Algebraic properties of polynomial iterates

Transcription

1 Algebrac propertes of polynomal terates Alna Ostafe Department of Computng Macquare Unversty

2 1 Motvaton 1. Better and cryptographcally stronger pseudorandom number generators (PRNG) as lnear constructons has been succefully attacked. We recall that an attack on a PRNG s an algorthm that observes several outputs of a PRNG and then able to contnue to generate the same sequence wth a nontrval probablty. 2. Possble new hash functons 3. Lnks wth tradtonal questons n the theory of algebrac dynamcal systems such as degree growth, perods, etc.

3 2 General settng p: prme number IF p : fnte feld of p elements. F = {F 1,..., F m }: system of m polynomals/ratonal functons n m varables over IF p How do we terate? Defne the k-th teraton of the ratonal functons F by the recurrence relaton F (0) = X, F (k) = F (F (k 1) 1,..., F (k 1) m ), where = 1,..., m and k = 1, 2,....

4 3 Our motvaton: Polynomal Pseudorandom Number Generators (PRNG) Consder the PRNG defned by a recurrence relaton n IF p un+1 = F(u n ), where u n = (u n,1,..., u n,m ) and F = (F 1 (X 1,..., X m ),..., F m (X 1,..., X m )). In partcular, for any n, k 0 and = 1,..., m we have u n+k = F (k) (u n ). IF p = fnte feld = sequence of vectors (w n ) s eventually perodc wth some perod τ p m.

5 4 Qualty of PRNG Unformty of dstrbuton of PRNG Estmatng exponental sums In our case: where S a (N) = N 1 n=0 e m =1 a u n,, e(z) = exp(2πz/p), a = (a 1,..., a m ) Z m. Informally: Trval bound S a (N) N (as t s a sum on N roots of unty). Motvaton: Good pseudorandom sequences should have small values of S a (N)!!! The smaller, the better!

6 5 How do we estmate S a (N)? Standard technque of estmatng S a (N) reduces t to estmatng the exponental sum v IF m p e m =1 a (F (k) (v) F (l) We can now try to apply Wel 1948: p x 1,...,x m =1 (v)) e (F (x 1,..., x m )) < Dpm 1/2, where F IF p [X 1,..., X m ] s a nonconstant polynomal of degree D.. Remarks: The degree plays an mportant role n the estmate of the exponental sum! We need lnear ndependence of the teratons F (k), k 1. In some specal cases elementary methods replace the use of the Wel bound and gve stronger bounds!

7 6 Degree growth If f s a unvarate polynomal of degree d, the degree growth deg f (k) = d k s fully controlled and exponental (f d 2). Queston: How about the multvarate case? Clearly the growth cannot be faster than exponental, but can t be slower? YES!!... and for us, generally speakng, the slower the better.

8 7 What systems do we study? Ostafe and Shparlnsk : F = {F 1,..., F m }: system of m ratonal functons n m varables over IF p of the form: F 1 (X 1,..., X m ) = X e 1 1 G 1(X 2,..., X m ) + H 1 (X 2,..., X m ),... F m 1 (X 1,..., X m ) = X e m 1 m 1 G m 1(X m ) + H m 1 (X m ), F m (X 1,..., X m ) = g m X e m m + h m, where e { 1, 1}, G, H IF p [X +1,..., X m ], g m, h m IF p. g e = 1 : G has a unque leadng monomal, G (X +1,..., X m ) = g X s, Xs,m m + G (X +1,..., X m ), g 0, deg Xj G < s,j, deg Xj H s,j, 1 < j m e = 1 : H has a unque leadng monomal H (X +1,..., X m ) = h X s, Xs,m m + H (X +1,..., X m ), wth h 0, deg Xj H < s,j, deg Xj G 2s,j, 1 < j m

9 8 Lemma 1 For the above systems, f e = 1, F (k) = X G,k (X +1,..., X m ) + H,k (X +1,..., X m ), and f e = 1, F (k) = X R,k + S,k X R,k 1 + S,k 1, = 1,..., m, k = 0, 1,..., where G,k, H,k, R,k, S,k IF(X +1,..., X m ), = 1,..., m 1. In both cases, deg F (k) = 1 (m )! km s,+1... s m 1,m + ψ (k), ψ (T ) Q[T ], deg ψ < m, = 1,..., m 1. Concluson: The degree grows polynomally n the number of teratons monotoncally (beyond a certan pont). Remark: The above effect does not occur n the unvarate case.

10 9 Why do we wn? we estmate the exponental sum for e = 1 for all = 1,..., m, v IF m p e m 1 =1 a (F (k) (v) F (l) (v)) where, as before, e(z) = exp(2πz/p) and F (k) F (l) = X (G,k G,l ) + H,k H,l. the case of ratonal functons wll follow more or less the same... but we have to take care of the poles!!! slow degree growth of the polynomals Remark: Slow, but not too slow!!!... so that G,k G,l s nontrval for k l the lnearty of the polynomals F n X Remark: We do not use the Wel bound. Instead we evaluate exponental sume wth lnear functons and estmate the number of zeros of G,k G,l, k l: any nontrval m-varate polynomal of degree D has at most Dp m 1 zeros, and thus we save p nstead of p 1/2 from the Wel bound.,

11 10 Wsh lst All results at some pont are based on an estmate of exponental sums wth L a,k,l (v) = m 1 =1 ( a F (k) (v) F (l) ) (v) We want L a,k,l (v) to be nontrval (.e. not to be a constant) exponental sums frendly: () of small degree (to apply the Wel bound) () lnear n some of the varables () to have a low dmensonal locus of sngularty (to apply the Delgne bound, or Delgnelke bounds by Katz, etc.) Propertes () and () have been exploted n the above constructons, a dream project s to fnd a system wth well-controlled property () and mprove prevous results.

12 11 Maxmal perods Ostafe 2010: Let (u n ) be defned as above wth perod τ p m. If e = 1 for all = 1,..., m, then τ = p m f and only f v IF m p where G (v) = 1, g m = 1, deg R = (m )(p 1), R = H G (2)... G (pm ) (mod I) s of degree at most p 1 n each varable and I s the deal generated by X p 1 X 1,..., X p m X m. Example: m = 2, p 3 (mod 4), F 1 = X 1 (f(x 2 ) 2 +(p+1)/4)+a and F 2 = X 2 +b, a, b IF p and f IF p [X] generates a permutaton. Queston: When do the systems of ratonal functons defned above acheve maxmal perod? Ostafe and Shparlnsk 2011 (n progress): When all e = 1, the maxmal perod reduces to descrbng the maxmal perod of a certan Möbus transformaton defned by the teratons of F.

13 12 Other algebrac propertes Can we fnd nontrval lower bounds for the number of monomals of the polynomals F (k), for any = 1,..., m, k = 1, 2,... over fnte felds? Fuchs and Zanner 2009: The number of monomals of the kth teratons of a unvarate ratonal functon over C tends to nfnty wth k. Remark: Ths s not necessarly the case of multvarate polynomals... Example: Let m = 4, F 1 = X 1 X 4 (X 3 X 1 + X 4 X 2 ), F 2 = X 2 +X 3 (X 3 X 1 +X 4 X 2 ), F 3 = X 3, F 4 = X 4. Then, for any k 1, F (k) 1 = X 1 kx 4 (X 3 X 1 + X 4 X 2 ) F (k) 2 = X 2 + kx 3 (X 3 X 1 + X 4 X 2 ). So, the degree growth s constant, as the number of monomals at every teraton (over any feld)!!!

14 13 There are stuatons (wth a bt of drty trcks wth the characterstc) when the degree grows exponentally, the number of monomals s constant, or even decreases to a monomal... at least over fnte felds. Example: Let m = 2, F 1 = X p 1 Xp 2, F 2 = X p 1 + Xp 2 over IF q, q = p m. Then F (k) = c,k X pk j, k = even, j k k {1, 2}, d,k (X pk 1 ± Xpk 2 ), k = odd where c,k, d,k IF q, = 1, 2. Queston: Can we acheve ths wthout cheatng?

15 14 The addtve complexty of the polynomals F (k), for any = 1,..., m, k = 1, 2,..., s the smallest number of + sgns n the formulas evaluatng these polynomals. Note that the addtve complexty can be much smaller than the number of monomals. Example: f(x, Y ) = (X 2 +2Y ) 100 (3X+Y 3 ) 200 +(X 300 +Y ) 10 s of total degree 3000 and thus has a very long representaton va the lst of coeffcents. However, the addtve complexty s 4 and thus has a very concse representaton/evaluaton. Construct polynomal systems wth exponentally degree growth, but wth polynomal addtve complexty growth of ther teratons. It s possble... Example: f(x) = (x b) d + b, f (k) (X) = (X b) dk + b, b IF q, d 2 If we have a Gröbner bass (GB) for {F 1,..., F m }, what can we say about the GB of {F (k) 1,..., F m (k) }? Can we say somethng about the dmenson of the deal generated by F (k) 1,..., F m (k)?

16 15 Let s talk about rreducblty!!! At some pont of the argument we need to estmate the number of zeros of some lnear combnatons of several dstnct terates F (k) (X) of F 1,..., F m. It s natural to start wth studyng the same terates. E.g.: can we say anythng nterestng about the polynomals F (k) (X)? Are they rreducble for any k? NO RESULTS! Let s start wth the unvarate case... stll no general results yet... except Gomez, Ncolas, Ostafe and Sadornl 2011, work n progress. Let s start wth quadratc polynomals... not too many results but there are some!!

17 16 Irreducblty of teratons For a feld IF, f IF[X] s called stable f f (k) rreducble for all k. Irreducblty s very common over Q. = over IF = Q a random polynomal s expected to be stable. Ahmad, Luca, Ostafe and Shparlnsk 2010: proved ths for quadratc polynomals. Over IF q rreducblty s rare: prob. 1/d for a random polynomal of degree d. = We expect very few stable polynomals (recall that deg f (k) grows fast). Gomez Perez and Pñera Ncolas 2010: estmate on the number of stable quadratc polynomals for odd q. Ahmad, Luca, Ostafe and Shparlnsk 2010: No stable quadratc polynomal over IF 2 n; t has nothng to do wth char = 2 as x 2 + t s stable over IF 2 (t).

18 17 Stablty testng of quadratc polynomals over IF q f(x) = ax 2 + bx + c IF q [X], a 0 γ = b/2a the unque crtcal pont of f Crtcal orbt of f: Orb(f) = {f (n) (γ) : n = 2, 3,...} t such that f (t) (γ) = f (s) (γ) for some postve nteger s < t. t f =the smallest value of t wth the above condton. Then: Orb(f) = {f (n) (γ) : n = 2,..., t f } Jones and Boston 2009: f IF q [X] s stable f and only f the adjusted crtcal orbt contans no squares. Orb(f) = { f(γ)} Orb(f)

19 18 Trvally #Orb(f) q and thus one can test f IF q [X] for stablty n q steps. Ostafe and Shparlnsk 2009: Theorem 2 For any odd q and any stable quadratc polynomal f IF q [X] we have t f = O ( q 3/4). Corollary 3 For any odd q, a quadratc polynomal f IF q [X] can be tested for stablty n tme q 3/4+o(1).

20 19 Stablty of arbtrary unvarate polynomals over IF q Ahmad, Luca, Ostafe and Shparlnsk 2010: even degree polynomals of the form F = g(f) where f s a quadratc polynomal, and g s any polynomal of degree d Theorem 4 For any odd q and any stable polynomal F = g(f) IF q [X], where f = ax 2 + bx + c IF q [X] and g IF q [X] of degree d, we have t F = O ( q 1 α d), where α d = log 2 2 log(4d). We can say a bt more...

21 20 Gomez, Ncolas, Ostafe and Sadornl n progress): 2011 (work f IF q [X] stable wth leadng coeffcent a IF q, q odd, deg f 2, (deg f, p) = 1 f nonconstant, γ roots of f, = 1,..., deg f 1 Then 1. f d = deg f s even, {a deg f 1 =1 f (n) (γ ) n > 1 } { ( 1) d 2a contans only non squares n IF q ; d 1 =1 f(γ ) } 2. f d = deg f s odd, { ( 1) (d 1) 2 d d 1 =1 f (n) (γ ) n 1 } contans only squares n IF q.

22 21 Questons: Is there an algorthm to check a quadratc polynomal for stablty n Q[X] n fntely many steps? Do there exst stable polynomals of degree kp, k 1, over a feld of characterstc p? Can we say somethng about the stablty of multvarate polynomals? Zaharescu 2005: some results about the rreducblty of terates of multvarate polynomals, but not n the usual way, only wth respect to one varable.