Generalized Fermat-Mersenne Number Theoretic Transform Vassil S. Dimitrov, Todor V. Cooklev, and Borislav D. Donevsky

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 41, NO., FEBRUARY Generalized Fermat-Mersenne Number Theoretic Transform Vassil S. Dimitrov, Todor V. Cooklev, and Borislav D. Donevsky Abstract-A generalization of the Fermat and Mersenne number transform is suggested. The transforms are defined over finite fields and rings. This paper establishes the conditions necessary for these numbers to be prime. The length of the transforms is a highly composite number. An algorithm for hding primitive roots of unity is also discussed. The proposed transforms are characterized by respectable Combinations of transform length, dynamic range and computational efficiency and can be used for fast convolution of integer sequences. S I. INTRODUCTION EVERAL papers appeared in the beginning of the seventies in which number-theoretic transforms (NTT's) were proposed as an alternative to the approaches with fast Fourier transform (FFT)[1]-[31]. There are many applications of the NTT: digital filtering [4]-[5], fast convolution [6]-[9], bilinear and other transforms [lo], image processing [ 111, decoding of Reed-Solomon codes [1], and solution of partial differential equations [13], to name just a few. Let Z, represent the ring of integers (0, l,...,p - 1). The NTT and its inverse over Z, are defined by the pair of relations: N-1 ~ ( k = ) Cx(n)ank, /c = 0, 1,..., N - 1 (la) n=o N-1 where all computations are modulo, and a suitable chosen integer p. N is a number of signal samples in the input and output sequences and the kernel of the transform a is a primitive Nth root of unity an 1 (mod p) and for every IC < N a' f 1 (mod p) () Every NTT is specified by three parameters-n, p, and a. There are several requirements which, when satisfied, will make the technique competitive: 1) the transform length (TL) N should be large enough in order for x(n) to accommodate practical signals; ) N must be a highly composite number Manuscript received April 30, 1991; revised May 8, 199, March 1, 1993 and September 16, V. S. Dimitrov is with the Technical University of Plovdiv, Plovdiv, Bulgaria. T. V. Cooklev is with the Faculty of Engineering, Tokyo Institute of Technology, Tokyo, Japan. B. D. Donevsky is with the IAMI, Technical University of Sofia, Sofia, Bulgaria. IEEE Log Number so that fast algorithms can be used; 3) the multiplication by powers of a must be simple operation and error-proof. This is easily accomplished in q-ary arithmetic if a is a power of q; 4) the modulus p determines the dynamic range and must be large enough; 5) p should have an attractive representation in q-ary arithmetic to facilitate arithmetic mod p; 6) in order to avoid overflow the modulus must be much larger than the TL. These requirements are our objectives and will be addressed in more details in the sequel. The problem for choosing N, p and a is complicated be the fact that these parameters are not independent. They can not be chosen separately. Theorem I [I], [3]: If the modulus p is composite, that is p = p;'. p;z.... p;k then N must divide the greatest common divisor (GCD) of (PI- 1, pa - 1,...,Pk- 1). If p is prime, then N must divide p - 1. NTTs with Fermat (Mersenne) numbers as a modulus are called Fermat (Mersenne) number transforms, or FNT (MNT). The Fermat and Mersenne numbers are defined correspondingly by ~,=~"+1, n=l,,3,... (3) M, = " - 1, n - prime. (4) Rader [] was the first to use the MNT for digital convolution. Agarwal and Bums [3] proposed the FNT. A major disadvantage of these transforms is that the relationship between the dynamic range and the transform length, as outlined in Theorem 1, is overly restrictive. To relax this restriction, several other NTT's have been considered. Pollard [14] used prime numbers of the form p = " - " + 1. (5) In [ 151 and [ 161, the authors have used numbers of the form 4q for q = 8, 1, 16, 18, 4, 3. Obviously, these numbers are a special case of (3, for m = n. Another intersting special case of (5) is m = n +, or p = 3. " + 1. Golomb et ul. [17] developed an algorithm for integer convolution over the finite field GF(3." + 1) and described a technique for finding prime numbers of the form 3. n + 1. From pure number-theoretic point of view, this problem is considered in [18]. The number 3. " + 1 have an interesting property-the necessary condition to be prime is that n has no prime factors greater than 3 [19]. Nussbaumer [0], [1] generalized the FNT and MNT and developed pseudo-fermat (pseudo-mersenne) number theoretic transforms, which are /94$ IEEE

2 134 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 41, NO., FEBRUARY 1994 transforms over the ring Z,, where p-is a divisor of composite Fermat (Mersenne) number. It should be noted that all of the above moduli can be written in the form p = " + s, where s is an arbitrary integer. Now we shall look for generalized Fermat-Mersenne (GFM) numbers, which will include the above numbers as special cases. Using these GFM numbers as moduli, the restrictive relationships between the transform length and the dynamic range can be significantly relaxed. Investigating the same problem in [] Lu and Lee proposed numbers of the form: p = qmt f (q - 1). (6) The numbers (6) have three free parameters-q, m, and t, which must be chosen properly in order for p to be prime. The authors of [] did not investigate the conditions when the numbers (6) are prime. In the present paper a different approach is used. We consider numbers of the form (from now on called GFM numbers): Lemma : The polynomial Q(z) = ~P-~+zp-~+...+z+1 divides the polynomial P(z) = zml + xm* zmp if and only if mi f mj (mod p) implies i # j. Theorem : The necessary conditions for Gq,p,n to be prime are: 1) pprime and ) n = pk (k-nonnegative integer). Now some interesting properties of the GFM numbers will be discussed. In the nineteenth century, Lucas [3] proved that all the prime divisors of the nth Fermat numbers F, have the form k."+ +l. The numbers Gq,, 71 have a similar property: Theorem 3: All prime divisors of Gq,P,pm greater than p have the form k. pa+' + 1. This theorem is a key to an algorithm for finding primes of the form Gq,p, ". B. Implementation of the Arithmetic Operations One of the most important reasons for the interest in the Fermat and Mersenne transforms is the simple binary representation of the Fermat and Mersenne numbers: (Z" + 1 )lo The following notations are used: M,, = G,p, = (P-l)" + (P-)" "+1 A, = M3," = 4" +" + 1 (8) T, = M5," = 16" + 8" + 4" + " + 1 S, = M7, 7L = 64" + 3" + 16" + 8" + 4" + " + 1. It is clear that Mp, 1 is a Mersenne number and M, k is a Fermat number. The NTT's over the ring ZM=, are called GFM N'IT's. The paper is organized as follows. In section 11, the necessary conditions for primality are established. These conditions will lead to an algorithm for finding prime numbers of the form (7). Then algorithms for the basic arithmetic operations modulo the GFM numbers are presented. Section I11 provides an algorithm for finding primitive roots of unity. In section IV some N7T's of practical importance are discussed. 11. ANALYSIS OF THE RING ZG~, p, ~ A. Necessary Conditions for Primality (zn = (11.. ' 1) = $1 (10) The subscripts denote the base of the number system. This simple binary representation not only greatly facilitates modular arithmetic, but makes possible error-free computation. The other numbers, which have been considered so far can be represented as follows: (3-"+1)10 = (110~~~01) - (p)o("-1)1) (11) 4" - " + 1 = (11...lo...01) - (1(")0("-1)1 ) (1) The numbers M,, ", which are in process of consideration, have the following binary representation: (Mp,n)10 = ( 10~~~010~~~010~~~01) - (13) - (1 o(n-1) (P 1) - 1 It will be demonstrated, that by exploiting the symmetry in the binary representation, the arithmetic operations can be performed efficiently. As outlined in [4], there are simple and efficient procedures for addition and multiplication in the residue arithmetic mod (n f s), where s 5 n/3. Four integer multiplications are necessary to compute the product of two integers mod (" f s). The symmetry in the As was pointed out in the introduction, the modulo does not have to be a prime number. However, the powers of the primitive element ai must be relative prime to the modulo for each integer i. So, to avoid additional restrictions, the modulo binary representation of the numbers M,," is chosen to be a prime number. Besides, N in this case is as large as possible, because it must divide p - 1. Now we are in a position to state the necessary conditions for the numbers Gq, to be prime. To do this, we need two lemmas (all Multiplication in the Ring ZM,, proofs are in the Appendix). Lemma I: If p is a prime, then for every t 1 and T 1 there will be a positive integer B,,t, such that qtpp+l = (Gq,P,Pr). BT,t + 1. makes possible the reduction of the number of multiplications to one, which is very significant. Let us define the function G(X) = X1 - Xs, where x = X I + ~(p-')". The following two theorems are trivial extensions of theorems and 3 from [4]. They are stated without proof.

3 ~ DIMITROV et al.:generalized FERMAT-MERSENNE NUMBER THEORETIC TRANSFORM 135 TABLE I COMPLEXITY OF ALGORITHMS FOR MODULAR MULTIPLICATIONS Theorem 4: G(X) = X (mod Mp, n) Theorem 5: Let X, Y E [0, Mp,n - 1 and s = (1--')x(P-l)n. Then -(P-l)n < G'(XY) < (P-l)n+l, where G' = G(G(... G(XY))...)). 7 k-times An efficient algorithm is given for multiplying X and Y mod Mp, n. 1. begin a. C:= X *Y; b. for i:= 1 to 3 do C:= G(C); c. if C < 0 then retum C - Mp, d. else if C > Mp, then retum C - Mp, e. else retum C. end; Example: Find the residue modulo 73 of the product of 7 and 70. X = 7010 = =, Y = 710 = , 73 = M3,3 Since the case IC = 0 (s = +1) corresponds to the Fermat numbers the suggested algorithm may be used for the computation of the FNT. The approach is an alternative to the technique of Leibowitz [7]. When s = -1 an efficient method for multiplication modulo Mersenne numbers is obtained AN ALGORITHM FOR FINDING NTH ROOTS OF UNITY IN NTT'S MOD kfp,n From algebra we know that if every element of a group is equal to a power of the primitive element this group is called cyclic and the primitive element is called a generator of the group. Recall that x a (mod p) then a is a quadratic residue mod p. According to the Euler's criterion for quadratic residues [8], if a(p-1)/ 1 (mod p) then a is a quadratic residue mod p, and if u(p-')/~ = -1 then a is a quadratic nonresidue mod p, where p is prime and p is not a divisor of a. If the length of the transform N is chosen to be a power of two, i.e. N = n, then highly efficient algorithm can be used. To specify completely the NTT, we must find a primitive element which generates the n-element cyclic subgroup in Z~LI,, n. Suppose g is a quadratic nonresidue mod Mp, n, i.e., We have g(mp.n-1)/ E -1 (mod M P, n ). (14) (Mp, - 1)/ = "-1((p-1)" - 1)/(n - 1) and " - 1 is odd, so Multiplicative step: X. Y = = = loolllollooooz ~ Partial result: = 5040 = = = = = = 78.8 Partial result: = -654 = 50+(-11) = = =11.8 Partial result: = 149 = 1 + ' = 1 10 = ~. 8 Final result: 11 = 310 The complexity of the above algorithm is one integer multiplication and three additions and shifts. Therefore, the complexity of the multiplication is crucial. There are three basic algorithms for the multiplication of m-bit integers-the direct one, with o(m) bit operations, the Karatsuba-Ofman algorithm [5] with 0(m1.585) operations, and the Schonhage- Strassen algorithm [6], which is the fastest one, with o(m log m log log m) operations. The numbers mod Mp, have o(pn) bits in their binary representation. The complexity of the suggested algorithm for modular multiplication is presented in Table I. If a is a generator of the n-element cyclic subgroup in the ring ZM~, then a" 1 (mod Mp, n) and, therefore, According to theorem 1 in [l] a"-' is the primitive element which generates the n-element cyclic subgroup. For sequences with less samples (N = P, 1 5 t 5 n) the primitive Nth root of unity will be gt = u(~"-~)~"-*. IV. SOME USEFUL GFM NIT'S Now we shall briefly review some NTT's of practical importance, which are special cases of the GFM NTT's. A. htt Over the Ring ZA,, It may easily be verified that Al = 7, A3 = 73 and Ag = 6657 are prime numbers. It is still unknown whether greater values for n, satisfying the condition An-prime exist or not. A7 and A81 can be factored as follows: A7 = A81 = In Table 11, the numbers A, (n < 16), a list of factors, and the maximum transform length (power of ) are given:

4 3l ~ 136 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11 ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 41, NO., FEBRUARY 1994 TABLE II TRANSFORM LENGTHS FOR Nn.5 MOD An n A, =4"+"+l FACTORIZATION MAX.TL(POWER OF TWO) prime prime prime According to Table 11, practical applications may find the case n = 9, Ag = 6657, because the maximum transform length, 51, is sufficient in many applications. The primitive element must satisfy (), which has cp(n) solutions, where cp is the Euler's totient function [8]. If N=56, cp(n) = 18, so there are 18 possible values for a, which may be found by a simple computer program. These values of a are presented in Table 111. B. NTT Over the Ring ZT, In this case the NTT is carried out with the numbers Tn as moduli. The necessary condition T, to be a prime is n = 5'+. If a then T, is too large. The case a = 0 corresponds to the MNT. We shall pay attention to the cases cy = 1 and cy =. 1) cy = 1. It is easily verified that T5 is composite: T5 = = According to Theorem 1, the maximum transform length is N,,, = GCD(600,1800) = 600. Note that 600 is a highly composite number, 600 = 3.3.5', and the Winograd algorithm [9] can be applied. At the same time, 600 samples are sufficient for many applications. Therefore, all objectives stated in section I are met. This NlT has been developed independently in [30] via an analysis of cyclotomic integers. ) a =. T5 is composite and can be factored: T5 = = Pi. P Using Euclid's algorithm we find N,, = GCD(P1-1, Pz - 1) = = 4 +. S3.41. This NlT can meet extreme requirements for precision and length. Again the Winograd algorithm for a length 4.3l. 53 can be used. C. NTT Over the Ring Zs, According to Theorem 1, n must have the form 7'+. Practical application may find the case a = 1. The number ST = is prime and the maximum transform length 57-1 is highly composite: S7-1 = Z Again the Winograd algorithm for a transform length 4 3'. 7 = 1008 can be applied. To the best of our knowledge, the number 5'7 is the greatest known prime number of the class.mp,n. Finding greater prime numbers which belong to this TABLE Roo~s OF UNITY IN class is an interesting problem for the computational number theory. The results of this section are summarized in Table IV. Note, that is of the form a.3b.5c.7d. In this case, Winograd or mixed radix [31,3] algorithms can be used. At this point a comparison to other existing transforms (Tables V-VII) is necessary. As is pointed by Agarwal and Burms in [3] the MNT is not of practical importance, because the TL it offers are small and are not highly composite. The NIT mod F4 is characterized by dynamic range of 17 bits and maximum TL of 16. This is not a good choice, because inevitably overflow must be satisfied. The modulus must be much greater than the TL. The suggested NTT's clearly meet all requirements stated in the introduction. They are useful alternatives to the other NTTs of practical importance, such as NTT mod 3, + 1. Some applications require a very large dynamic range. In this case, transforms mod 5'7 and T5 should be used. V. CONCLUSIONS In this paper, a generalization of the Fermat and Mersenne number theoretic transform was presented. A class of numbers called generalized Fermat-Mersenne numbers leads to these transforms. An algorithm for finding prime numbers belonging to this class was given and an algorithm for finding primitive roots of unity was also included. These Nm's provide practical sequence lengths and dynamic range. The generalized Fermat-Mersenne NTT's are a useful alternative to existing transforms. VI. APPENDIX A. Proof of Lemma 1. Let t = 1. Then qpp+l = (qpr - 1). G,,~,~'. SO E I ~ =, ~ qpp - 1.

5 DIMITROV er al.:generalized FERMAT-MERSENNE NUMBER THEORETIC TRANSFORM 137 TABLE IV SOME NEW "ITS OF PRACTICAL IMPORTANCE MODULUS G FACTORS OF G TRANSFORM LENGTH OF G DYNAMIC RANGE (BITS) A '. 33 = A7 593,71119, = T ' = TZ '3.5' = s '.3'. 7 = TABLE V PARAMETERS OF FERMAT [3] MODLILUS MAX. TRANSFORM LENGTH DYNAMIC RANGE (BITS) F4 = F~ = l 33.& = MODULUS 3 ' ' TABLE VII PARAMETERS OF THE NlT MOD 3 '" MAX. TL (POWER OF TWO) DYNAMIC RANGE (BITS) ' TABLE VI PARAMETERS OF THE "IT MOD 4" - " + 1[16] n MODULUS MAX. TRANSFORM LENGTH DYNAMIC RANGE (BITS) *-3+l C. Proof of Theorem. This theorem for the special case q = has been proven in [33]. The proof, which will be given here, uses only elementary number theory. It is believed that our proof is simpler but nevertheless valid for the general case. The proof of the first condition is a very simple. Let us suppose that p is a composite, or p = plp. Then If t 1, then B. Proof of Lemma. Necessity: mi $ mj (mod p) implies i # j. Let mi = kip + si, for i = 1,..., p and let w be a root of Q(z). Then WP = 1. and and, therefore Gq, p, is a composite too. Thus p must be prime. The proof of the second condition is not so obvious. Let n = pr. n1, and n1 = pk + a, where 1 5 a 5 p - 1. Now we use Lemma 1: The condition mi $ mj (mod p) means that all numbers ~i are distinct, and since 0 5 ~i 5 p + 1 the numbers ~i are a permutation of the set (0, 1,..., p - 1) and therefore P(w) = 0. Suficiency: Let us consider the polynomial P C(z) = czpt. It is clear that C(w) = P(w) = 0, therefore every root of Q(z) is a root of C(z). But the degree of C(z) is less than or equal to p - 1, and if there is a pair (i, j) for i # j, such that mi mj (mod p) then Q(z)/C(z), which is impossible. i=l To complete the proof it is enough to prove that where a I b means a divides b. We define a set A = (icr, i = 0,..., p - l), where a and p are mutually prime. For every two distinct elements ai and aj from A we have ai f aj (mod p), because a; - aj = (i - j)a f 0 (mod p). And now in Lemma, the substitutions z = qpr and mi = ia are made. Thus (C4) is proved. A conclusion is made that if n has a factor of the form pk f a (15 a 5 p - l), then Gq, p, cannot be a prime. This completes the proof of Theorem.

6 ~ 138 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS-11: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 41, NO., FEBRUARY 1994 D. Proof of Theorem 3 Let T be a prime divisor of Gq, P, Pe. Then qpa+ z 1 (mod T). Let us define a number theoretic function ord (a. 711): if (a, m) = 1 then ord(w. vi) = r; else ord(a. m) = 0 where z is the least integer, such that flx -- 1 (mod m). It will be proved, that ord(q. T) = yn+l. It i? well known [8] that if ord(a, m) = T, and an -- 1 (mod 7n), then.f is a divisor of n. If ord(q, r) = f, then t I yo+, therefore f = p, where,o L Q + 1. It will be proved that fi = (Y + 1. Let us suppose that 0 < CY + 1. Then we have the following congruence relationships: qp i 1 (mod 1.) (D1) qp = 1 (mod 1.) qzp 3 1 (mod r) 1s 1 (modr) (D) (D3) (D5) The addition of the congruences D4, 05 leads to Gy,p,pa 3 y (mod T ) But Gq, p, pa 0 (mod T ) and y and r are primes, so p = T, which contradicts to the condition T > p. Hence, [I = a + 1. From Fermat s little theorem we have qr- 1 (mod T), therefore T = k. pn Q.E.D. ACKNOWLEDGMENT The authors are indebted to the anonymous reviewers for their comments and suggestions, which improved the quality of the presentation. REFERENCES [l] J. M. Pollard, The fast Fourier transform over finite fields, Math. Comput., vol. 5, pp , [] C. M. Rader, Discrete convolutions via Meraenne transform, IEEE Trans. Computers, vol. C-1, pp , 197. [3] R. C. Agarwal and C. S. Burrus, Fast convolution using Fermat number transform with application to digital filtering, IEEE Truns. Acaust. Speech, Signal Proc., vol., pp , [4] W. Li and A. M. Peterson, FIR filtering by the modified Fen number transform, IEEE Trans. Acoust. Speech, Signal Proc., vol. 38, pp , [5] Y. C. Lee, B. K. Min, and M. Suk, Realization of adaptive digital filtering using the Fermat number transform, IEEE Truns. Acoust. Speech, Signal Proc., vol. ASSP-33, pp , B. Martens and M. C. 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Dimitrov was bom in Plovdiv, Bulgaria in He received the M.Sc. degree in computer science from the Technical University of Sofia, Bulgaria. Now he is working towards the Ph.D. degree of mathematics at the Mathematical Institute of Bulgarian Academy of Sciences. His research interests include fast algorithms for digital signal processing, computational number theory, computational complexity, parallel computing, computer arithmetic and related topics..

7 DIMITROV et al.:generalized FERMAT-MERSENNE NUMBER THEORETIC TRANSFORM 139 T. V. Cooklev (S 9) was bom in Plovdiv, Bulgaria in 1966 He graduated from the Technical University of Sofia, Bulgana in 1988 Dunng 1991 he was bnefly on the faculty at the same University. At the moment, he is studying towards the Ph D degree in engineenng at Tokyo Institute of Technology, Japan. He is a recipient of a Monbusho scholarship His research activities are in digital filters, algonthms for digital signal processing, circuits and systems and related topics Mr Cooklev is a student member of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan and student member of IEEE. Borislav Donevsky was bom in Pleven, Bulgaria, on July 10, He received the B.Sc. degree in communications and electronics, and the M.S. and Ph.D. degrees in electncal engineering in 1961, 1963 and 1974, respectively, from the Technical University of Sofia, Bulgana. In 1964 he joined the Department of Applied Mathematics and Informatics of the Technical University of Sofia as an Assistant Professor, and since 1981 has been an Associate Professor of Applied Mathematics there His main research interests are in digital signal processing and computer anthmetics. He is a member of the Amencan Mathematical Society. Donevsky is the author of the books. Numerical Methods by the Calculator, Technica, Sofia, 198; Fourier Series, Technica, Sofia, He was also the coauthor of the books: The Application of the Graph Theory for the Analysis and Synthesis of the Electronic Circuits, Technica, Sofia, 1979; Digital Filters, Technica, Sofia, 1981; FFT, Technical University, Sofia, He translated from English to Bulganan the books: Manual for Operational Amplrfier Users, J. D Lenk, Reston Publishing CO, 1976; Operational Amplifiers, G B. Clayton, Buttenvorth, 1979; Experiments with Operational Amplrfier, G. B. Clayton, The Macmillan Press Ltd., 1975; 110 Waveform Generator Projects for the Home Constructor, R M. Marston, TAB, 1978; Signals and Systems, A. V. Oppenheim, A. S. Willsky, I T Young, Prentice- Hall, 1983; Art of Electronics, T C. Hayes, P. Horowitz, Cambridge University Press, 1989