# On the coefficients of the polynomial in the number field sieve

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1 On the coefficients of the polynomial in the number field sieve Yang Min a, Meng Qingshu b,, Wang Zhangyi b, Li Li a, Zhang Huanguo b a International School of Software, Wuhan University, Hubei, China, b Computer School, Wuhan University, Hubei, China, Abstract Polynomial selection is very important in number field sieve. If the yield of a pair of polynomials is closely correlated with the coefficients of the polynomials, we can select polynomials by checking the coefficients first. This can speed up the selection of good polynomials. In this paper, we aim to study the correlation between the polynomial coefficients and the yield of the polynomials. By theoretical analysis and experiments, we find that a polynomial with the ending coefficient containing more small primes is usually better in yield than the one whose ending coefficient contains less. One advantage of the ending coefficient over the leading coefficient is that the ending coefficient is bigger and can contain more small primes in root optimizing stage. Using the complete discrimination system, we also analyze the condition on coefficients to obtain more real roots. Key words: cryptography, integer factorization, number field sieve, polynomial selection, coefficients 1 Introduction The general number field sieve[1,2] is known as the asymtotically fastest algorithm for factoring large integers. It is based on the observation that if a 2 = b 2 mod N and a b, gcd(a b, N) will give a proper factor of N with at This work is supported in part by the National Natural Science Foundation of China(No , No , No ) and the Fundamental Research Funds of Central Universities Corresponding author address: (Meng Qingshu ).

2 least a half chance. The number field sieve starts by choosing two irreducible and coprime polynomials f(x) and g(x) over Z which share a common root m modulo N. Let F (x, y) = y d 1 f(x/y) and G(x, y) = y d 2 g(x/y) be the homogenized polynomials corresponding to f(x) and g(x) respectively, where d 1 and d 2 are the degree of f(x) and g(x) respectively. We want to find many coprime pairs (a, b) Z 2 such that the polynomials values F (a, b) and G(a, b) are simultaneously smooth with respect to some upper bound B and the pair (a, b) is called a relation. An integer is smooth with respect to bound B (or B-smooth) if none of its prime factors are larger than B. If we find enough number of relations, by finding linear dependency[3,4] we can construct: (a,b) S (a bα 1 ) = β 2 1, where f(α 1 ) = 0, β 1 Z[α 1 ] (a,b) S (a bα 2 ) = β 2 2, where g(α 2 ) = 0, β 2 Z[α 2 ]. As there exist maps such that ϕ 1 (α 1 ) = m mod N and ϕ 2 (α 2 ) = m mod N, we have ϕ 1 (β 2 1) = ϕ 2 (β 2 2). We can obtain the square root β 1 and β 2 from β 2 1 and β 2 2 respectively using method in [5]. If we let ϕ 1 (β 1 ) = x and ϕ 2 (β 2 ) = y, then y 2 = x 2 mod N, and we have constructed a congruent squares and so may attempt to factor N by computing gcd(x y, N). In order to obtain enough relations, selecting a polynomial with high probability of being smooth is very important. A good polynomial not only can decrease sieving time, but also can reduce the expected matrix size[6]. The polynomial selection is now a hot research area. Based on base-m method and with translation and rotation technique[6], non-skewed or skewed polynomial can be constructed, where one polynomial f(x) is nonlinear and the other g(x) is monic and linear. If the linear polynomial is nonmonic, the size of nonlinear polynomial can be greatly reduced[7,1]. The two methods above are called linear method. Montgomery[8] proposed the nonlinear method, where the two polynomials are both nonlinear. Recently several papers[9 11] address nonlinear polynomials construction problem. The concept of evolutionary cryptography was proposed in [12] and its idea was used to design cryptographic functions[13,14] or cryptographic algorithm[15]. The capabilities of evolutionary cryptosystem against linear and differential cryptanalysis were given in [16,17] respectively. To search for good polynomials for number field sieve with evolutionary computing, we need to solve the difficulty that the space of candidate polynomials is too huge. With the idea similar to [13], we study if there exists any correlation between the polynomial coefficients and the yield of the polynomials. If they are closely related, we can search for polynomials by checking the coefficients first before calculating its alpha value[6]. Therefore we can search for good polynomials in a 2

3 smaller space. This would speed up the selection of polynomials. By theoretical analysis and experiments, we find that the yield of polynomial is closely related to the coefficients of the polynomial. The polynomial with ending coefficient containing more small primes usually have better yields. We also find that the number of real roots can be determined by partial coefficients of the polynomial if it is skewed. The rest of the paper is organized as follows. In Section 2 we review elements related to the yield of a polynomial. In Section 3 we recite the number of real roots of a rational polynomial. In Section 4 we analyze the effect of the ending coefficient and leading coefficient on the yield respectively. In Section 5 we analyze the effects of coefficients on the number of real roots and on the yield. Finally we make a conclusion in Section 6. 2 Elements related to smoothness of a polynomial An integer is said to be B-smooth if the integer can be factored into factors bounded by B. By Dickman function, given the smooth bound B, the less the integer is, the more likely the integer is B-smooth. In number field sieve, we want the homogenous form F (x, y) = a d x d + + a 1 xy d 1 + a 0 y d of the polynomial f(x) = a d x d + + a 1 x + a 0 to be small. In [6], the size and root property are used to describe the quantity. By size we refer to the magnitude of the values taken by F (x, y). By root property we refer to the distribution of the roots of F (x, y) modulo small p k, for p prime and k 1. If F (x, y) has many roots modulo small p k, values taken by F (x, y) behave as if they are smaller than they actually are. That is, on average, the likelihood of F (x, y) values being smooth is increased. It has always been well understood that size affects the yield of F (x, y). In [18], the number of real roots, the order of Galois group of f 1 (x)f 2 (x) were taken into account. By the number of real roots, if a/b is near a real root, the value F (a, b) will be small and will be smooth with high chance. By the order of Galois group of f 1 f 2, it is better to chose polynomial for which the order of Galois group of f 1 f 2 are small, because they provide more free relations. Obviously, if the coefficients of f(x) are small, F (x, y) would have good size property. In order to obtain polynomial with small coefficients, we can search extensively, or let the linear polynomial be nonmonic as suggested in [1,7]. In order to obtain good root property, usually it is required that the leading coefficient contains many small prime as its factors[6]. The paper[19,20] discussed other ways to improve root property. As for the number of the real roots, it is left as random. 3

4 3 The number of real roots of a polynomial In [21,22],the number of real roots or roots distribution of a rational polynomial is given by CDS(complete discrimination system). In degree 3, take polynomial f(x) = ax 3 + bx 2 + cx + d as example. The CDS is = 18abcd 4b 3 d + b 2 c 2 4ac 3 27a 2 d 2. The root distribution is as follows. (1) If > 0, the equation has three distinct real roots. (2) If = 0, the equation has a multiple root and all its roots are real. (3) If < 0, the equation has one real root and two nonreal complex conjugate roots. In degree 4, take f(x) = a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0, (a 4 0) as example. Its CDS is as follows: D 2 = 3a 2 3 8a 2 a 4, D 3 = 16a 2 4a 0 a 2 18a 2 4a 2 1 4a 4 a a 4 a 1 a 3 a 2 6a 4 a 0 a a 2 2a 2 3 3a 1 a 3 3, D 4 = 256a 3 4a a 2 4a a 2 4a 1 a 2 0a 3 27a 4 3a 2 0 6a 4 a 2 3a 0 a 2 1+a 2 2a 2 1a 2 3 4a 4 a 3 2a a 2 a 0 a 3 3a a 4 a 2 a 2 0a a 4 a 2 2a 0 a 3 a 1 +18a 4 a 2 a 3 1a 3 4a 3 2a 0 a 2 3 4a 3 3a a 4 a 4 2a 0 128a 2 4a 2 2a a 2 4a 2 a 0 a 2 1, E = 8a 2 4a 1 + a 3 3 4a 4 a 3 a 2. The numbers of real and imaginary roots and multiplicities of repeated roots in all cases are given as follows: (1) D 4 > 0 D 3 > 0 D 2 > 0 {1, 1, 1, 1}, (2) D 4 > 0 (D 3 0 D 2 0) {}, (3) D 4 < 0 {1, 1}, (4) D 4 = 0 D 3 > 0 {2, 1, 1}, (5) D 4 = 0 D 3 < 0 {2}, (6) D 4 = 0 D 3 = 0 D 2 > 0 E = 0 {2, 2}, (7) D 4 = 0 D 3 = 0 D 2 > 0 E 0 {3, 1}, (8) D 4 = 0 D 3 = 0 D 2 < 0 {}, (9) D 4 = 0 D 3 = 0 D 2 = 0 {4}, where the column on the right side describes the situations of the roots. For example, {1, 1, 1, 1} means four real simple roots and {2, 1, 1} means one real double root plus two real simple roots. In degree 5, take f(x) = x 5 + px 3 + qx 2 + rx + s as example. Its CDS is as follows: D 2 = p 4

5 D 3 = 40rp 12p 3 45q 2 D 4 = 12p 4 r 4p 3 q prq 2 88r 2 p 2 40qp 2 s+125ps 2 27q 4 300qrs+160r 3 D 5 = 1600qsr ps 3 q+2000ps 2 r 2 4p 3 q 2 r 2 +16p 3 q 3 s 900rs 2 p q 2 p 2 s pq 2 r q 2 rs 2 +16p 4 r p 5 s 2 128r 4 p 2 27q 4 r q 5 s+256r s 4 72p 4 rsq + 560r 2 p 2 sq 630prq 3 s E 2 = 160r 2 p q 2 r 2 48rp 5 +60q 2 p 2 r+1500rpsq+16q 2 p qp 3 s+625s 2 p q 3 s F 2 = 3q 2 8rp The numbers of real and imaginary roots and multiplicities of repeated roots of polynomial in all cases are given as follows: (1) D 5 > 0 D 4 > 0 D 3 > 0 D2 > 0 {1, 1, 1, 1, 1} (2) D 5 > 0 (D 4 0 D 3 0 D 2 0) {1} (3) D 5 < 0 {1, 1, 1} (4) D 5 = 0 D 4 > 0 {2, 1, 1, 1} (5) D 5 = 0 D 4 < 0 {2, 1} (6) D 5 = 0 D 4 = 0 D 3 > 0 E 2 0 {2, 2, 1} (7) D 5 = 0 D 4 = 0 D 3 > 0 E 2 = 0 {3, 1, 1} (8) D 5 = 0 D 4 = 0 D 3 < 0 E 2 0 {1} (9) D 5 = 0 D 4 = 0 D 3 < 0 E 2 = 0 {3} (10) D 5 = 0 D 4 = 0 D 3 = 0 D 2 0 F 2 0 {3, 2} (11) D 5 = 0 D 4 = 0 D 3 = 0 D 2 0 F 2 = 0 {4, 1} (12) D 5 = 0 D 4 = 0 D 3 = 0 D 2 = 0 {5} 4 The yield and the ending coefficient Let f(x) = a d x d + + a 1 x + a 0 be a nonlinear polynomial. Let F (x, y) = a d x d + + a 1 xy d 1 + a 0 y d be the homogenous form of polynomial f(x). Let p q be the number of roots of the homogeneous polynomial F modulo p and let α(f ) = small prime p p (1 p q p + 1 ) log p p 1. In order to make α(f ) small, we can increase the value of p q. Equation F (x, y) = 0 mod p has three kind of roots. (1) If p b and p a d, the pair (a,b) is called the projective root. (2) If p a and p a 0, the pair (a,b) is called the zero root. (3) The rest of pairs (a,b) satisfying F (a, b) = 0 mod p are called ordinary roots, or simply roots. 5

6 Correspondingly, there are three methods to increase the value of p q. The first method is already known that the leading coefficient a d containing many small primes can increase the number of projective roots. For example, the leading coefficient usually is the multiple of 60[7]. As for the ordinary roots,we refer to [19,20].We propose the third method that if the ending coefficient contains many small primes, the number of zero roots can be increased. As f(x) is skewed with a d << a 0, a 0 can contain more small primes than a d can. The number of pair (a,b) satisfying second case would be much more than the one in the first case, especially after the optimization of root properties. In order to check whether the above analysis is right, we do many experiments. In our experiments, we let N be an integer of about 30 digits. In Experiment 1, the polynomials are generated by base-m method as described in [6], but without the optimization step. Experiment 1: (1) Generate polynomial as [6]. For each leading coefficient a d below a bound, we examine m ( N a d ) 1 d. Check the magnitude of a d 1, and of a d 2 compared to m, by computing the integral and non-integral parts of N a d m d m d 1 = a d 1 + a d 2 m + O(m 2 ). If these are sufficiently small, accept a d and m, and we get a polynomial f(x) by the expansion of N in base m and with leading coefficient a d. (2) Collect relations. For each above polynomial, skew the sieving area with skewness= d a0 a d. Randomly choose enough pair of coprime (a,b) in sieving area and check if they form relations. For each polynomial,we denote the number of relations by num rel. (3) For each polynomial, there is a row corresponding to it. It includes the following items: the number of relations num rel, the number of small primes contained in a d below a predefined bound, denoted by num ad, the number of small primes in a 0 below a predefined bound, denoted by num a0. A file is formed. (4) Sort the above file in ascending order with num rel as key word. From the sorted file, we can find the parameter num a0 and num ad both are in ascending order, but not strictly. In order to obtain an obvious impression, we divide the sorted file by rows into many length-equal parts, each of which contains equal number of rows and calculate the sum of the parameters num ad and num a0 in each part respectively. 6

7 Table 1 The trend of three parameter(256 rows/block) Block num num ad num a num root Table 1 lists the sums of num ad and num a0 respectively, where the parameters are as follows. N = ( an integer in example 3 [9] ). The degree of the nonlinear polynomial is 3. Sieving area is 2A A, where A = 4000, coprime pair (a, b) are chosen randomly from sieving area in a way like for(a=-a s;a< A s;a+=rand()%6+1) for(b=1;b< A/s;b+=rand()%6+1), where s = 6 a 0 /a 3. From Table 1 we find that the ending coefficients correlate with the yields of polynomials as the leading coefficient does. For polynomial of degree 4 or 5, we get similar results. For nonmonic linear polynomial generated as suggested in [1,7], we can get similar results. For nonlinear polynomials as suggested in[9], we don t do the experiments. We conjecture the results should be similar. Remark 1: If the size of N, the integer to be factored, gets larger, the skewness of the polynomial would become larger, and a 0 can contain more small primes than a d can. This can be used in optimization of root properties. The advantage of ending coefficient over leading coefficient will be more obvious in term of relation yield. 5 The number of real roots and the coefficients A polynomial with more real roots are preferable in number field sieve because if a/b is near a real root, the value F (a, b) will be small and will be smooth with high possibility. Usually the number of real roots is left as random in polynomial generation. In [21,22],the number of real roots or roots distribution of a rational polynomial is given by CDS. From CDS, the number of real roots should depends on all coefficients of the polynomial. However, the polynomial for NFS is not randomly generated. It has many special properties. For example, it is skewed and usually its first 3 coefficients a d, a d 1 and a d 2 are small compared to m while the size of the rest coefficients is similar to or bigger than that of m. From this, the ratio abs(a d 3 /a d 2 ) will bigger than d a 0 /a d while the ratio abs(a 0 /a 1 ) is smaller than d a 0 /a d. Based on all these features and CDS we can analyze the correlation between the number of real roots and the coefficients. 7

8 In degree 3, from the expression of, the sign of should depend on the item 4ac 3 or 27a 2 d 2, but not depend on 4b 3 d since b is usually small. If the ratio abs(d/c) is not big, the sign of mainly be determined by 4ac 3. If c is small enough, > 0, which means the polynomial f(x) has 3 real roots. Similarly if the ratio abs(d/c) is big, < 0, which means the polynomial f(x) has only one real root. In Experiment 2, on about more than a half cases the polynomial has 1 real root and on less than a half cases the polynomial has 3 real roots, where the ratio abs(d/c) is small and c is negative and small enough. In degree 4, from the expression of D 4, each monomial s exponent sum is equal to 6 and highest exponent of a 3, a 2, a 1 are 4 and the exponent of a 0 is 3. If the polynomial is skewed, the coefficients a 1, a 0 or the ratio abs(a 0 /a 1 ) determine the sign of D 4. As the ratio abs(a 0 /a 1 ) is usually smaller than 4 a 0 /a 4, we have abs(27a 2 4a 4 1) > abs(256a 3 4a 3 0) and D 4 < 0 with large chance. That is, in most case the polynomial has 2 real roots. To obtain 4 real roots, the coefficient a 2 should be negative and small enough and the absolute value of a 1 should be of similar size with that of a 2. In Experiment 2, on about 80 percent of cases the polynomial has 2 real roots and on about 20 percent of cases the polynomial has 4 real roots, where the absolute values of a 2 and a 1 are of similar size. The case with zero real roots should be avoided. For degree =5, from the expression of D 5, the monomials exponent sums are not equal. It is not obvious to determine its sign. However, the sign of D 5 should mainly depend on the sum of 4p 3 q 2 r p 3 q 3 s + 16p 4 r p 5 s 2. To obtain 5 real roots, first the variable p should be negative and small enough such that D 2 > 0, D 3 > 0 and D 4 > 0. Secondly, let p < 0, q < 0, r > 0, s > 0, or let p < 0, s < 0, q > 0, r > 0 such that D 5 > 0. The analysis above in case of degree 3 should be useful in choosing polynomial in nonlinear method, where a polynomial with degree 3 is already enough for practical purpose. Experiment 2: (1) Generate polynomial as step 1 of experiment 1. (2) Collect relation as step 2 of experiment 1. Denote the number of relation by num rel. (3) For each polynomial, there is a row corresponding to it. The row has num rel, the number of real roots num root, and all coefficients as its items. We form a file now. (4) Sort the above file in ascending order with num root as the key word. From the sorted file, we observe the correlation between num root and the polynomial coefficients. In case degree =3, if the coefficients of degree 1 is below some value, there will be 3 real roots for most cases. In case 8

9 degree=4, if coefficient of degree 2 is below some value, there will be 4 real roots for most cases. In case degree=5, only a few polynomials have 5 real roots. (5) Sort the file above in ascending order with num rel as the key word. From the sorted file, we observe the correlation between num root and num rel. We can find num root is also in ascending order, but not strictly. In order to obtain an obvious impression, we divide the sorted file by rows into many length-equal parts, each of which contains equal number of rows and calculate the sum of the parameters num root in each part. Table 1 lists the sum of num root, where the parameters are the same as in experiment 1. From Table 1 we find that increasing the number of roots can increase the yield in degree 3. For degree 4 or 5, we get similar results, but not strong as the case in degree 3. For polynomial generated as suggested by Kleinjung in [7], where the linear polynomial is nonmonic, the results is similar. As for the nonlinear polynomial, we don t do the experiments, but we conjecture results should be similar if the polynomials are skewed. Remark 2: Usually the number of real roots is left as random. However, based on our analysis, we can adjust the value of the related coefficients in polynomial optimizing stage such that the polynomial have more real roots. This is useful because it was stated that increasing the number of real roots could increase the yield[18]. The relationship between the size of real roots and the coefficients is another good research topic because if the size of real roots is near the size of skewness the yield will be high. 6 Conclusion Studying the correlation between the yield of a polynomial and its coefficients is important because it take less computation if we can choose polynomial by checking its coefficients first. In this paper, we study the correlation between the yield of a polynomial and its coefficients. The theoretical analysis and the experiments both show that the ending coefficient containing more small primes will increase the yield of the polynomial. As the ending coefficient is much bigger than the leading coefficient, the ending coefficient can contain more small primes in root optimization step. This is one advantage of ending coefficient over leading coefficient. Using CDS we analyze the coefficients condition to obtain more real roots. This is a necessary consideration in choose polynomial as increasing the number of real roots can increase the yield of the polynomial pair. As the yield will be high when the size of real roots is near the size of the skewness, the relation between the size of real roots and the 9

10 coefficients should be under research. References [1] J. P. Buhler, H.W. Lenstra, JR., C. Pomerance, Factoring Integers With The Number Field Sieve, in A. K. Lenstra and H. W. Lenstra, Jr. (eds.), The Development of the Number Field Sieve, LNCS 1554, 50-94, [2] C. Pomerance, The Number Field Sieve, Proceedings of Symposia in Applied Mathematics, Vol.48, , [3] P. L. Montgomery, A Block Lanczos Algorithm for Finding Dependencies over GF(2), Eurocrypt 95, LNCS921, , [4] D. Coppersmith, Solving Homogeneous Linear Equations over GF(2) via Block Wiedemann Algorithm, Mathematics of Computation. 62, , [5] P. Nguyen, A Montgomery-like Square Root for the Number Field Sieve, Proceedings ANTS III, Springer-Verlag, LNCS 1423,151-68, [6] B. Murthy, Polynomial Selection for the Number Field Sieve Integer Factorisation Algorithm, Ph.D. thesis, The Australian National University, [7] T. Kleinjung, On Polynomial Selection For The General Number Field Sieve, Mathematics of Computation, Vol.75, No.256, , [8] P. L. Montgomery, Small Geometric Progressions Modulo n, manuscript (1995). [9] N. Koo, G.H. Jo, and S. Kwon, On Nonlinear Polynomial Selection and Geometric Progression (mod N) for Number Field Sieve. [10] T. Prest, P. Zimmermann, Non-linear Polynomial Selection For The Number Field Sieve, Journal of Symbolic Computation, Vol.47, Issue 4, , [11] R. S. Williams, Cubic Polynomials in the Number Field Sieve, Master Thesis, Texas Tech University, [12] Huanguo Zhang, Xiutao Feng, Zhongpin Qin, Yuzhen Liu, Research on Evolutionary Cryptosystems and Evolutionary DES, Journal of computer, Vol.26,No.12, , [13] Qingshu Meng, Huanguo Zhang, Zhangyi Wang, Zhongpin Qin, Wenling Peng, Designing Bent Functions Using Evolving Method, Acta Electronica Sinica, Vol.32, No.11, , [14] Min Yang, Qingshu Meng, Huanguo Zhang, Evolutionary Design of Trace Form Bent Functions in Cryptography, International Journal of Information and Computer Security, Vol.3, NO.1, 47-59,

11 [15] Huanguo Zhang, Qingshu Meng, Sequence Generator Based on Time-Varying Binary Combiner, Acta Electronica Sinica, Vol.32,No.4, , [16] Huanguo Zhang, Chunlei Li, Ming Tang, Evolutionary Cryptography Against Multidimensional Linear Cryptanalysis, Science China: Information Science, Vol.54, No.12, , [17] Huanguo Zhang, Chunlei Li, Ming Tang, Capability of Evolutionary Cryptosystem Against Differentil Cryptanalysis. Science China: Information Science, Vol. 54,No.10, , [18] M. Elkenbracht-Huizing,An Implementation of the Number Field Sieve, Experimental Mathematics, Vol.5, No.3, , [19] J.E. Gower,Rotations and Translations of Number Field Sieve Polynomials, Advances in Cryptology - ASIACRYPT 2003, LNCS2894, , [20] Shi Bai, Polynomial Selection for the Number Field Sieve, Ph.D thesis, the Australian National University,2011. [21] Lu. YANG, Recent Advances on Determining the Number of Real Roots of Parametric Polynomials, Journal of Symbolic Computation, Vol. 28, , [22] function. 11

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