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def short_pad_attack(c1, c2, e, n): PRxy.<x,y> = PolynomialRing(Zmod(n)) PRx.<xn> = PolynomialRing(Zmod(n)) PRZZ.<xz,yz> = PolynomialRing(Zmod(n)) g1 = x^e - c1 g2 = (5*x+y)^e - c2 q1 = g1.change_ring(PRZZ) q2 = g2.change_ring(PRZZ) h = q2.resultant(q1) h = h.univariate_polynomial() h = h.change_ring(PRx).subs(y=xn) h = h.monic() kbits = n.nbits()//(2*e*e) diff = h.small_roots(X=2^kbits, beta=0.4)[0] return diff def related_message_attack(c1, c2, diff, e, n): PRx.<x> = PolynomialRing(Zmod(n)) g1 = x^e - c1 g2 = (5*x+diff)^e - c2 def gcd(g1, g2): while g2: g1, g2 = g2, g1 % g2 return g1.monic() return -gcd(g1, g2)[0] if __name__ == '__main__': n = 20361372240024088786698455948788052559208001789410016096382703853157107986024860262721685000417719260611935731634077852127432140361792767202581631816544546972750034494061276779878409544779707914261679633764772575040304712361634318086289783951555842021028438799649252652041211341825451500751760872572402250747982495384263677669526575825183733353800694161425360299521143726681387485097281832219009682768523304737252763907939642212542959846630464628135025203489075698699980715986689341069964387779523254203021424865355054215122316160201073604105317768112281914334065349420946717116563634883368316247495042216330408372176714499012778410160478384503335610321108263706243329745785632599707740534386988945259578897614317582546751658480917188464178997026284336861027299289073045677754342746386408505695243800685323283852020325044649604548575089927541935884800327121875191739922436199496098842684301207745090701158839031935190703347091 e = 3 c1 = 3639847731266473012111996909765465259684540134584180368372338570948892196816095838781423020996407457408188225238520927483809091079993151555076781372882518810174687150067903870448436299501557380508793238254471833275507634732947964907461619182112787911133054275872120243558556697900528427679352181961312958660881800731678134481664074711076672290178389996403357076809805422591851145306425951725627843352207233693810474618882394140691334742086008967260117740486955640068190440609984095657695423536016475468229419187489359563800737261212975921663803729112420222039005478830477455592167092520074509241894829304209406713781082959299623674294927249556083486223036858674077173104518013601628447504500606447821540687465361616447631579976579754996021653630804073535352129315413118764836270751250405649683786487251823247828947202336680538849571498780353357272103697510910576879383751704763858882439578045020243015928994208017750848637513 c2 = 7030777127779173206633582847346001157991477456002191926122836599155148909465054067800807615361108442560942058865403188672629297039703065927801771646334817871335134889139894648729527452541098449842202838983982508551750669662540615534327150829869964429006130891731472099912937717406120443380283548571270317421722042835639732966975812764084015221255115940508456442279902250677665136380988902682370875602145833135937210740790528756301051981994351553247852018355526641012434670664732924491790949235519600899289515495046353559475806935200029321563549553167235419039924276406059858659476329718809657072997385947262654743181242885709558209249589482036673428723035300722280229192727192487772217518673838209646300548275957450994828221329299666216457961746189885356929698674294944243729739850927111231235060005119781652245234537583181232715964191675241206562888107252569566488402724441835466680342239244581162530424964324562530832713397 diff = short_pad_attack(c1, c2, e, n) print("difference of two messages is %d" % diff) p = related_message_attack(c1, c2, diff, e, n) print("p:", p) print("q:", 5*p + diff)
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