EXPONENTIAL SUMS OVER THE SEQUENCES OF PRN S PRODUCED BY INVERSIVE GENERATORS

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1 Annales Univ. Sci. Budapest. Sect. Comp EXPONENTIAL SUMS OVER THE SEQUENCES OF PRN S PRODUCED BY INVERSIVE GENERATORS Sergey Varbanets Odessa Ukraine Communicated by Imre Kátai Received February 0 018; accepted March Abstract. The inversive congruential method for generating uniform pseudorandom numbers is a particulary attractive alternative to linear congruential generators which show many undesirable regularities. Exponential sums on inversive congruential pseudorandom numbers are estimates. 1. Introduction The character of equidistribution the sequence of pseudorandom numbers abbreviate PRN s is defined by the discrepancy of this sequence. Usually the bound of discrepancy for the sequence of PRN s that is generated by the congruential generator is estimated by using the Turán Erdős Koksma inequality the core of which is the exponential sum with elements of this sequence in exponent. In the works of R.G. Stoneham [5] and H. Niederreiter [] [4] a certain exponential sums are investigated which are intimately connected with the linear congruential PRN s produced by the linear congruential recursion y n+1 ay n + b mod M n = y 1 M where a b M y 0 Z a 1 M > 1 b y 0 0. Key words and phrases: Inversive congruential pseudorandom numbers exponential sum equidistribution. 010 Mathematics Subect Classification: Primary 11L07; Secondary 45B67 11T3 11T71 11K45.

2 6 S. Varbanets H. Niederreiter [4] proved the following assertion Theorem. Let h Z h M = 1 a M = 1 and assume that a belongs to the exponent l modulo M. Then for 1 N τ 1 hyn πi Mτ e M < l π log τ where τ is the least period length of the sequence {y n }. The well-known deficiencies of the linear congruential sequences of PRN s such as the relatively coarse lattice structure of these sequences and as consequence a predictability of elements of the linear congruential generators of PRN s. Let fx be an integral-value function and let {y n } be a sequence produced by the congruential recursion with initial value y 0. y n+1 fy n mod M Consider the sequence {x n } x n = y n n = This sequence calls the M sequence of PRN s if it is an equidistribution on [0 1 statistical independence and has a large period length. In 1986 Eichenauer and Lehn [1] and then Niederreiter [] have studied a recursive sequence generated by the recursive relation { ay 1 n + b mod p if y n p = 1 y n+1 b mod b if y n 0 mod p with some coefficients a Z p b Z p y 0 p = 1 yn 1 for y n modulo p. is a multiplicative inverse Such generator of PRN s calls the inversive generator modulo p. For the case M = p m m > 1 we also can consider similar generator if only for all n = the values y n satisfy the condition y n p = 1. This condition holds if a p = 1 b 0 mod p. In the sequel we shall always assume also without of explicit mention that this condition accomplishes. In the present paper we study some exponential sums over the sequence of inversive congruential PRN s {y n } produced by one congruence I II y n+1 ay 1 n + bn mod p m y n+1 ay 1 n 1 y 1 n + bn mod p m

3 Exponential sums over the sequences of PRN s 7 where bn = b + c 1 n + p µ F n b = p ν0 b 0 ν 0 > 0 b 0 p = 1 c 1 = cy 0 ν p c = µ 0 > ν 0 for I and c 1 = c for II; a p = y 0 p = y 1 p = 1 µ > max ν 0 µ 0 F n Z[n]. The generator of PRN s I respectively II calls the inversive generator with a variable shift respectively the inversive generator of second order with a variable shift. The aim of this paper to obtain non-trivial bounds for the exponential sums of type S l h; N = K l h 1 h ; N = y n l e y n l hy πi n p m = 1 ; l = I II; e πi h 1 yn+h y 1 n p m l = I II. Here we note that the subscription y n l implies the satisfaction of y n to recursion l l = I or II. Notations. The letter p denotes a prime number p 5. For m N the notation Z M respectively Z M denotes the complete respectively reduced system of residue modulo M. We write gcda b = a b for the greatest common divisor of a and b. For z Z z p = 1 let z 1 be the multiplicative inverse modulo p m. Through ν p A we denote the p-adic valuation of A Z >0 throughout the sequel for brevity we write e q n = e πi n q for integer n.. Auxiliary results Now we consider some lemmas which will be necessary furthermore. Lemma 1 see [7]. Let {y n [} be] the sequence produced by recursion I under m its conditions. Then for k > ν the following representations modulo p m hold. y k = Ak 0 + A k 1 y A k 0 B k 0 + B k 1 y B r k y0 r y k+1 = r 1 yr 1 Ck 0 + C k 1 y C r k y0 r D k 0 + D k 1 y D k r 1 yr 1 0

4 8 S. Varbanets Moreover we have A k A k r B k+1 C k+1 C k+1 r = ad k = bc k r ; + bc k = C k = r; = ab k+1 + ba k+1 + F k + C k 1 = r 1; + F k + 3A k+1 = r 1; = ba k+1 r D k+1 = A k+1 = r 1; F u = cu + p µ F 0 u c = c 1 y 1 0. Corollary. For all k we have modulo p ν kb a 1 kk+1 b + kk + 1c y 0 y k = 1 + kk+1 a 1 b + ka 1 b y 0 + k 1a k 1 cy0 a k+1 + kk+1 a k b + k + 1a k b y 0 + k + 1a k cy0 y k+1 = ka k b + a k + kk+1 a k 1 b + a k + kk + 1c y 0 Lemma see [6]. Let y n be produced by inversive generator of type II. Then the following representations a k + c k 1 b kk 1 y 0 + a k 1 bk + a k 1 cφ 1 k y 0 y 1 y 3k 1 = kk 1 a k 1 b + a k 1 cφ k + k 1a k 1 by 0 + a k 1 y 0 y 1 ka k b + a k y 0 + kk 1 a k 1 b + a k 1 cφ 3 k y 0 y 1 y 3k = a k + kk 1 a k 1 b y 0 + ka k 1 b + a k 1 cφ 1 k y 0 y 1 kk 1 a k b + a k cφ k ka by 0 a k y 0 y 1 y 3k+1 = ka k b + a k y 0 + kk 1 a k 1 b + a k 1 cφ 3 k y 0 y 1 hold where Φ 1 k = F + F F 3k 1 Φ k = F 4 + F F 3k Φ 3 k = F 3 + F F 3k F = + p µ1 F 1 F 1 n Z[n] µ 1 1.

5 Exponential sums over the sequences of PRN s 9 Lemma 3 see [6] Lemma. Let h 1 h k l be positive integers and let ν p h 1 +h = α ν p h 1 k +h l = β δ = min α β. Then for every = 3... we have ν p h 1 k 1 + h l 1 δ. Moreover for every polynomial Gu = A 1 u + A u + p t G 1 u Z[u] we have h 1 Gk + h Gl = A 1 h 1 k + h l + A h 1 k + h l + p t+s G k l where s min ν p h 1 + h ν p h 1 k + h l h 1 h k l Z G u v Z[u v]. Lemma 4 see [6] Lemma 3. Let p > be a prime number m be a positive integer m 0 = [ ] m fx gx hx be polynomials over Z and moreover fx = A 1 x + A x + gx = B 1 x + B x + hx = C l x + C l+1 x l+1 + l 1 ν p A = λ ν p B = µ ν p C = ν k = λ < λ 3 0 = µ 1 < µ < µ 3 ν p C l = 0 ν p C > 0 l + 1. Then the following bounds occur x Z p m e m fx { p m+k if ν p A 1 k 0 if ν p A 1 < k; e m fx + gx 1 p m Ipm m0 x Z p m { 1 if l = 1 e m hx 0 if l > 1 x Z p m where Ip m m0 is a number of solutions of the congruence y f y g y 1 y 1 mod p m m0 y Z p m m 0.

6 30 S. Varbanets 3. Main results For the sequence type I by Lemma 1 we have Lemma 5. Let {y n } be the sequence produced by recursion I under its conditions. Then for k there are valid the following representations modulo p ν y k = y 0 1 a 1 y 0 + k b 1 a 1 y 0 + a 1 b y 0 + cy k a 1 y 0 1 a 1 y0 b + a 1 cy0 := E0 + E 1 k + E k ; y k+1 = ay0 1 + b + cy 0 + k b 1 ay 0 + b + cy k 1 ay b 1 4 a 1 b y0 1 ac := F 0 + F 1 k + F k ; 0 From here the following assertion follows immediately: Corollary. The period length τ of the sequence {y n } is equal to p m ν0 if a is a quadratic non-residue modulo p. And hence the maximal period length takes out for ν p b = ν 0 = 1. For the sequence type II by Lemma we have Lemma 6. Let {y n } be the sequence produced by recursion II under its conditions. Then for k the following representations modulo p ν y 3k 1 = ay0 1 y k 1b ay0 y by 1 1 ay 0 y k 1 b 1 y a 1 y 0 y1 y 3k = y 0 + a 1 by0y 1 + b y0y a 1 b y 0 y k b + a 1 b y 0 y 1 1 a 1 b y0 a 1 by 0y 1 b y 3 0y 1 1 a 1 b y 0 y k a 1 b y 0 y 1 1 a 1 b y 0 + b y 3 0y a 1 b y 0 y 1 y 3k+1 = y 1 a 1 b y1 1 + kb b y0 1 a 1 y1 + 1 y 1 0 y 1 + hold. + k b 1 y a 1 b y1

7 Exponential sums over the sequences of PRN s 31 Corollary. If at least one of the congruences a y 0y 1 mod p ν0 a y 0 y 1 mod p ν0 is violated then the sequence II has a period length τ equal to 3p m ν0. If a is a cubic non-residue modulo p and y = y 0 mod p ν we have the same period length. And hence the maximal period length takes out for ν p b = ν 0 = 1. Now from representations {y n } obtained above for the sequences I and II we infer by Lemma 4 Theorem 1. Let {y n } be the sequence of PRN s produced by recursion l l = I II where a is a quadratic non-residue modulo p. Then hold. S l h; N = y n l e hy πi n p m p m+ν 0 Z p = 1 h p = 1 Theorem. Let h 1 h be arbitrary integers with s = ν p gcdh 1 h p m s m ν 0. Then for the sequence {y n } produced by recursion l l = I II and with a maximal period length τ = 3p m ν0 we have K l h 1 h ; N = y n l e πi h 1 yn+h y 1 n p m p m+ν 0 +s. To prove the estimates for above sums it is enough to split the summation over n for two parts n 0 mod and n 1 mod for l = I and respectively for three parts n 1 mod 3 n 0 mod 3 and n 1 mod 3 for l = II and then apply Lemma 4. Finally note that the more complicated sums over the sequences of PRN s of type I and II may be investigated. References [1] Eichenauer J. and J. Lehn A non-linear congruential pseudorandom number generator Statist. Hefte [] Niederreiter H. Some new exponential sums with applications to pseudo-random numbers In: Topics in Number Theory Debrecen 1974 Colloq. Math. Soc. János Bolyai North-Holland Amsterdam

8 3 S. Varbanets [3] Niederreiter H. On the cycle structure of linear recurring sequences Math. Scand [4] Niederreiter H. On the distribution of pseudo-random numbers generated by the linear congruential method III Math. Comp [5] Stoneham R.G. On the uniform ε-distribution of residue within the periods of rational fractions with applications to normal numbers Acta Arith [6] Varbanets Pavel and Sergey Varbanets Inversive generator of the second order with a variable shift for the sequence of PRNs Annales Univ. Sci. Budapest. Sect. Comp [7] Tran Kim Thanh Tran The Vinh and Varbanets Sergey Generalization of inversive congruential generator with a variable shift 11th CHAOS 018 International Conference Proceedings 5-8 June 018 Rome Italy 018 to appear. Sergey Varbanets Department of Computer Algebra and Discrete Mathematics I.I. Mechnikov Odessa National University Odessa 6506 Ukraine varb@sana.od.ua