On asymptotic behavior of composite integers n = pq Yasufumi Hashimoto

Transcription

1 Journal of Mah-for-Indusry Vol1009A On asymoic behavior of comosie inegers n = q Yasufumi Hashimoo Received on March Absrac In his aer we sudy he asymoic behavior of he number of comosie inegers wrien by roducs of wo rimes Such inegers are someimes called by he RSA inegers because hese are used in he RSA cryosysems The number of all such inegers has been already sudied by Landau Sahe Selberg ec Furhermore he number of inegers wih n = q and < q < c for a fied c > 1 was recenly sudied by Decker and Moree The aim of his aer is o eend Decker-Moree s resul and he main heorem describes he asymoic formula of he number of inegers wih < q < f for a fied increasing funcion f Keywords comosie ineger n = q rime number heorem RSA cryosysem I is well known ha 1 Inroducion #{: rime < } log as The asymoic formula above is called by he rime number heorem For he number of comosie inegers wih r 1 disinc rime facors Landau [10] roved he following asymoic formula see also [4] #{n = 1 r < 1 r : rimes} log log r1 r 1! log as Noe ha his asymoic formula has been imroved by Sahe [1] Selberg [13] Hensley [5] and Hildebrand- Tenenbaum [6] In his aer we sudy he disribuion of comosie inegers n = q wih wo rimes q Such inegers are someimes called by he RSA inegers because hese are used in he RSA cryosysem [11] whose securiy is based on he difficuly o facor n In general i is no easy o facor huge comosie inegers feasibly wihou quanum comuers see [15] However i is known ha he RSA is weak when q saisfy some secial condiions One of such condiions is for he difference beween and q In fac he comuaional ask of he Ferma facoring algorihm deends on he difference q see eg [8] Also Weger [16] found ha he secre key in he RSA cryosysem should be larger as he difference q is smaller Conversely when one of q is much larger han he oher RSA wih such q is called by he unbalanced RSA see [14] Boneh-Durfee [1] oined ou ha he secre key should be large enough In his sense i is imoran o sudy he number of comosie inegers n = q saisfying some condiions of and q for he racical use of he rime numbers In fac he following asymoic formula has been eerimenally known among he cryologiss 1 #{n = q < q < c n < } α log as where α > 0 is a consan deending on c > 1 Recenly Decker and Moree [] roved 1 ure mahemaically and found ha α = log c In he resen aer we obain he following resul as an eension of he work in [] Theorem 1 Le f g : R >1 R >1 be increasing funcions such ha f > and gfg = #{n = q q: rimes < q < f n < } log log log log g 1 f M for M > 0 log l log f cl for l > 1 c > 0 log c f c for c > 1 log c 3 δ δ1 1 δ log f cδ for c > 0 1/ < δ < 1 and RH holds as where RH is he Riemann hyohesis In his aer we avoid he case where f = o 1/ because he esimaion of he error erms wrien by A and B in he roof of Theorem 1 is difficul However he disribuion of n for such f is imoran in he analyic number heory In fac i relaes o he roblem o coun rime numbers in shor inervals Esecially when 45

2 46 Journal of Mahemaics for Indusry Vol1009A-6 f = consan our roblem is almos same o he famous rime-air roblem sudied by Hardy and Lilewood [3] Alhough here are eerimenal resuls see also [9] for he recen research and conjecures for he rimeair roblem i sill remains as an unsolved roblem a he resen ime Le I is known ha Proof of Theorem 1 π :=#{: rime < } = < 1 ψ := log < π =li R 1 ψ = R where li := log 1 d / log as and he reminder erms R 1 and R are as follows { Oe clog 1/ uncondiionally R 1 R = O 1/ϵ if RH holds Noe ha here are sharer esimaes of he reminder erms for he uncondiional case see eg [7] However we do no use hem in his aer Pu π f := 1 ψ f := log q q: rime <q<f q< q: rime <q<f q< We now esimae ψ f o rove Theorem 1 We see ha ψ f = <g log πf <g ψf log π g < 1/ log π ψ < 1/ < 1/ g < 1/ ψ Divide he above by ψ f = A B where A := <g log lif <g log li g < 1/ log li < 1/ < 1/ f g < 1/ B := <g log R 1 f <g R f log R 1 g < 1/ log R 1 R < 1/ < 1/ g < 1/ R We furhermore divide A by A = A 1 A where A 1 := A := g 1/ g 1/ g 1/ = g 1/ g 1/ g 1/ lifd li d g 1/ g 1/ lid lifdr li dr f log d log d log d g 1/ g fdr 1 dr 1 1/ lidr dr 1 f R log f R 1 d R log log R 1 d R log R 1 d We now sar esimaing A 1 A and B 1 Esimae of A 1 I is easy o see ha 1/ 1/ g 1/ g [ ] 1/ li d = li log log g = 1/ li 1/ log log 1/ gli log log g log d =[ log log ] 1/ g g = log log 1/ log log g li d = [ li ] 1/ = 1/ li 1/ log g A 1 = lif f log d log log log g 1 gli g =:A 11 A 1 A 13

3 Yasufumi Hashimoo 47 We esimae A 11 A 1 and A 13 in he above wih he condiions of he growh of f 11 The case of f M for any M > 0 For A 11 and A 13 we see ha g A 11 = lif f d gfg = log A 13 =gli g = g g Since g 1/M1 for f M we have log A 1 = log log g 1 log M for any M > 0 Thus we obain log A 1 A 1 log log g 1 as 1 The case f c l for l > 1 and c > 0 Firs consider he case of f = c l o l for c > 0 and g = /c 1/l1 o 1/l1 In his case we see ha g f A 11 = lif d log g l log d = O log log A 1 = log log g 1 l 1 log = log log log c o1 1 = log l o A 13 =gli = O g log A 1 A 1 log l as 13 The case of f c for c > 1 In his case g = /c 1/ o 1/ g A 11 = lif f log log A 1 = log log g 1 = log 1 d = log c o1 log log c o1 = log c log o log A 13 =gli = g log o log Thus we ge A 1 log c log log o log as 14 The case of f c δ for c > 0 and 0 < δ < 1 In his case we see ha g = 1/ c/ δ/ o δ/ Now we esimae each erm as follows log A 1 = log log g 1 = log 1 log 1 c δ1 o δ1 1 log o1 δ1 c o = log 1 δ1 We also have Since = g g =c δ1 g we obain log o log =[li c δ ] g log o1 δ1 lif f d log li c δ cδ log lio δ oδ log g cδ δ1 d o log log d d =gli g cg δ c1 δ g cδ δ log O δ1 g δ δ1 O log d o log =gli fg og δ c 1 δ 1 δ δ1 δ1 log o log δ log d gli fg og δ δ1 A 13 = o log A 1 c 3 δ δ1/ 1 δ log Esimae of A 1 When RH is no assumed as Since f is increasing f akes osiive values for > 1 And we see ha R 1 R /log l for any l 1

4 48 Journal of Mahemaics for Indusry Vol1009A-6 g 1/ d f A = O log L O 1/ O g log L d log log g f log L d O log log g log L This means ha o f M for any M > 0 A = o f c l for l 1 c > 0 log When RH holds If he Riemann hyohesis is rue i holds ha R 1 R 1/ϵ for any ϵ > 0 Consider he case of f I is no difficul o see ha A = g 1/ 3/4ϵ 3 Esimae of B 1/ O 1/ϵ d O 3/ϵ d g O 1/ϵ d 31 When RH is no assumed Since R 1 R /log L for any L > 0 we have B = <g log f O log f L log O log log L g < 1/ log L1 < 1/ fg log fg L log log L <g g < 1/ O log log L1 < 1/ log /g L O log L1 log L1 d 3 When RH holds If he Riemann hyohesis is rue i holds ha R 1 R 1/ϵ for any ϵ > 0 Consider he case where f B = <g O log f 1/ϵ g < 1/ O < 1/ O 3/4ϵ 4 Concluding he roof log 1/ϵ 1/ϵ log 1/ϵ Combining he resuls in Secion 1 and 3 we have ψ f log log log g 1 f M for M > 0 log l f c l for l > 1 and c > 0 log c f c for c > 1 log c 3 δ δ1 1 δ log f cδ for c > 0 and 1/ < δ < 1 and RH holds as Since π f = dψ f log he desired resul follows immediaely References [1] D Boneh and G Durfee Cryanalysis of RSA wih rivae key d less han N 09 Eurocry 99 Lecure Noes in Comuer Science Sringer [] A Decker and P Moree Couning RSA-inegers arxivmah/ [3] GH Hardy and E Lilewood Some roblems of ariio numerorum III: On he eression of a number as a sum of rimes Aca Mah [4] GH Hardy and EM Wrigh An inroducion o he heory of numbers Fifh ediion Oford Universiy Press 1979 [5] D Hensley The disribuion of round numbers Proc London Mah Soc

5 Yasufumi Hashimoo 49 [6] A Hildebrand and G Tenenbaum On he number of rime facors of an ineger Duke Mah J [7] A Ivìc The Riemann zea-funcion The heory of he Riemann zea-funcion wih alicaions A Wiley- Inerscience Publicaion New York 1985 [8] S Kazenbeisser Recen Advances in RSA Cryograhy Advances in Informaion Securiy 3 Kluwer Academic Publishers 001 [9] J Korevaar and H Riele Average rime-air couning formula arxivmah/ [10] E Landau Handbuch der Lehre von der Bereilung der Primzahlen Vol 1 Leizig 1909 [11] RL Rives A Shamir and L Adleman A mehod for obaining digial signaures and ublic-key cryosysems Comm ACM [1] LG Sahe On a roblem of Hardy and Ramanujan on he disribuion of inegers having a given number of rime facors J Indian Mah Soc ; [13] A Selberg Noe on a aer by L G Sahe J Indian Mah Soc [14] A Shamir RSA for aranoids RSA Laboraories CryoByes 1 no [15] PW Shor Polynomial-ime algorihms for rime facorizaion and discree logarihms on a quanum comuer SIAM J Comu [16] B de Weger Cryanalysis of RSA wih small rime difference Al Algebra Eng Commun Comu Yasufumi Hashimoo Insiue of Sysems Informaion Technologies and Nanoechnologies 7F -1- Momochihama Fukuoka JAPAN hasimooaisiorj