Exponential sums and the distribution of inversive congruential pseudorandom numbers with prime-power modulus

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1 ACTA ARITHMETICA XCII1 (2000) Exponential sus and the distribution of inversive congruential pseudorando nubers with prie-power odulus by Harald Niederreiter (Vienna) and Igor E Shparlinski (Sydney) 1 Introduction Let p 3 be a prie and 1 an integer We write U = (Z/p Z) for the group of reduced residue classes odulo p, where we drop the dependence on p in the notation for siplicity (we ay think of p as a fixed prie) Then U = (p 1)p 1 It will often be convenient to identify eleents of Z/p Z with the corresponding eleents of the least residue syste odulo p For given a, b Z/p Z we consider a ap ψ : U Z/p Z of the for (1) ψ(w) = aw 1 + b for w U It is easy to see that ψ is a perutation of U if and only if gcd(a, p) = 1 and b 0 (od p) These conditions will be assued fro now on If we start fro an initial value u 0 U, then the recurrence relation (2) u n+1 = ψ(u n ) for n = 0, 1, generates a sequence u 0, u 1, of eleents of U It is obvious that this sequence is purely periodic with least period length τ (p 1)p 1 Detailed studies of the possible values of τ can be found in [1] and [4] If u 0, u 1, is a sequence generated by (1) and (2), then it is of interest for the application entioned below to establish upper bounds for the onential sus (3) χ(u n ), where χ is a nontrivial additive character of Z/p Z and 1 N τ In the case = 1, and with a slight change of forula (1) to arrive at ore 1991 Matheatics Subject Classification: 11K38, 11K45, 11L07, 11T23, 65C10 [89]

2 90 H Niederreiter and I E Shparlinski interesting perutations ψ of U 1, a nontrivial upper bound for the corresponding onential sus was first proved in [10] (see also [12]) In the present paper we treat the case 2 in which the details of the ethod are quite different The onential sus (3) are relevant in the analysis of a well-known faily of pseudorando nubers If u 0, u 1, is a sequence of eleents of U as above, then the nubers u 0 /p, u 1 /p, in the interval [0, 1) for a sequence of inversive congruential pseudorando nubers with odulus p For p 3 and 2, the case we are concerned with here, this ethod of pseudorando nuber generation was introduced in [4] In practice, one works with a large power p of a sall prie p For surveys of results on inversive congruential pseudorando nubers we refer to [2], [8, Chapter 8], [9] It is clear that upper bounds on the onential sus (3) yield results on the distribution of the inversive congruential pseudorando nubers u 0 /p, u 1 /p, A quantitative version of such a result in the for of a discrepancy bound will be given in Section 4 This is the first nontrivial discrepancy bound for parts of the period of inversive congruential pseudorando nubers with prie-power odulus An analogous result for prie oduli was first established in [10] Related results on the distribution in parts of the period for pseudorando nubers generated by nonlinear ethods can be found in [5], [6], [11], [12] 2 Auxiliary results If ψ is the perutation of U, 1, given by (1) and r is an arbitrary integer, then let ψ r denote the rth power of ψ in the group of perutations of U We have the licit forula in Lea 1 below Here and in the following, it will often be convenient to write u/v for an ression uv 1 in a ultiplicative abelian group Lea 1 For any integer r 0 there exist c r, e r Z/p Z such that ψ r (w) = (bc r e r )w + ac r c r w e r for all w U Moreover, for even r we have c r 0 (od p) and e r 0 (od p) and for odd r we have c r 0 (od p) and e r 0 (od p) P r o o f For r = 0 we can take c 0 = 0 and e 0 = 1 The general case follows by straightforward induction on r and the additional properties of c r and e r are obtained along the way If u 0, u 1, is a sequence generated by (1) and (2), then for 1 k we let τ k be the least period length of the sequence u 0, u 1, considered odulo p k (so that τ = τ )

3 Exponential sus and pseudorando nubers 91 Lea 2 If c r 0 (od p k ) for soe r 1 and 1 k, then τ k divides r P r o o f Fro c r 0 (od p k ) it follows by Lea 1 that e r 0 (od p) and hence ψ r (w) w (od p k ) for all w U Then r is a period length of the sequence u 0, u 1, considered odulo p k, and so τ k divides r Lea 3 Let p 3 be a prie, let be a positive integer, and let f and g be arbitrary integers Put gcd(f, p ) = p l Then p 1 ( 2πi(fz 2 ) + gz) = 0 if g 0 (od p l ) and p 1 p 2πi(fz 2 + gz) = p (+l)/2 if g 0 (od p l ) p P r o o f This follows fro Lea 6 in [3] For 1 r τ 1 and a nontrivial additive character χ of Z/p Z we introduce the onential su (4) σ r = χ(ψ r (w) w) Note that χ is deterined by an integer h 0 (od p ), in the sense that 2πihv (5) χ(v) = for all v Z/p Z p Put gcd(h, p ) = p d with 0 d <, so that we can write h = p d h 0 with an integer h 0 0 (od p) By Lea 1 we have σ r = ( cr (a + bw w 2 ) ) c r w e r χ Let gcd(c r, p ) = p k with k 0, then Lea 2 shows that k < Thus, we can write c r = p k c with an integer c 0 (od p) Then (6) σ r = ( 2πip d+k p ch 0(a + bw w 2 ) ) p k cw e r It is trivial that (7) σ r = U = (p 1)p 1 if d + k For d + k < we obtain the following bound Lea 4 With the notation above we have σ r 2p (+d+k)/2 if d + k <

4 92 H Niederreiter and I E Shparlinski P r o o f In (6) we put w = sp d k +t with 0 s < p d+k and t U d k Then (8) σ r = p d+k 2πich0 a + bt t2 p d k p k ct e r t U d k If k = 0, then t ct e r is a perutation of U d by Lea 1, hence carrying out this substitution in the su above yields σ r = p d 2πich0 p d ((a + bc 1 e r c 2 e 2 r)v 1 c 2 v) v U d The last onential su is always bounded by 2p ( d)/2, naely by a result in [13, p 97] for d 2 and by the Weil bound for Kloosteran sus (see [7, Theore 545]) for d = 1 Therefore the result of the lea follows for k = 0 Next we consider the case k d k Then fro (8) we get σ r = p d+k 2πich0 p d k t2 bt e r t U d k Furtherore, 2πich0 p d k t2 bt e r t U d k p d k 1 = 2πich0 p d k z2 p bz d k 1 1 e r u U d 2k 2πich0 p d k 1 pz2 bz e r Now Lea 3 applied to the last two sus shows that the first su has absolute value p ( d k)/2 and the second su has absolute value at ost p ( d k)/2, and so the lea is again established Finally, we consider the case 1 k < d k In (8) we put t = zp d 2k + u, 0 z < p k, u U d 2k Then p d k 2πich0 a + bu u2 σ r = p d k p k cu e r = p k 1 ( 2πich0 u U d 2k p k 1 p d k (b 2u)p d 2k z p 2 2d 4k z 2 ) p k cu e r 2πich0 a + bu u2 p d k p k cu e r ( 2πich0 p k p d 2k z 2 ) + (2u b)z e r

5 Exponential sus and pseudorando nubers 93 By Lea 3, each inner su is 0 since d 2k > 0 and 2u b 2u 0 (od p) for all u U d 2k Thus, we have σ r = 0 3 The bound for onential sus For a sequence u 0, u 1, generated by (1) and (2) with least period length τ and for integers h and N with 1 N τ we consider the onential su S N (h) = ( 2πihun Theore 1 Let p 3 be a prie, let 2 be an integer, and let h be an integer with gcd(h, p ) = p d, 0 d < Then S N (h) < 49 p N 1/2 p (+d)/4 for 1 N τ 16 τ p P r o o f With the notation in (5) we can write S N (h) = χ(u n ) Note that u n = ψ n (u 0 ) for all integers n 0, and we use this identity to define u n for all negative integers n It is easy to see that for any integer k we have (9) S N (h) χ(u n+k ) 2 k For an integer K 1 put { {k Z : (K 1)/2 k (K 1)/2} if K is odd, R(K) = {k Z : K/2 + 1 k K/2} if K is even Then k R(K) k K 2 /4 If we use (9) for all k R(K), then we get (10) K S N (h) W + K 2 /2 with W = = k R(K) k R(K) χ(u n+k ) χ(ψ k (u n )) ) k R(K) χ(u n+k )

6 94 H Niederreiter and I E Shparlinski By the Cauchy Schwarz inequality we obtain W 2 N N k R(K) k R(K) χ(ψ k (u n )) 2 χ(ψ k (w)) N χ(ψ k (w) ψ l (w)) k,l R(K) KNp + 2N χ(ψ k (w) ψ l (w)) k,l R(K) k>l Recalling that ψ is a perutation of U, we can now write χ(ψ k (w) ψ l (w)) = χ(ψ k l (ψ l (w)) ψ l (w)) = χ(ψ k l (w) w), and so (11) W 2 KNp + 2KN 2 K 1 r=1 σ r, where σ r is as in (4) and we assue K τ Fro Lea 2, equation (7), and Lea 4 we derive (12) K 1 r=1 d 1 σ r 2p (+d)/2 k=0 d 1 2p (+d)/2 k=0 p k/2 N k + (p 1)p 1 K 1 r=1 τ d r p k/2 (M k M k+1 ) + (p 1)p 1 K, τ d where N k, resp M k, is the nuber of r, 1 r K 1, with gcd(c r, p ) = p k, resp c r 0 (od p k ) For 1 k and each r counted by M k we have τ k r by Lea 2 By using either [4, Lea 6] or noting that every value odulo p k gives rise to p k distinct values odulo p, we see that (13) τ p k τ k for 1 k Therefore M k K/τ k Kp k /τ for 1 k 1

7 It follows that d 1 k=0 Exponential sus and pseudorando nubers 95 p k/2 (M k M k+1 ) d 1 = M 0 + (p k/2 p (k 1)/2 )M k p ( d 1)/2 M d k=1 ( K p 1/2 ( < ) Kp p 1/2 τ ) d 1 k=1 ( p k/2 M k < K ) Kp p 1/2 τ k=1 Together with (12) and (13) this yields K 1 ( σ r < ) p p 1/2 τ Kp(+d)/2 + p 1 p p τ Kpd r=1 ( p + p 1 ) p 1/2 p 3/2 τ Kp(+d)/2 < 354 p τ Kp(+d)/2 Substituting this bound in (11), we obtain We put Then W 2 < KNp p τ K2 Np (+d)/2 K = p /2 W 2 < 808 p τ K2 Np (+d)/2 p k/2 We reark that if τ < K, then the bound in Theore 1 is trivial because S N (h) N τ < p /2 < 49 p p /4 16 p /2 < 49 p N 1/2 p (+d)/4 16 τ So we can assue K τ, and siilarly we can assue N 1/ p/4

8 96 H Niederreiter and I E Shparlinski because otherwise S N (h) N < 49 p N 1/2 p (+d)/4 16 τ Then K p / N 1/2 p /4 Fro (10) we conclude S N (h) W K + K 2 < p 808 N 1/2 p (+d)/ < ( and this yields the desired result τ ) p N 1/2 p (+d)/4, τ 147 N 1/2 p /4 4 The discrepancy bound Let u 0 /p, u 1 /p,, u /p be inversive congruential pseudorando nubers with odulus p and 1 N τ The discrepancy D N of these nubers is defined by D N = sup J [0,1) A(J, N) N J, where the supreu is extended over all subintervals J of [0, 1), A(J, N) is the nuber of points u n /p in J for 0 n N 1, and J is the length of J Theore 2 Let p 3 be a prie and 2 an integer Then the discrepancy D N of inversive congruential pseudorando nubers with odulus p satisfies p D N < N 1/2 p /4 (18 log N + 151) for 1 N τ τ P r o o f By the Erdős Turán inequality in the for given in [14, p 214], for any integer H 1 we have (14) D N 1 H H ( 1 N πh + 1 ) S N (h), H + 1 h=1 where S N (h) is as in Theore 1 We apply this bound with 1/2 3τ H = p N 1/2 p /4 We can assue H 1 since otherwise the discrepancy bound in the theore

9 h=1 Exponential sus and pseudorando nubers 97 is trivial By Theore 1 we obtain H 1 h S N(h) < 49 p 1 N 1/2 p /4 16 τ Siilarly we get 49 ( p 16 τ < 11 ( p 2 τ d=0 p d/4 H h=1 p d h 1 h ) 1/2 1 N 1/2 p /4 (1 + log H) d=0 ) 1/2 N 1/2 p /4 ( log N ) H S N (h) < 11 p N 1/2 p /4 H 2 τ h=1 p 3d/4 Using (14) and the special for of H, we conclude p D N < N 1/2 p /4 3τ p ( N 1/2 p /4 τ 2π log N + 1 ) π + 1, and after siple calculations we derive the desired result Theore 2 yields a nontrivial discrepancy bound in the case where N is at least of the order of agnitude p /2 log 2 τ We note that, in principle, the ethod in this paper works also for the case p = 2 which is convenient for practical ipleentations of pseudorando nuber generators, but that soe odifications have to be ade in the details It is also of interest to extend our results to inversive congruential pseudorando nubers with an arbitrary coposite odulus References [1] W-S C h o u, The period lengths of inversive congruential recursions, Acta Arith 73 (1995), [2] J Eichenauer-Herrann, E Herrann and S Wegenkittl, A survey of quadratic and inversive congruential pseudorando nubers, in: Monte Carlo and Quasi-Monte Carlo Methods 1996, H Niederreiter et al (eds), Lecture Notes in Statist 127, Springer, New York, 1998, [3] J Eichenauer-Herrann and H Niederreiter, On the discrepancy of quadratic congruential pseudorando nubers, J Coput Appl Math 34 (1991),

10 98 H Niederreiter and I E Shparlinski [4] J Eichenauer-Herrann and A Topuzoǧlu, On the period length of congruential pseudorando nuber sequences generated by inversions, ibid 31 (1990), [5] F Griffin, H Niederreiter and I E Shparlinski, On the distribution of nonlinear recursive congruential pseudorando nubers of higher orders, in: Proc 13th Sypos on Appl Algebra, Algebraic Algoriths, and Error-Correcting Codes, Hawaii, 1999, Lecture Notes in Coput Sci, Springer, Berlin, to appear [6] J Gutierrez, H Niederreiter and I E Shparlinski, On the ultidiensional distribution of inversive congruential pseudorando nubers in parts of the period, Monatsh Math, to appear [7] R Lidl and H Niederreiter, Finite Fields, Addison-Wesley, Reading, MA, 1983; reprint, Cabridge Univ Press, Cabridge, 1997 [8] H Niederreiter, Rando Nuber Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992 [9], New developents in unifor pseudorando nuber and vector generation, in: Monte Carlo and Quasi-Monte Carlo Methods in Scientific Coputing, H Niederreiter and PJ-S Shiue (eds), Lecture Notes in Statist 106, Springer, New York, 1995, [10] H Niederreiter and I E Shparlinski, On the distribution of inversive congruential pseudorando nubers in parts of the period, preprint, 1998 [11],, On the distribution and lattice structure of nonlinear congruential pseudorando nubers, Finite Fields Appl 5 (1999), [12],, On the distribution of pseudorando nubers and vectors generated by inversive ethods, Appl Algebra Engrg Co Coput, to appear [13] H Salié, Über die Kloosteranschen Suen S(u, v; q), Math Z 34 (1932), [14] J D Vaaler, Soe extreal functions in Fourier analysis, Bull Aer Math Soc (NS) 12 (1985), Institute of Discrete Matheatics Departent of Coputing Austrian Acadey of Sciences Macquarie University Sonnenfelsgasse 19 Sydney, NSW 2109 A-1010 Vienna, Austria Australia E-ail: niederreiter@oeawacat E-ail: igor@copqeduau Received on (3621)