Burgess On character sums and prmtve roots Proceedngs of the London Mathematcal Socety, 3():79 92, 962 7

8 [3] E Rodney Canfeld, Paul Erdös, and Carl Pomerance On a problem of oppenhem concernng factorsato numerorum Journal of Number Theory, 7(): 28, 983 [4] Whtfeld D e and Martn E Hellman New drectons n cryptography Informaton Theory, IEEE Transactons on, 22(6): , 976 [5] Eran Gat and Shaf Goldwasser Probablstc search algorthms wth unque answers and ther cryptographc applcatons In Electronc Colloquum on Computatonal Complexty (ECCC), volume 8, page 36, 20 [6] Hendrk W Lenstra and Carl Pomerance A rgorous tme bound for factorng ntegers Journal of the Amercan Mathematcal Socety, 5(3):483 56, 992 [7] Vctor Shoup Searchng for prmtve roots n fnte felds Mathematcs of Computaton, 58(97): , 992 [8] Vctor Shoup A computatonal ntroducton to number theory and algebra Cambrdge unversty press, ECCC ISSN