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New Complexity Trade-Offs for the (Multiple) Number Field Sieve Algorithm in Non-Prime Fields

  • Palash Sarkar
  • Shashank SinghEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9665)

Abstract

The selection of polynomials to represent number fields crucially determines the efficiency of the Number Field Sieve (NFS) algorithm for solving the discrete logarithm in a finite field. An important recent work due to Barbulescu et al. builds upon existing works to propose two new methods for polynomial selection when the target field is a non-prime field. These methods are called the generalised Joux-Lercier (GJL) and the Conjugation methods. In this work, we propose a new method (which we denote as \(\mathcal {A}\)) for polynomial selection for the NFS algorithm in fields \(\mathbb {F}_{Q}\), with \(Q=p^n\) and \(n>1\). The new method both subsumes and generalises the GJL and the Conjugation methods and provides new trade-offs for both n composite and n prime. Let us denote the variant of the (multiple) NFS algorithm using the polynomial selection method “X” by (M)NFS-X. Asymptotic analysis is performed for both the NFS-\(\mathcal {A}\) and the MNFS-\(\mathcal {A}\) algorithms. In particular, when \(p=L_Q(2/3,c_p)\), for \(c_p\in [3.39,20.91]\), the complexity of NFS-\(\mathcal {A}\) is better than the complexities of all previous algorithms whether classical or MNFS. The MNFS-\(\mathcal {A}\) algorithm provides lower complexity compared to NFS-\(\mathcal {A}\) algorithm; for \(c_p\in (0, 1.12] \cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is the same as that of the MNFS-Conjugation and for \(c_p\notin (0, 1.12] \cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is lower than that of all previous methods.

Keywords

Number Field Sieve (NFS) NFS Algorithm Polynomial Selection Conjugation Method Medium Prime Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Let \(\mathfrak {G}=\langle {\mathfrak {g}} \rangle \) be a finite cyclic group. The discrete log problem (DLP) in \(\mathfrak {G}\) is the following. Given \(({\mathfrak {g}},{\mathfrak {h}})\), compute the minimum non-negative integer \(\mathfrak {e}\) such that \({\mathfrak {h}}={\mathfrak {g}}^{\mathfrak {e}}\). For appropriately chosen groups \(\mathfrak {G}\), the DLP in \(\mathfrak {G}\) is believed to be computationally hard. This forms the basis of security of many important cryptographic protocols.

Studying the hardness of the DLP on subgroups of the multiplicative group of a finite field is an important problem. There are two general algorithms for tackling the DLP on such groups. These are the function field sieve (FFS) [1, 2, 16, 18] algorithm and the number field sieve (NFS) [11, 17, 19] algorithm. Both these algorithms follow the framework of index calculus algorithms which is currently the standard approach for attacking the DLP in various groups.

For small characteristic fields, the FFS algorithm leads to a quasi-polynomial running time [6]. Using the FFS algorithm outlined in [6, 15], Granger et al. [12] reported a record computation of discrete log in the binary extension field \(\mathbb {F}_{2^{9234}}\). FFS also applies to the medium characteristic fields. Some relevant works along this line are reported in [14, 18, 25].

For medium to large characteristic finite fields, the NFS algorithm is the state-of-the-art. In the context of the DLP, the NFS was first proposed by Gordon [11] for prime order fields. The algorithm proceeded via number fields and one of the main difficulties in applying the NFS was in the handling of units in the corresponding ring of algebraic integers. Schirokauer [26, 28] proposed a method to bypass the problems caused by units. Further, Schirokauer [27] showed the application of the NFS algorithm to composite order fields. Joux and Lercier [17] presented important improvements to the NFS algorithm as applicable to prime order fields.

Joux, Lercier, Smart and Vercauteren [19] later showed that the NFS algorithm is applicable to all finite fields. Since then, several works [5, 13, 20, 24] have gradually improved the NFS in the context of medium to large characteristic finite fields.

The efficiency of the NFS algorithm is crucially dependent on the properties of the polynomials used to construct the number fields. Consequently, polynomial selection is an important step in the NFS algorithm and is an active area of research. The recent work [5] by Barbulescu et al. extends a previous method [17] for polynomial selection and also presents a new method. The extension of [17] is called the generalised Joux-Lercier (GJL) method while the new method proposed in [5] is called the Conjugation method. The paper also provides a comprehensive comparison of the trade-offs in the complexity of the NFS algorithm offered by the various polynomial selection methods.

The NFS based algorithm has been extended to multiple number field sieve algorithm (MNFS). The work [8] showed the application of the MNFS to medium to high characteristic finite fields. Pierrot [24] proposed MNFS variants of the GJL and the Conjugation methods. For more recent works on NFS we refer to [4, 7, 22].

Our contributions: In this work, we build on the works of [5, 17] to propose a new method of polynomial selection for NFS over \(\mathbb {F}_{p^n}\). The new method both subsumes and generalises the GJL and the Conjugation methods. There are two parameters to the method, namely a divisor d of the extension degree n and a parameter \(r\ge k\) where \(k=n/d\).

For \(d=1\), the new method becomes the same as the GJL method. For \(d=n\) and \(r=k=1\), the new method becomes the same as the Conjugation method. For \(d=n\) and \(r>1\); or, for \(1<d<n\), the new method provides polynomials which leads to different trade-offs than what was previously known. Note that the case \(1<d<n\) can arise only when n is composite, though the case \(d=n\) and \(r>1\) arises even when n is prime. So, the new method provides new trade-offs for both n composite and n prime.

Following the works of [5, 24] we carry out an asymptotic analysis of new method for the classical NFS as well as for MNFS. For the medium and the large characteristic cases, the results for the new method are exactly the same as those obtained for existing methods in [5, 24]. For the boundary case, however, we obtain some interesting asymptotic results. Letting \(Q=p^n\), the subexponential expression \(L_Q(a,c)\) is defined to be the following:
$$\begin{aligned} L_Q(a,c)= & {} \exp \left( (c+o(1))(\ln Q)^a(\ln \ln Q)^{1-a}\right) . \end{aligned}$$
(1)
Write \(p=L_Q(2/3,c_p)\) and let \(\theta _0\) and \(\theta _1\) be such that the complexity of the MNFS-Conjugation method is \(L_Q(1/3,\theta _0)\) and the complexity of the MNFS-GJL method is \(L_Q(1/3,\theta _1)\). As shown in [24], \(L_Q(1/3,\theta _0)\) is the minimum complexity of MNFS1 while for \(c_p > 4.1\), complexity of new method (MNFS-\(\mathcal {A}\)) is lower than the complexity \(L_Q(1/3,\theta _1)\) of MNFS-GJL method.
The classical variant of the new method, (i.e., NFS-\(\mathcal {A}\)) itself is powerful enough to provide better complexity than all previously known methods, whether classical or MNFS, for \(c_p\in [3.39,20.91]\). The MNFS variant of the new method provides lower complexity compared to the classical variant of the new method for all \(c_p\).
Fig. 1.

Complexity plot for MNFS boundary case

The complexity of MNFS-\(\mathcal {A}\) with \(k=1\) and using linear sieving polynomials can be written as \(L_Q(1/3,\mathbf {C}(c_p,r))\), where \(\mathbf {C}(c_p,r)\) is a function of \(c_p\) and a parameter r. For every integer \(r\ge 1\), there is an interval \([\epsilon _0(r),\epsilon _1(r)]\) such that for \(c_p\in [\epsilon _0(r),\epsilon _1(r)]\), \(\mathbf {C}(c_p,r)<\mathbf {C}(c_p,r^{\prime })\) for \(r\ne r^{\prime }\). Further, for a fixed r, let C(r) be the minimum value of \(\mathbf {C}(c_p,r)\) over all \(c_p\). We show that C(r) is monotone increasing for \(r\ge 1\); \(C(1)=\theta _0\); and that C(r) is bounded above by \(\theta _1\) which is its limit as r goes to infinity. So, for the new method the minimum complexity is the same as MNFS-Conjugation method. On the other hand, as r increases, the complexity of MNFS-\(\mathcal {A}\) remains lower than the complexities of all the prior known methods. In particular, the complexity of MNFS-\(\mathcal {A}\) interpolates nicely between the complexity of the MNFS-GJL and the minimum possible complexity of the MNFS-Conjugation method. This is depicted in Fig. 1. In Fig. 4 of Sect. 8.1, we provide a more detailed plot of the complexity of MNFS-\(\mathcal {A}\) in the boundary case.

The complete statement regarding the complexity of MNFS-\(\mathcal {A}\) in the boundary case is the following. For \(c_p\in (0,1.12]\cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is the same as that of MNFS-Conjugation; for \(c_p\notin (0,1.12]\cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is lower than that of all previous methods. In particular, the improvements for \(c_p\) in the range (1.12, 1.45) is obtained using \(k=2\) and 3; while the improvements for \(c_p>3.15\) is obtained using \(k=1\) and \(r>1\). In all cases, the minimum complexity is obtained using linear sieving olynomials.

2 Background on NFS for Non-Prime Fields

We provide a brief sketch of the background on the variant of the NFS algorithm that is applicable to the extension fields \(\mathbb {F}_{Q}\), where \(Q=p^n\), p is a prime and \(n>1\). More detailed discussions can be found in [5, 17].

Following the structure of index calculus algorithms, NFS has three main phases, namely, relation collection (sieving), linear algebra and descent. Prior to these, is the set-up phase. In the set-up phase, two number fields are constructed and the sieving parameters are determined. The two number fields are set up by choosing two irreducible polynomials f(x) and g(x) over the integers such that their reductions modulo p have a common irreducible factor \(\varphi (x)\) of degree n over \(\mathbb {F}_{p}\). The field \(\mathbb {F}_{p^n}\) will be considered to be represented by \(\varphi (x)\). Let \({\mathfrak {g}}\) be a generator of \(\mathfrak {G}=\mathbb {F}_{p^n}^\star \) and let q be the largest prime dividing the order of \(\mathfrak {G}\). We are interested in the discrete log of elements of \(\mathfrak {G}\) to the base \({\mathfrak {g}}\) modulo this largest prime q.

The choices of the two polynomials f(x) and g(x) are crucial to the algorithm. These greatly affect the overall run time of the algorithm. Let \(\alpha ,\beta \in \mathbb {C}\) and \(m \in \mathbb {F}_{p^{n}}\) be the roots of the polynomials \(f(x),\;g(x)\) and \(\varphi (x)\) respectively. We further let l(f) and l(g) denote the leading coefficient of the polynomials f(x) and g(x) respectively. The two number fields and the finite field are given as follows.
$$\begin{aligned} \mathbb {K}_1=\mathbb {Q}(\alpha )=\frac{\mathbb {Q}[x]}{\langle f(x) \rangle } \text {, } \mathbb {K}_2= \mathbb {Q}(\beta )=\frac{Q[x]}{\langle g(x) \rangle } \text { and } \mathbb {F}_{p^{n}}= \mathbb {F}_{p}(m)=\frac{\mathbb {F}_{p}[x]}{\langle \varphi (x) \rangle }. \end{aligned}$$
Thus, we have the following commutative diagram shown in Fig. 2, where we represent the image of \(\xi \in \mathbb {Z}(\alpha )\) or \(\xi \in \mathbb {Z}(\beta )\) in the finite field \(\mathbb {F}_{p^{n}}\) by \(\overline{\xi }\). Actual computations are carried out over these number fields and are then transformed to the finite field via these homomorphisms. In fact, instead of doing the computations over the whole number field \(\mathbb {K}_i\), one works over its ring of algebraic integers \(\mathcal {O}_i\). These integer rings provide a nice way of constructing a factor basis and moreover, unique factorisation of ideals holds over these rings.
The factor basis \(\mathcal {F}=\mathcal {F}_1 \cup \mathcal {F}_2\) is chosen as follows.
$$\begin{aligned} \mathcal {F}_1= \left\{ \begin{array}{@{}c@{}} \text { prime ideals }\mathfrak {q}_{1,j} \text { in } \mathcal {O}_1\text {, either having norm less than }B \\ \text { or lying above the prime factors of }l(f) \end{array} \right\} \\ \mathcal {F}_2= \left\{ \begin{array}{@{}c@{}} \text { prime ideals } \mathfrak {q}_{2,j} \text { in }\mathcal {O}_2\text {, either having norm less than }B \\ \text { or lying above the prime factors of }l(g) \end{array} \right\} \end{aligned}$$
where B is the smoothness bound and is to be chosen appropriately. An algebraic integer is said to be B-smooth if the principal ideal generated by it factors into the prime ideals of norms less than B. As mentioned in the paper [5], independently of choice of f and g, the size of the factor basis is \(B^{1+o(1)}\). For asymptotic computations, this is simply taken to be B. The work flow of NFS can be understood by the diagram in Fig. 2.
Fig. 2.

A work-flow of NFS.

A polynomial \(\phi (x) \in \mathbb {Z}[x]\) of degree at most \(t-1\) (i.e. having t coefficients) is chosen and the principal ideals generated by its images in the two number fields are checked for smoothness. If both of them are smooth, then
$$\begin{aligned} \phi (\alpha ) \mathcal {O}_1 = \prod _j {\mathfrak {q}_{1,j}}^{e_j} \text{ and } \phi (\beta ) \mathcal {O}_2 = \prod _j {\mathfrak {q}_{2,j}}^{e^\prime _j} \end{aligned}$$
(2)
where \(\mathfrak {q}_{1,j}\) and \(\mathfrak {q}_{2,j}\) are prime ideals in \(\mathcal {F}_1\) and \(\mathcal {F}_2\) respectively. For \(i=1,2\), let \(h_i\) denote the class number of \(\mathcal {O}_i\) and \(r_i\) denote the torsion-free rank of \(\mathcal {O}_i^\star \). Then, for some \(\varepsilon _{i,j} \in \mathfrak {q}_{i,j}\) and units \(u_{i,j} \in \mathcal {O}_{i}^\star \), we have
$$\begin{aligned} \log _g \overline{\phi \left( \alpha \right) } \equiv \sum _{j=1}^{r_1} \lambda _{1,j} \left( \,\phi \left( \alpha \right) \, \right) \Lambda _{1,j} + \sum _{j} e_j X_{1,j} \quad (\mathrm{mod}\; q), \end{aligned}$$
(3)
$$\begin{aligned} \log _g \overline{\phi \left( \beta \right) } \equiv \sum _{j=1}^{r_2} \lambda _{2,j} \left( \,\phi \left( \beta \right) \, \right) \Lambda _{2,j} + \sum _{j} e_j^\prime X_{2,j} \quad (\mathrm{mod}\; q), \end{aligned}$$
(4)
where for \(i=1,2\) and \(j=1\ldots r_i\), \(\Lambda _{i,j}=\log _q \overline{u_{i,j}}\) is an unknown virtual logarithm of the unit \(u_{i,j}\), \(X_{i,j}= h_i^{-1} \log _g \overline{\varepsilon _{i,j}}\) is an unknown virtual logarithm of prime ideal \(\mathfrak {q}_{i,j}\) and \(\lambda _{i,j}: \mathcal {O}_i \mapsto \mathbb {Z}/q\mathbb {Z}\) is Schirokauer map [19, 26, 28]. We skip the details of virtual logarithms and Schirokauer maps, as these details will not affect the polynomial selection problem considered in this work.
Since \(\overline{ \phi \left( \alpha \right) } = \overline{\phi \left( \beta \right) }\), we haveThe relation given by (5) is a linear equation modulo q in the unknown virtual logs. More than \((\#\mathcal {F}_1+\#\mathcal {F}_2+ r_1 + r_2)\) such relations are collected by sieving over suitable \(\phi (x)\). The linear algebra step solves the resulting system of linear equations using either the Lanczos or the block Wiedemann algorithms to obtain the virtual logs of factor basis elements.

After the linear algebra phase is over, the descent phase is used to compute the discrete logs of the given elements of the field \(\mathbb {F}_{p^{n}}\). For a given element \(\mathfrak {y}\) of \(\mathbb {F}_{p^{n}}\), one looks for an element of the form \(\mathfrak {y}^{i}{\mathfrak {g}}^{j}\), for some \(i,j\in \mathbb {N}\), such that the principal ideal generated by preimage of \(\left( \mathfrak {y}^{i}{\mathfrak {g}}^{j}\right) \) in \(\mathcal {O}_1\), factors into prime ideals of norms bounded by some bound \(B_1\) and of degree at most \(t-1\). Then the special-\(\mathfrak {q}\) descent technique [19] is used to write the ideal generated by the preimage as a product of prime ideals in \(\mathcal {F}_1\), which is then converted into a linear equation involving virtual logs. Putting the value of virtual logs, obtained after linear algebra phase, the value of \(\log _{\mathfrak {g}}(\mathfrak {y})\) is obtained. For more details and recent work on the descent phase, we refer to [13, 19].

3 Polynomial Selection and Sizes of Norms

It is evident from the description of NFS that the relation collection phase requires polynomials \(\phi (x)\in \mathbb {Z}[x]\) whose images in the two number fields are simultaneously smooth. For ensuring the smoothness of \(\phi (\alpha )\) and \(\phi (\beta )\), it is enough to ensure that their norms viz, \(\mathrm{Res}(f,\phi )\) and \(\mathrm{Res}(g,\phi )\) are B-smooth. We refer to [5] for further explanations.

Using the Corollary 2 of Kalkbrener’s work [21], we have the following upper bound for the absolute value of the norm.
$$\begin{aligned} \left| \mathrm{Res}(f,\phi )\right| \le \kappa \left( \deg f, \deg \phi \right) ||f ||_\infty ^{\deg \phi } ||\phi ||_\infty ^{\deg f} \end{aligned}$$
(6)
where \(\kappa (a,b)= \left( {\begin{array}{c}a+b\\ a\end{array}}\right) \left( {\begin{array}{c}a+b-1\\ a\end{array}}\right) \) and \(||f ||_\infty \) is maximum of the absolute values of the coefficients of f.
Following [5], let E be such that the coefficients of \(\phi \) are in \(\left[ -\frac{1}{2}E^{2/t},\frac{1}{2}E^{2/t}\right] \). So, \(||\phi ||_\infty \approx E^{2/t}\) and the number of polynomials \(\phi (x)\) that is considered for the sieving is \(E^2\). Whenever \(p=L_Q(a,c_p)\) with \(a > \frac{1}{3}\), we have the following bound on the \(\mathrm{Res}(f,\phi ) \times \mathrm{Res}(g,\phi )\) (for details we refer to [5]).
$$\begin{aligned} |\mathrm{Res}(f,\phi ) \times \mathrm{Res}(g,\phi ) |\approx \big ( ||f ||_\infty ||g ||_\infty \big )^{t-1} E^{(\deg f +\deg g)2/t}. \end{aligned}$$
(7)
For small values of n, the sieving polynomial \(\phi (x)\) is taken to be linear, i.e., \(t=2\) and then the norm bound becomes approximately \(||f ||_\infty ||g ||_\infty E^{(\deg f +\deg g)}\).
The methods for choosing f and g result in the coefficients of one or both of these polynomials to depend on Q. So, the right hand side of (7) is determined by Q and E. All polynomial selection algorithms try to minimize the RHS of (7). From the bound in (7), it is evident that during polynomial selection, the goal should be to try and keep the degrees and the coefficients of both f and g to be small. Ensuring both degrees and coefficients to be small is a nontrivial task and leads to a trade-off. Previous methods for polynomial selections provide different trade-offs between the degrees and the coefficients. Estimates of Q-E trade-off values have been provided in [5] and is based on the CADO factoring software [3]. Table 1 reproduces these values where Q(dd) represents the number of decimal digits in Q.
Table 1.

Estimate of Q-E values [5].

Q(dd)

100

120

140

160

180

200

220

240

260

280

300

Q(bits)

333

399

466

532

598

665

731

798

864

931

997

E(bits)

20.9

22.7

24.3

25.8

27.2

28.5

29.7

30.9

31.9

33.0

34.0

As mentioned in [5, 13], presently the following three polynomial selection methods provide competitive trade-offs.

  1. 1.

    JLSV1: Joux, Lercier, Smart, Vercauteren method [19].

     
  2. 2.

    GJL: Generalised Joux Lercier method [5, 23].

     
  3. 3.

    Conjugation method [5].

     

Brief descriptions of these methods are given below.

JLSV1. Repeat the following steps until f and g are obtained to be irreducible over \(\mathbb {Z}\) and \(\varphi \) is irreducible over \(\mathbb {F}_p\).

  1. 1.

    Randomly choose polynomials \(f_0(x)\) and \(f_1(x)\) having small coefficients with \(\deg (f_1) < \deg (f_0) = n\).

     
  2. 2.

    Randomly choose an integer \(\ell \) to be slightly greater than \(\lceil \sqrt{p}\rceil \).

     
  3. 3.

    Let (uv) be the rational reconstruction of \(\ell \) in \(\mathbb {F}_p\), i.e., \(\ell \equiv u/v \mod p\).

     
  4. 4.

    Define \(f(x)=f_0(x)+ \ell f_1(x)\) and \(g(x)=vf_0(x)+uf_1(x)\) and \(\varphi (x)=f(x)\mod p\).

     

Note that \(\deg (f)=\deg (g)=n\) and both \(||f||_\infty \) and \(||g ||_\infty \) are \(O\left( p^{1/2}\right) =O\left( Q^{1/(2n)}\right) \) and so (7) becomes \(E^{4n/t}Q^{(t-1)/n}\) which is \(E^{2n}Q^{1/n}\) for \(t=2\).

GJL. The basic Joux-Lercier method [17] works for prime fields. The generalised Joux-Lercier method extends the basic Joux-Lercier method to work over composite fields \(\mathbb {F}_{p^n}\).

The heart of the GJL method is the following idea. Let \(\varphi (x)\) be a monic polynomial \(\varphi (x)=x^n+\varphi _{n-1}x^{n-1}+\cdots +\varphi _1x+\varphi _0\) and \(r\ge \mathrm{deg}(\varphi )\) be an integer. Let \(n=\mathrm{deg}(\varphi )\). Given \(\varphi (x)\) and r, define an \((r+1)\times (r+1)\) matrix \(M_{\varphi ,r}\) in the following manner.
$$\begin{aligned} M_{\varphi ,r}= & {} \left[ \begin{array}{ccccccc} p \\ &{} \ddots \\ &{} &{} \ddots \\ &{} &{} &{} p \\ \varphi _{0} &{} \varphi _{1} &{} \cdots &{} \varphi _{n-1} &{} 1 \\ &{} \ddots &{} \ddots &{} &{} \ddots \\ &{} &{} \varphi _{0} &{} \varphi _{1} &{} \cdots &{} \varphi _{n-1} &{} 1 \end{array} \right] \end{aligned}$$
(8)
The first \(n\times n\) principal sub-matrix of \(M_{\varphi ,r}\) is \(\mathrm{diag}[p,p,\ldots ,p]\) corresponding to the polynomials \(p,px,\ldots ,px^{n-1}\). The last \(r-n+1\) rows correspond to the polynomials \(\varphi (x),x\varphi (x),\ldots ,x^{r-n}\varphi (x)\).
Apply the LLL algorithm to \(M_{\varphi ,r}\) and let the first row of the resulting LLL-reduced matrix be \([g_{0},g_1,\ldots ,g_{r-1},g_{r}]\). Define
$$\begin{aligned} g(x)= & {} g_0 + g_1x + \cdots + g_{r-1}x^{r-1} + g_{r}x^{r}. \end{aligned}$$
(9)
The notation
$$\begin{aligned} g= & {} \mathrm{LLL}\left( M_{\varphi ,r}\right) \end{aligned}$$
(10)
will be used to denote the polynomial g(x) given by (9). By construction, \(\varphi (x)\) is a factor of g(x) modulo p.

The GJL procedure for polynomial selection is the following. Choose an \(r\ge n\) and repeat the following steps until f and g are irreducible over \(\mathbb {Z}\) and \(\varphi \) is irreducible over \(\mathbb {F}_p\).

  1. 1.

    Randomly choose a degree \((r+1)\)-polynomial f(x) which is irreducible over \(\mathbb {Z}\) and having coefficients of size \(O(\ln (p))\) such that f(x) has a factor \(\varphi (x)\) of degree n modulo p which is both monic and irreducible.

     
  2. 2.

    Let \(\varphi (x)=x^n+\varphi _{n-1}x^{n-1}+\cdots +\varphi _1x+\varphi _0\) and \(M_{\varphi ,r}\) be the \((r+1)\times (r+1)\) matrix given by (8).

     
  3. 3.

    Let \(g(x)=\mathrm{LLL}\left( M_{\varphi ,r}\right) \).

     

The polynomial f(x) has degree \(r+1\) and g(x) has degree r. The procedure is parameterised by the integer r.

The determinant of M is \(p^n\) and so from the properties of the LLL-reduced basis, the coefficients of g(x) are of the order \(O\left( p^{n/(r+1)}\right) = O\left( Q^{1/(r+1)}\right) \). The coefficients of f(x) are \(O(\ln p)\).

The bound on the norm given by (7) in this case is \(E^{2(2r+1)/t}Q^{(t-1)/(r+1)}\) which becomes \(E^{2r+1}Q^{1/(r+1)}\) for \(t=2\). Increasing r reduces the size of the coefficients of g(x) at the cost of increasing the degrees of f and g. In the concrete example considered in [5] and also in [24], r has been taken to be n and so M is an \((n+1)\times (n+1)\) matrix.

Conjugation. Repeat the following steps until f and g are irreducible over \(\mathbb {Z}\) and \(\varphi \) is irreducible over \(\mathbb {F}_p\).

  1. 1.

    Choose a quadratic monic polynomial \(\mu (x)\), having coefficients of size \(O(\ln p)\), which is irreducible over \(\mathbb {Z}\) and has a root \(\mathfrak {t}\) in \(\mathbb {F}_p\).

     
  2. 2.

    Choose two polynomials \(g_0(x)\) and \(g_1(x)\) with small coefficients such that \(\deg g_1 < \deg g_0 = n\).

     
  3. 3.

    Let (uv) be a rational reconstruction of \(\mathfrak {t}\) modulo p, i.e., \(\mathfrak {t}\equiv u/v\mod p\).

     
  4. 4.

    Define \(g(x)=v g_0(x) + u g_1(x)\) and \(f(x)=\mathrm{Res}_y \big (\mu (y),g_0(x)+y\;g_1(x)\big )\).

     

Note that \(\deg (f)=2n\), \(\deg (g)=n\), \(||f||_\infty = O(\ln p)\) and \(||g||_\infty = O(p^{1/2})=O(Q^{1/(2n)})\). In this case, the bound on the norm given by (7) is \(E^{6n/t}Q^{(t-1)/(2n)}\) which becomes \(E^{3n}Q^{1/(2n)}\) for \(t=2\).

4 A Simple Observation

For the GJL method, while constructing the matrix M, the coefficients of the polynomial \(\varphi (x)\) are used. If, however, some of these coefficients are zero, then these may be ignored. The idea is given by the following result.

Proposition 1

Let n be an integer, d a divisor of n and \(k=n/d\). Suppose A(x) is a monic polynomial of degree k. Let \(r\ge k\) be an integer and set \(\psi (x)=\mathrm{LLL}(M_{A,r})\). Define \(g(x)=\psi (x^d)\) and \(\varphi (x)=A(x^d)\). Then
  1. 1.

    \(\mathrm{deg}(\varphi )=n\) and \(\mathrm{deg}(g)=rd\);

     
  2. 2.

    \(\varphi (x)\) is a factor of g(x) modulo p;

     
  3. 3.

    \(||g||_{\infty } = p^{n/(d(r+1))}\).

     

Proof

The first point is straightforward. Note that by construction A(x) is a factor of \(\psi (x)\) modulo p. So, \(A(x^d)\) is a factor of \(\psi (x^d)=g(x)\) modulo p. This shows the second point. The coefficients of g(x) are the coefficients of \(\psi (x)\). Following the GJL method, \(||\psi ||_{\infty }=p^{k/(r+1)}=p^{n/(d(r+1))}\) and so the same holds for \(||g||_{\infty }\). This shows the third point.    \(\square \)

Note that if we had defined \(g(x)=\mathrm{LLL}(M_{\varphi ,rd})\), then \(||g||_{\infty }\) would have been \(p^{n/(rd+1)}\). For \(d>1\), the value of \(||g||_{\infty }\) given by Proposition 1 is smaller.

A Variant. The above idea shows how to avoid the zero coefficients of \(\varphi (x)\). A similar idea can be used to avoid the coefficients of \(\varphi (x)\) which are small. Suppose that the polynomial \(\varphi (x)\) can be written in the following form.
$$\begin{aligned} \varphi (x)= & {} \varphi _{i_1}x^{i_1}+\cdots +\varphi _{i_k}x^{i_k}+x^n + \sum _{j\notin \{i_1,\ldots ,i_k\}} \varphi _jx^j \end{aligned}$$
(11)
where \(i_1,\ldots ,i_k\) are from the set \(\{0,\ldots ,n-1\}\) and for \(j\in \{0,\ldots ,n-1\}\setminus \{i_1,\ldots ,i_k\}\), the coefficients \(\varphi _{j}\) are all O(1). Some or even all of these \(\varphi _j\)’s could be zero. A \((k+1)\times (k+1)\) matrix M is constructed in the following manner.
$$\begin{aligned} M= & {} \left[ \begin{array}{ccccc} p \\ &{} \ddots \\ &{} &{} \ddots \\ &{} &{} &{} p \\ \varphi _{i_1} &{} \varphi _{i_2} &{} \cdots &{} \varphi _{i_k} &{} 1 \end{array} \right] \end{aligned}$$
(12)
The matrix M has only one row obtained from \(\varphi (x)\) and it is difficult to use more than one row. Apply the LLL algorithm to M and write the first row of the resulting LLL-reduced matrix as \([g_{i_1},\ldots ,g_{i_k},g_n]\). Define
$$\begin{aligned} g(x)= & {} (g_{i_1}x^{i_1}+\cdots +g_{i_k}x^{i_k}+g_nx^n) + \sum _{j\notin \{i_1,\ldots ,i_k,n\}} \varphi _j x^j. \end{aligned}$$
(13)
The degree of g(x) is n and the bound on the coefficients of g(x) is determined as follows. The determinant of M is \(p^k\) and by the LLL-reduced property, each of the coefficients \(g_{i_1},\ldots ,g_{i_k},g_n\) is \(O(p^{k/(k+1)})=O(Q^{k/(n(k+1))})\). Since \(\varphi _j\) for \(j\notin \{i_1,\ldots ,i_k\}\) are all O(1), it follows from (13) that all the coefficients of g(x) are \(O(Q^{k/(n(k+1))})\) and so \(||g||_{\infty }=O(Q^{k/(n(k+1))})\).

5 A New Polynomial Selection Method

In the simple observation made in the earlier section, the non-zero terms of the polynomial g(x) are powers of \(x^d\). This creates a restriction and does not turn out to be necessary to apply the main idea of the previous section. Once the polynomial \(\psi (x)\) is obtained using the LLL method, it is possible to substitute any degree d polynomial with small coefficients for x and still the norm bound will hold. In fact, the idea can be expressed more generally in terms of resultants. Algorithm \(\mathcal {A}\) describes the new general method for polynomial selection.

The following result states the basic properties of Algorithm \(\mathcal {A}\).

Proposition 2

The outputs f(x), g(x) and \(\varphi (x)\) of Algorithm \(\mathcal {A}\) satisfy the following.

  1. 1.

    \(\mathrm{deg}(f)=d(r+1)\); \(\mathrm{deg}(g)=rd\) and \(\mathrm{deg}(\varphi )=n\);

     
  2. 2.

    both f(x) and g(x) have \(\varphi (x)\) as a factor modulo p;

     
  3. 3.

    \(||f||_{\infty }=O(\ln (p))\) and \(||g||_{\infty }=O(Q^{1/(d(r+1))})\).

     
Consequently,
$$\begin{aligned} |\mathrm{Res}(f,\phi ) \times \mathrm{Res}(g,\phi ) |\approx & {} \left( ||f ||_\infty ||g ||_\infty \right) ^{t-1} \times E^{2(\deg f +\deg g)/t} \nonumber \\= & {} O\left( E^{2d(2r+1)/t}\times Q^{(t-1)/(d(r+1))}\right) . \end{aligned}$$
(14)

Proof

By definition \(f(x)=\mathrm{Res}_y\left( A_1(y), C_0(x) + y\,C_1(x) \right) \) where \(A_1(x)\) has degree \(r+1\), \(C_0(x)\) has degree d and \(C_1(x)\) has degree \(d-1\), so the degree of f(x) is \(d(r+1)\). Similarly, one obtains the degree of \(\varphi (x)\) to be n. Since \(\psi (x)\) is obtained from \(A_2(x)\) as \(\mathrm{LLL}(M_{A_2,r})\) it follows that the degree of \(\psi (x)\) is r and so the degree of g(x) is rd.

Since \(A_2(x)\) divides \(A_1(x)\) modulo p, it follows from the definition of f(x) and \(\varphi (x)\) that modulo p, \(\varphi (x)\) divides f(x). Since \(\psi (x)\) is a linear combination of the rows of \(M_{A_2,r}\), it follows that modulo p, \(\psi (x)\) is a multiple of \(A_2(x)\). So, \(g(x)=\mathrm{Res}_y\left( \psi (y), C_0(x) + y\,C_1(x) \right) \) is a multiple of \(\varphi (x)=\mathrm{Res}_y\left( A_2(y), C_0(x) + y\,C_1(x) \right) \) modulo p.

Since the coefficients of \(C_0(x)\) and \(C_1(x)\) are O(1) and the coefficients of \(A_1(x)\) are \(O(\ln p)\), it follows that \(||f||_{\infty }=O(\ln p)\). The coefficients of g(x) are O(1) multiples of the coefficients of \(\psi (x)\). By third point of Proposition 1, the coefficients of \(\psi (x)\) are \(O(p^{n/(d(r+1))})=Q^{1/(d(r+1))}\) which shows that \(||g||_{\infty }=O(Q^{1/(d(r+1))})\).    \(\square \)

Proposition 2 provides the relevant bound on the product of the norms of a sieving polynomial \(\phi (x)\) in the two number fields defined by f(x) and g(x). We note the following points.

  1. 1.

    If \(d=1\), then the norm bound is \(E^{2(2r+1)/t}Q^{(t-1)/(r+1)}\) which is the same as that obtained using the GJL method.

     
  2. 2.

    If \(d=n\), then the norm bound is \(E^{2n(2r+1)/t}Q^{(t-1)/(n(r+1))}\). Further, if \(r=k=1\), then the norm bound is the same as that obtained using the Conjugation method. So, for \(d=n\), Algorithm \(\mathcal {A}\) is a generalisation of the Conjugation method. Later, we show that choosing \(r>1\) provides asymptotic improvements.

     
  3. 3.

    If n is a prime, then the only values of d are either 1 or n. The norm bounds in these two cases are covered by the above two points.

     
  4. 4.

    If n is composite, then there are non-trivial values for d and it is possible to obtain new trade-offs in the norm bound. For concrete situations, this can be of interest. Further, for composite n, as value of d increases from \(d=1\) to \(d=n\), the norm bound nicely interpolates between the norm bounds of the GJL method and the Conjugation method.

     
Existence of \(\mathbb {Q}\) -automorphisms: The existence of \(\mathbb {Q}\)-automorphism in the number fields speeds up the NFS algorithm in the non-asymptotic sense [19]. Similar to the existence of \(\mathbb {Q}\)-automorphism in GJL method, as discussed in [5], the first polynomial generated by the new method, can have a \(\mathbb {Q}\)-automorphism. In general, it is difficult to get an automorphism for the second polynomial as it is generated by the LLL algorithm. On the other hand, we can have a \(\mathbb {Q}\)-automorphism for the second polynomial also in the specific cases. Some of the examples are reported in [10].

6 Non-asymptotic Comparisons and Examples

We compare the norm bounds for \(t=2\), i.e., when the sieving polynomial is linear. In this case, Table 2 lists the degrees and norm bounds of polynomials for various methods. Table 3 compares the new method with the JLSV1 and the GJL method for concrete values of n, r and d. This shows that the new method provides different trade-offs which were not known earlier.

As an example, we can see from Table 3 that the new method compares well with GJL and JLSV1 methods for \(n=4\) and Q of 300 dd (refer to Table 1). As mentioned in [5], when the differences between the methods are small, it is not possible to decide by looking only at the size of the norm product. Keeping this in view, we see that the new method is competitive for \(n=6\) as well. These observations are clearly visible in the plots given in the Fig. 3. From the Q-E pairs given in Table 1, it is clear that the increase of E is slower than that of Q. This suggests that the new method will become competitive when Q is sufficiently large.
Fig. 3.

Product of norms for various polynomial selection methods

Table 2.

Parameterised efficiency estimates for NFS obtained from the different polynomial selection methods.

Methods

\(\deg f\)

\(\deg g\)

\(||f ||_\infty \)

\(||g ||_\infty \)

\(||f ||_\infty ||g ||_\infty E^{(\deg f +\deg g)}\)

JLSV1

n

n

\(Q^{\frac{1}{2n}}\)

\(Q^{\frac{1}{2n}}\)

\(E^{2n}Q^{\frac{1}{n}}\)

GJL (\(r\ge n\))

\(r+1\)

r

\(O(\ln p)\)

\(Q^{\frac{1}{r+1}}\)

\(E^{2r+1}Q^{\frac{1}{r+1}}\)

Conjugation

2n

n

\(O(\ln p)\)

\(Q^{\frac{1}{2n}}\)

\(E^{3n}Q^{\frac{1}{2n}}\)

\(\mathcal {A}\) (d|n, \(r\ge n/d\))

\(d(r+1)\)

dr

\(O(\ln p)\)

\(Q^{\frac{1}{d(r+1)}}\)

\(E^{d(2r+1)}Q^{1/(d(r+1))}\)

Table 3.

Comparison of efficiency estimates for composite n with \(d=2\) and \(r=n/2\).

\(\mathbb {F}_{Q}\)

method

\((\deg f,\deg g)\)

\(||f ||_\infty \)

\(||g ||_\infty \)

\(||f ||_\infty ||g ||_\infty E^{(\deg f +\deg g)}\)

\(\mathbb {F}_{p^{4}}\)

GJL

(5, 4)

\(O(\ln p)\)

\(Q^{\frac{1}{5}}\)

\(E^9 Q^{\frac{1}{5}}\)

JLSV1

(4, 4)

\(Q^{\frac{1}{8}}\)

\(Q^{\frac{1}{8}}\)

\(E^8 Q^{\frac{1}{4}}\)

\(\mathcal {A}\)

(6, 4)

\(O(\ln p)\)

\(Q^{\frac{1}{6}}\)

\(E^{10} Q^{\frac{1}{6}}\)

\(\mathbb {F}_{p^{6}}\)

GJL

(7, 6)

\(O(\ln p)\)

\(Q^{\frac{1}{7}}\)

\(E^{13} Q^{\frac{1}{7}}\)

JLSV1

(6, 6)

\(Q^{\frac{1}{12}}\)

\(Q^{\frac{1}{12}}\)

\(E^{12} Q^{\frac{1}{6}}\)

\(\mathcal {A}\)

(8, 6)

\(O(\ln p)\)

\(Q^{\frac{1}{8}}\)

\(E^{14} Q^{\frac{1}{8}}\)

\(\mathbb {F}_{p^{8}}\)

GJL

(9, 8)

\(O(\ln p)\)

\(Q^{\frac{1}{9}}\)

\(E^{17} Q^{\frac{1}{9}}\)

JLSV1

(8, 8)

\(Q^{\frac{1}{16}}\)

\(Q^{\frac{1}{16}}\)

\(E^{16} Q^{\frac{1}{8}}\)

\(\mathcal {A}\)

(10, 8)

\(O(\ln p)\)

\(Q^{\frac{1}{10}}\)

\(E^{18} Q^{\frac{1}{10}}\)

\(\mathbb {F}_{p^{9}}\)

GJL

(10, 9)

\(O(\ln p)\)

\(Q^{\frac{1}{10}}\)

\(E^{19} Q^{\frac{1}{10}}\)

JLSV1

(9, 9)

\(Q^{\frac{1}{18}}\)

\(Q^{\frac{1}{18}}\)

\(E^{18} Q^{\frac{1}{9}}\)

\(\mathcal {A}\)

(12, 9)

\(O(\ln p)\)

\(Q^{\frac{1}{12}}\)

\(E^{21} Q^{\frac{1}{12}}\)

Next we provide some concrete examples of polynomials f(x), g(x) and \(\varphi (x)\) obtained using the new method. The examples are for \(n=6\) and \(n=4\). For \(n=6\), we have taken \(d=1,2,3\) and 6 and in each case r was chosen to be \(r=k=n/d\). For \(n=4\), we consider \(d=2\) with \(r=k=n/d\) and \(r=k+1\); and \(d=4\) with \(r=k\). These examples are to illustrate that the method works as predicted and returns the desired polynomials very fast. We have used Sage [29] and MAGMA computer algebra system [9] for all the computations done in this work.

Example 1

Let \(n=6\), and p is a 201-bit prime given below.
$$\begin{aligned} p={\scriptstyle 1606938044258990275541962092341162602522202993782792835361211} \end{aligned}$$
Taking \(d=1\) and \( r=n/d\), we get
$$\begin{aligned} \;\;f(x)=x^7 + {\scriptstyle 18}\,x^6 + {\scriptstyle 99}\,x^5 - {\scriptstyle 107}\,x^4 - {\scriptstyle 3470}\,x^3 - {\scriptstyle 15630}\,x^2 - {\scriptstyle 30664}\,x - {\scriptstyle 23239} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 712965136783466122384156554261504665235609243446869}\,x^6+ {\scriptstyle 16048203858903}\\&{\scriptstyle 260691766216702652575435281807544247712} \,x^5 +{\scriptstyle 14867720774814154920358989}\\&{\scriptstyle 0852868028274077107624860184} \,x^4+{\scriptstyle 7240853845391439257955648357229262561} \\&{\scriptstyle 71920852986660372}\,x^3+{\scriptstyle 194693204195493982969795038496468458378024972218}\\&{\scriptstyle 5345772}\,x^2+{\scriptstyle 2718971797270235171234259793142851416923331519178675874} \;x \\&+ {\scriptstyle 1517248296800681060244076172658712224507653769252953211} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^6+{\scriptstyle 671560075936012275401828950369729286806144005939695349290760}\,x^5+\\&{\scriptstyle 774705834624554066737199160555511502088270323481268337340514}\,x^4+ {\scriptstyle 1100} \\&{\scriptstyle 646447552671580437963861085020431145126151057937318479717}\,x^3+{\scriptstyle 27131646}\\&{\scriptstyle 3864123658232870095113273120009266491174096472632727}\,x^2+{\scriptstyle 4101717389506}\\&{\scriptstyle 73951225351009256251353058695601874372080573092}\,x+{\scriptstyle 1326632804961027767}\\&{\scriptstyle 272334662693578855845363854398231524390607} \end{aligned}$$
Note that \(||g ||_\infty \approx 2^{180}\). Taking \(d=2\) and \( r=n/d\), we get
$$\begin{aligned} \;\;f(x)= x^8 - x^7 - {\scriptstyle 5}\,x^6 - {\scriptstyle 50}\,x^5 - {\scriptstyle 181}\,x^4 - {\scriptstyle 442}\,x^3 - {\scriptstyle 801}\,x^2 - {\scriptstyle 633}\,x -{\scriptstyle 787} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 833480932500516492505935839185008193696457787}\,x^6+ {\scriptstyle 2092593616641287655}\\&{\scriptstyle 065740032896986343580698615}\,x^5+{\scriptstyle 1298540899568952261791537743468335194}\\&{\scriptstyle 3188533320}\,x^4+{\scriptstyle 21869741590966357897620167461539967141532970622}\,x^3+{\scriptstyle 6} \\&{\scriptstyle 4403097224634262677273803471992671747860968564}\,x^2+{\scriptstyle 558647116952815842}\\&{\scriptstyle 83909455665521092749502793807}\,x+{\scriptstyle 921778354059077827252784356704871327}\\&{\scriptstyle 10722661831} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^6+{\scriptstyle 225577566898041285405539226183221508226286589225546142714057}\,x^5+\\&{\scriptstyle 726156673723889082895351451739733545328394720523246272955173}\,x^4+{\scriptstyle 10214}\\&{\scriptstyle 78132054694721578888994001730764934454660630543688348056}\,x^3+{\scriptstyle 674978102}\\&{\scriptstyle 55620874288201802771995130845407860934811815878391}\,x^2+{\scriptstyle 632426210761786}\\&{\scriptstyle 622105494194314937817927439372918029042718843}\,x+{\scriptstyle 104093530686601670252}\\&{\scriptstyle 6455143725415379604742339065421793844038} \end{aligned}$$
Note that \(||g ||_\infty \approx 2^{156}\). Taking \(d=3\) and \( r=n/d\), we get
$$\begin{aligned} f(x)= & {} x^9 -{\scriptstyle 4}\,x^8 - {\scriptstyle 54}\,x^7 - {\scriptstyle 174}\,x^6 - {\scriptstyle 252}\,x^5 - {\scriptstyle 174}\,x^4 - {\scriptstyle 76}\,x^3 -{\scriptstyle 86}\,x^2 -{\scriptstyle 96}\,x -{\scriptstyle 42} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 2889742364508381557593312392497801006712}\,x^6+{\scriptstyle 83633695370646306085610}\\&{\scriptstyle 87765146274738509}\,x^5+{\scriptstyle 10828078806524085705506412783408772941877}\,x^4+\\&{\scriptstyle 41812824889730400169000397417267197701179}\,x^3+{\scriptstyle 1497421347777532476213}\\&{\scriptstyle 31508897969482387354}\,x^2+{\scriptstyle 240946716989443210293442965552611305592194}\,x\\&+{\scriptstyle 151696455655104744403073743333940426598833} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^6+{\scriptstyle 265074577705978624915342871970538348132010154368109244143774}\,x^5\\&+{\scriptstyle 21159801273629654486978970226092134077566675973129512551886}\,x^4+{\scriptstyle 10}\\&{\scriptstyle 63445611445684266941289540827947199397416276334188055837892}\,x^3+{\scriptstyle 1459}\\&{\scriptstyle 587283058054365639950761731919998074021438242745336103973}\,x^2+{\scriptstyle 145654}\\&{\scriptstyle 3437800571643325638648207188371117923539168263210522995}\,x+{\scriptstyle 378129170}\\&{\scriptstyle 960510211491600303623674471468414144797178846977007} \end{aligned}$$
Note that \(||g ||_\infty \approx 2^{137}\). Taking \(d=6\) and \( r=n/d\), we get
$$\begin{aligned} f(x)= & {} x^{12} +{\scriptstyle 3}\,x^{10} +{\scriptstyle 10}\,x^9 +{\scriptstyle 53}\,x^8 +{\scriptstyle 112}\,x^7 + {\scriptstyle 163}\,x^6 \\&+{\scriptstyle 184}\,x^5 +{\scriptstyle 177}\,x^4 +{\scriptstyle 166}\,x^3 +{\scriptstyle 103}\,x^2 +{\scriptstyle 72}\,x +{\scriptstyle 48} \end{aligned}$$
$$\begin{aligned} g(x)= & {} -{\scriptstyle 666878138402353195498832669848}\,x^6-{\scriptstyle 1867253271074924746011849188889}\,x^5\\&-{\scriptstyle 5601759813224774238035547566667}\,x^4-{\scriptstyle 6668753801765210948063915265053}\,x^3\\&-{\scriptstyle 4268003536420067847037882226971}\,x^2-{\scriptstyle 6935516090029480629033212906363}\,x\\&-{\scriptstyle 7469013084299698984047396755556} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^6+{\scriptstyle 356485336847074091920944597187811284411849047991334266185684}\,x^5+\\&{\scriptstyle 1069456010541222275762833791563433853235547143974002798557052}\,x^4+{\scriptstyle 175}\\&{\scriptstyle 488639976380184062760893597893819537042246173878495567205}\,x^3+{\scriptstyle 1069456}\\&{\scriptstyle 010541222275762833791563433853235547143974002798557050}\,x^2+{\scriptstyle 1069456010}\\&{\scriptstyle 541222275762833791563433853235547143974002798557054}\,x+{\scriptstyle 14259413473882}\\&{\scriptstyle 96367683778388751245137647396191965337064742736}\\ \end{aligned}$$
In this case we get \(||g ||_\infty \approx 2^{102}\).

Example 2

Let \(n=4\), and p is a 301-bit prime given below.
$$\begin{aligned} p= & {} {\scriptstyle 203703597633448608626844568840937816105146839366593625063614044935438}\\&{\scriptstyle 1299763336706183493607} \end{aligned}$$
Taking \(d=2\) and \(r=n/d\), we get
$$\begin{aligned} f(x)= x^6 +{\scriptstyle 2}\,x^5 +{\scriptstyle 10}\,x^4 +{\scriptstyle 11}\,x^3 +{\scriptstyle 8}\,x^2 +{\scriptstyle 3}\,x +{\scriptstyle 5} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 1108486244023576208689360410176300373132220654590976786482134}\,x^4+{\scriptstyle 20}\\&{\scriptstyle 50762938144982289360096083705563965935573667103554994528044}\,x^3+{\scriptstyle 5523}\\&{\scriptstyle 467580377021934753091786207648479867036209679151793015319}\,x^2+{\scriptstyle 456222}\\&{\scriptstyle 7246514756745388645848004531501269616133890841445574058}\,x+{\scriptstyle 441498133}\\&{\scriptstyle 6353445726063731376031348106734815555088175006533185} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^4+{\scriptstyle 1305623360698284685175599277707343457576279146188242586245210199}\\&{\scriptstyle 344777856138293049165536292}\,x^3+{\scriptstyle 1630663764713242722426772175575945319}\\&{\scriptstyle 640665655794962932653634545690570677252853972689997048}\,x^2+{\scriptstyle 1955704168}\\&{\scriptstyle 7282007596779450734445471817050521654016832790620588920363634983674148} \\&{\scriptstyle 96214457800}\,x+{\scriptstyle 163066376471324272242677217557594531964066565579496293}\\&{\scriptstyle 2653634545690570677252853972689997047} \end{aligned}$$
In this case we have \(||g ||_\infty \approx 2^{201}\). If we take \(r=n/d+1\), we get
$$\begin{aligned} f(x)= x^8 +{\scriptstyle 16}\,x^7 +{\scriptstyle 108}\,x^6 +{\scriptstyle 398}\,x^5 +{\scriptstyle 865}\,x^4 +{\scriptstyle 1106}\,x^3 +{\scriptstyle 820}\,x^2 +{\scriptstyle 328}\,x +{\scriptstyle 55} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 348482147842083865380881347784399925335728557}\,x^6+{\scriptstyle 5536103979982210590}\\&{\scriptstyle 186016445459289773029045618}\,x^5+{\scriptstyle 3381254505070666477453052572333514580}\\&{\scriptstyle 1290667783}\,x^4+{\scriptstyle 96062171957261124763428590648958745188735445330}\,x^3+{\scriptstyle 1}\\&{\scriptstyle 24085795781307363759935898131887563792535489069}\,x^2+{\scriptstyle 73090839973729169}\\&{\scriptstyle 966964061428402316131911130808}\,x+{\scriptstyle 16093810783274309055350481972028841}\\&{\scriptstyle 649178007790} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^4+{\scriptstyle 5128690964597943246501962358998676237033930846168967447990334244}\\&{\scriptstyle 55696319185673262765599428}\,x^3+{\scriptstyle 1802408796932749487444974790576022081}\\&{\scriptstyle 708344659229207911271845827650035713383268427662416444}\,x^2+{\scriptstyle 1553341208}\\&{\scriptstyle 0263216762891646375525736686031169799908288433475579574772861500238438}\\&{\scriptstyle 04262435184}\,x+{\scriptstyle 263801507553366513494386082876419210598165405378517676}\\&{\scriptstyle 874745554282946755826248639365618168} \end{aligned}$$
In this case we have \(||g ||_\infty \approx 2^{156}\). If we take \(d=4\) and \( r=d/n\), we have
$$\begin{aligned} f(x)=x^8 -{\scriptstyle 3}\,x^7 -{\scriptstyle 33}\,x^6 -{\scriptstyle 97}\,x^5 -{\scriptstyle 101}\,x^4 +{\scriptstyle 3}\,x^3 +{\scriptstyle 73}\,x^2 -{\scriptstyle 35}\,x -{\scriptstyle 8} \end{aligned}$$
$$\begin{aligned} g(x)= & {} {\scriptstyle 684862886024125973911391867198415841436877278}\,x^4+{\scriptstyle 1925808392957060519}\\&{\scriptstyle 248933705295588974774910731}\,x^3+{\scriptstyle 1668247862726425714278449912696271875}\\&{\scriptstyle 703468525}\,x^2+{\scriptstyle 40961560447538961485182385700123093758271763}\,x+{\scriptstyle 124094}\\&{\scriptstyle 5506932934545337541838097173133338033453} \end{aligned}$$
$$\begin{aligned} \varphi (x)= & {} x^4+{\scriptstyle 3001292991290566658187708046113162326822746963576576248059013380}\\&{\scriptstyle 7217067092452460559896554}\,x^3+{\scriptstyle 900387897387169997456312413833948698046}\\&{\scriptstyle 82408907297287441770401421651201277357381679689656}\,x^2+{\scriptstyle 15006464956452}\\&{\scriptstyle 8332909385402305658116341137348178828812402950669036085335462262302799}\\&{\scriptstyle 482756}\,x+{\scriptstyle 30012929912905666581877080461131623268227469635765762480590}\\&{\scriptstyle 133807217067092452460559896553} \end{aligned}$$
In this case also we have \(||g ||_\infty \approx 2^{150}\).

7 Asymptotic Complexity Analysis

The goal of the asymptotic complexity analysis is to express the runtime of the NFS algorithm using the L-notation and at the same time obtain bounds on p for which the analysis is valid. Our description of the analysis is based on prior works predominantly those in [5, 17, 19, 24].

For \(0<a<1\), write
$$\begin{aligned} p= & {} L_Q(a,c_p), \text{ where } c_p=\frac{1}{n}\left( \frac{\ln Q}{\ln \ln Q}\right) ^{1-a} \text{ and } \text{ so } n=\frac{1}{c_p}\left( \frac{\ln Q}{\ln \ln Q}\right) ^{1-a}.\nonumber \\ \end{aligned}$$
(15)
The value of a will be determined later. Also, for each \(c_p\), the runtime of the NFS algorithm is the same for the family of finite fields \(\mathbb {F}_{p^n}\) where p is given by (15).

From Sect. 3, we recall the following.

  1. 1.

    The number of polynomials to be considered for sieving is \(E^2\).

     
  2. 2.

    The factor base is of size B.

     
Sparse linear algebra using the Lanczos or the block Wiedemann algorithm takes time \(O(B^2)\). For some \(0<b<1\), let
$$\begin{aligned} B= & {} L_Q(b,c_b). \end{aligned}$$
(16)
The value of b will be determined later. Set
$$\begin{aligned} E= & {} B \end{aligned}$$
(17)
so that asymptotically, the number of sieving polynomials is equal to the time for the linear algebra step.
Let \(\pi =\Psi (\Gamma ,B)\) be the probability that a random positive integer which is at most \(\Gamma \) is B-smooth. Let \(\Gamma =L_Q(z,\zeta )\) and \(B=L_Q(b,c_b)\). Using the L-notation version of the Canfield-Erdös-Pomerance theorem,
$$\begin{aligned} \left( \Psi (\Gamma ,B)\right) ^{-1}= & {} L_Q\left( z-b,(z-b)\frac{\zeta }{c_b}\right) . \end{aligned}$$
(18)
The bound on the product of the norms given by Proposition 2 is
$$\begin{aligned} \Gamma= & {} E^{\frac{2}{t}d(2r+1)}\times Q^{\frac{t-1}{d(r+1)}}. \end{aligned}$$
(19)
Note that in (19), \(t-1\) is the degree of the sieving polynomial. Following the usual convention, we assume that the same smoothness probability \(\pi \) holds for the event that a random sieving polynomial \(\phi (x)\) is smooth over the factor base.
The expected number of polynomials to consider for obtaining one relation is \(\pi ^{-1}\). Since B relations are required, obtaining this number of relations requires trying \(B\pi ^{-1}\) trials. Balancing the cost of sieving and the linear algebra steps requires \(B\pi ^{-1}=B^2\) and so
$$\begin{aligned} \pi ^{-1}= & {} B. \end{aligned}$$
(20)
Obtaining \(\pi ^{-1}\) from (18) and setting it to be equal to B allows solving for \(c_b\). Balancing the costs of the sieving and the linear algebra phases leads to the runtime of the NFS algorithm to be \(B^2=L_Q(b,2c_b)\). So, to determine the runtime, we need to determine b and \(c_b\). The value of b will turn out to be 1/3 and the only real issue is the value of \(c_b\).

Lemma 1

Let \(n=kd\) for positive integers k and d. Using the expressions for p and \(E(=B)\) given by (15) and (16), we obtain the following.
$$\begin{aligned} \left. \begin{array}{rcl} E^{\frac{2}{t}d(2r+1)} &{} = &{} L_Q\left( 1-a+b,\frac{2c_b(2r+1)}{c_pkt}\right) ; \\ Q^{\frac{t-1}{d(r+1)}} &{} = &{} L_Q\left( a,\frac{kc_p(t-1)}{(r+1)}\right) . \end{array} \right\} \end{aligned}$$
(21)

Proof

The second expression follows directly from \(Q=p^n\), \(p=L_Q(a,c_p)\) and \(n=kd\). The computation for obtaining the first expression is the following.
$$\begin{aligned} E^{\frac{2}{t}d(2r+1)}= & {} L_Q\left( b,c_b \frac{2}{t}d(2r+1)\right) \\= & {} \exp \left( c_b \frac{2}{t}(2r+1)\frac{n}{k}(\ln Q)^b(\ln \ln Q)^{1-b}\right) \\= & {} \exp \left( c_b \frac{2}{c_pkt}(2r+1)\left( \frac{\ln Q}{\ln \ln Q}\right) ^{1-a}(\ln Q)^b(\ln \ln Q)^{1-b}\right) \\= & {} L_Q\left( 1-a+b,\frac{2c_b(2r+1)}{c_pkt}\right) . \end{aligned}$$
   \(\square \)

Theorem 1

(Boundary Case). Let k divide n, \(r\ge k\), \(t\ge 2\) and \(p=L_Q(2/3,c_p)\) for some \(0<c_p<1\). It is possible to ensure that the runtime of the NFS algorithm with polynomials chosen by Algorithm \(\mathcal {A}\) is \(L_Q(1/3,2c_b)\) where
$$\begin{aligned} c_b= & {} \frac{2r+1}{3c_pkt} + \sqrt{\left( \frac{2r+1}{3c_pkt}\right) ^2 + \frac{kc_p(t-1)}{3(r+1)}}. \end{aligned}$$
(22)

Proof

Setting \(2a=1+b\), the two L-expressions given by (21) have the same first component and so the product of the norms is
$$\begin{aligned} \Gamma =L_Q\left( a,\frac{2c_b(2r+1)}{c_pkt}+\frac{kc_p(t-1)}{(r+1)}\right) . \end{aligned}$$
Then \(\pi ^{-1}\) given by (18) is
$$\begin{aligned} L_Q\left( a-b,(a-b)\left( \frac{2(2r+1)}{c_pkt}+\frac{kc_p(t-1)}{c_b(r+1)}\right) \right) . \end{aligned}$$
From the condition \(\pi ^{-1}=B\), we get \(b=a-b\) and
$$\begin{aligned} c_b= & {} (a-b)\left( \frac{2(2r+1)}{c_pkt}+\frac{kc_p(t-1)}{c_b(r+1)}\right) . \end{aligned}$$
The conditions \(a-b=b\) and \(2a=1+b\) show that \(b=1/3\) and \(a=2/3\). The second equation then becomes
$$\begin{aligned} c_b = \frac{1}{3}\left( \frac{2(2r+1)}{c_pkt}+\frac{kc_p(t-1)}{c_b(r+1)}\right) . \end{aligned}$$
(23)
Solving the quadratic for \(c_b\) and choosing the positive root gives
$$\begin{aligned} c_b = \frac{2r+1}{3c_pkt} + \sqrt{\left( \frac{2r+1}{3c_pkt}\right) ^2 + \frac{kc_p(t-1)}{3(r+1)}}. \end{aligned}$$
   \(\square \)

Corollary 1

(Boundary Case of the Conjugation Method [5]). Let \(r=k=1\). Then for \(p=L_Q(2/3,c_p)\), the runtime of the NFS algorithm is \(L_Q(1/3,2c_b)\) with
$$\begin{aligned} c_b = \frac{1}{c_pt} + \sqrt{\left( \frac{1}{c_pt}\right) ^2 + \frac{c_p(t-1)}{6}}. \end{aligned}$$

Allowing r to be greater than k leads to improved asymptotic complexity. We do not perform this analysis. Instead, we perform the analysis in the similar situation which arises for the multiple number field sieve algorithm.

Theorem 2

(Medium Characteristic Case). Let \(p=L_Q(a,c_p)\) with \(a>1/3\). It is possible to ensure that the runtime of the NFS algorithm with the polynomials produced by Algorithm \(\mathcal {A}\) is \(L_Q(1/3,(32/3)^{1/3})\).

Proof

Since \(a>1/3\), the bound \(\Gamma \) on the product of the norms can be taken to be the expression given by (7). The parameter t is chosen as follows [5]. For \(0<c<1\), let \(t=c_tn((\ln Q)/(\ln \ln Q))^{-c}\). For the asymptotic analysis, \(t-1\) is also assumed to be given by the same expression for t. Then the expressions given by (21) become the following.
$$\begin{aligned} \begin{array}{rcl} E^{\frac{2}{t}d(2r+1)} = L_Q\left( b+c,\frac{2c_b(2r+1)}{kc_t}\right) ;&\,&Q^{\frac{t-1}{d(r+1)}} = L_Q\left( 1-c,\frac{kc_t}{r+1}\right) . \end{array} \end{aligned}$$
(24)
This can be seen by substituting the expression for t in (21) and further by using the expression for n given in (15).
Setting \(2c=1-b\), the first components of the two expressions in (24) become equal and so
$$\begin{aligned} \Gamma = L_Q\left( b+c,\frac{2c_b(2r+1)}{kc_t}+\frac{kc_t}{r+1}\right) . \end{aligned}$$
Using this \(\Gamma \), the expression for \(\pi ^{-1}\) is
$$\begin{aligned} \pi ^{-1}=L_Q\left( c,c\left( \frac{2(2r+1)}{kc_t}+\frac{kc_t}{c_b(r+1)}\right) \right) . \end{aligned}$$
We wish to choose \(c_t\) so as to maximise the probability \(\pi \) and hence to minimise \(\pi ^{-1}\). This is done by setting \(2(2r+1)/(kc_t) = (kc_t)/(c_b(r+1))\) whence \(kc_t=\sqrt{2c_b(r+1)(2r+1)}\). With this value of \(kc_t\),
$$\begin{aligned} \pi ^{-1}= L_Q\left( c,\frac{2c\sqrt{2c_b(r+1)(2r+1)}}{c_b(r+1)}\right) . \end{aligned}$$
Setting \(\pi ^{-1}\) to be equal to \(B=L_Q(b,c_b)\) yields \(b=c\) and
$$\begin{aligned} c_b= & {} \left( \frac{2c\sqrt{2c_b(r+1)(2r+1)}}{c_b(r+1)}\right) . \end{aligned}$$
From \(b=c\) and \(2c=1-b\) we obtain \(c=b=1/3\). Using this value of c in the equation for \(c_b\), we obtain \(c_b=(2/3)^{2/3}\times ((2(2r+1))/(r+1))^{1/3}\). The value of \(c_b\) is the minimum for \(r=1\) and this value is \(c_b=(4/3)^{1/3}\).    \(\square \)

Note that the parameter a which determines the size of p is not involved in any of the computation. The assumption \(a>1/3\) is required to ensure that the bound on the product of the norms can be taken to be the expression given by (7).

Theorem 3

(Large Characteristic). It is possible to ensure that the runtime of the NFS algorithm with the polynomials produced by Algorithm \(\mathcal {A}\) is \(L_Q(1/3,(64/9)^{1/3})\) for \(p\ge L_Q(2/3,(8/3)^{1/3})\).

Proof

Following [5], for \(0<e<1\), let \(r=c_r/2((\ln Q)/(\ln \ln Q))^e\). For the asymptotic analysis, the expression for \(2r+1\) is taken to be two times this expression. Substituting this expression for r in (21), we obtain
$$\begin{aligned} \left. \begin{array}{rcl} E^{\frac{2}{t}d(2r+1)} &{} = &{} L_Q\left( 1-a+b+e,\frac{2c_b c_r}{c_pkt}\right) ; \\ Q^{\frac{t-1}{d(r+1)}} &{} = &{} L_Q\left( a-e,\frac{2kc_p(t-1)}{c_r}\right) . \end{array} \right\} \end{aligned}$$
(25)
Setting \(1+b=2(a-e)\), we obtain \(\displaystyle \Gamma = L_Q\left( \frac{1+b}{2}, \frac{2c_b c_r}{c_pkt} + \frac{2kc_p(t-1)}{c_r}\right) \) and so the probability \(\pi ^{-1}\) is given by
$$\begin{aligned} L_Q\left( \frac{1-b}{2}, \frac{1-b}{2}\times \left( \frac{2c_r}{c_pkt} + \frac{2kc_p(t-1)}{c_rc_b}\right) \right) . \end{aligned}$$
The choice of \(c_r\) for which the probability \(\pi \) is maximised (and hence \(\pi ^{-1}\) is minimised) is obtained by setting \(c_r/(c_pk) = \sqrt{(t(t-1))/c_b}\) and the minimum value of \(\pi ^{-1}\) is
$$\begin{aligned} L_Q\left( \frac{1-b}{2}, \frac{1-b}{2}\times \left( 4\sqrt{\frac{t-1}{tc_b}}\right) \right) . \end{aligned}$$
Setting this value of \(\pi ^{-1}\) to be equal to B, we obtain
$$\begin{aligned} b = (1-b)/2;&c_b = \frac{1-b}{2}\times \left( 4\sqrt{\frac{t-1}{tc_b}}\right) . \end{aligned}$$
The first equation shows \(b=1/3\) and using this in the second equation, we obtain \(c_b=(4/3)^{2/3}((t-1)/t)^{1/3}\). This value of \(c_b\) is minimised for the minimum value of t which is \(t=2\). This gives \(c_b=(8/9)^{1/3}\).

Using \(2(a-e)=1+b\) and \(b=1/3\) we get \(a-e=2/3\). Note that \(r\ge k\) and so \(p\ge p^{k/r}=L_Q(a,(c_pk)/r)=L_Q(a-e,(2c_pk)/c_r)\). With \(t=2\), the value of \((c_pk)/c_r\) is equal to \((1/3)^{1/3}\) and so \(p\ge L_Q(2/3,(8/3)^{1/3})\).    \(\square \)

Theorems 2 and 3 show that the generality introduced by k and r do not affect the overall asymptotic complexity for the medium and large prime case and the attained complexities in these cases are the same as those obtained for previous methods in [5].

8 Multiple Number Field Sieve Variant

As the name indicates, the multiple number field sieve variant uses several number fields. The discussion and the analysis will follow the works [8, 24].

There are two variants of multiple number field sieve algorithm. In the first variant, the image of \(\phi (x)\) needs to be smooth in at least any two of the number fields. In the second variant, the image of \(\phi (x)\) needs to be smooth in the first number field and at least one of the other number fields.

We have analysed both the variants of multiple number field sieve algorithm and found that the second variant turns out to be better than the first one. So we discuss the second variant of MNFS only. In contrast to the number field sieve algorithm, the right number field is replaced by a collection of V number fields in the second variant of MNFS. The sieving polynomial \(\phi (x)\) has to satisfy the smoothness condition on the left number field as before. On the right side, it is sufficient for \(\phi (x)\) to satisfy a smoothness condition on at least one of the V number fields.

Recall that Algorithm \(\mathcal {A}\) produces two polynomials f(x) and g(x) of degrees \(d(r+1)\) and dr respectively. The polynomial g(x) is defined as \(\mathrm{Res}_y(\psi (y),C_0(x)+yC_1(x))\) where \(\psi (x)=\mathrm{LLL}(M_{A_2,r})\), i.e., \(\psi (x)\) is defined from the first row of the matrix obtained after applying the LLL-algorithm to \(M_{A_2,r}\).

Methods for obtaining the collection of number fields on the right have been mentioned in [24]. We adapt one of these methods to our setting. Consider Algorithm \(\mathcal {A}\). Let \(\psi _1(x)\) be \(\psi (x)\) as above and let \(\psi _2(x)\) be the polynomial defined from the second row of the matrix \(M_{A_2,r}\). Define \(g_1(x)=\mathrm{Res}_y(\psi _1(y),C_0(x)+yC_1(x))\) and \(g_2(x)=\mathrm{Res}_y(\psi _2(y),C_0(x)+yC_1(x))\). Then choose \(V-2\) linear combinations \(g_i(x)=s_ig_1(x)+t_ig_2(x)\), for \(i=3,\ldots ,V\). Note that the coefficients \(s_i\) and \(t_i\) are of the size of \(\sqrt{V}\). All the \(g_i\)’s have degree dr. Asymptotically, \(||\psi _2||_{\infty }=||\psi _1||_{\infty }=Q^{1/(d(r+1))}\). Since we take \(V=L_Q(1/3)\), all the \(g_i\)’s have their infinity norms to be the same as that of g(x) given by Proposition 2.

For the left number field, as before, let B be the bound on the norms of the ideals which are in the factor basis defined by f. For each of the right number fields, let \(B^{\prime }\) be the bound on the norms of the ideals which are in the factor basis defined by each of the \(g_i\)’s. So, the size of the entire factor basis is \(B+VB^{\prime }\). The following condition balances the left portion and the right portion of the factor basis.
$$\begin{aligned} B= & {} VB^{\prime }. \end{aligned}$$
(26)
With this condition, the size of the factor basis is \(B^{1+o(1)}\) as in the classical NFS and so asymptotically, the linear algebra step takes time \(B^2\). As before, the number of sieving polynomials is \(E^2=B^2\) and the coefficients of \(\phi (x)\) can take \(E^{2/t}\) distinct values.
Let \(\pi \) be the probability that a random sieving polynomial \(\phi (x)\) gives rise to a relation. Let \(\pi _1\) be the probability that \(\phi (x)\) is smooth over the left factor basis and \(\pi _2\) be the probability that \(\phi (x)\) is smooth over at least one of the right factor bases. Further, let \(\Gamma _1=\mathrm{Res}_x(f(x),\phi (x))\) be the bound on the norm corresponding to the left number field and \(\Gamma _2=\mathrm{Res}_x(g_i(x),\phi (x))\) be the bound on the norm for any of the right number fields. Note that \(\Gamma _2\) is determined only by the degree and the \(L_{\infty }\)-norm of \(g_i(x)\) and hence is the same for all \(g_i(x)\)’s. Heuristically, we have
$$\begin{aligned} \begin{array}{rcl} \pi _1 &{} = &{} \Psi (\Gamma _1,B); \\ \pi _2 &{} = &{} V\Psi (\Gamma _2,B^{\prime }); \\ \pi &{} = &{} \pi _1\times \pi _2. \end{array} \end{aligned}$$
(27)
As before, one relation is obtained in about \(\pi ^{-1}\) trials and so B relations are obtained in about \(B\pi ^{-1}\) trials. Balancing the cost of linear algebra and sieving, we have as before \(B=\pi ^{-1}\).
The following choices of B and V are made.
$$\begin{aligned} \begin{array}{rcl} E = B &{} = &{} L_Q\left( \frac{1}{3},c_b\right) ; \\ V &{} = &{} L_Q\left( \frac{1}{3},c_v\right) ; \text{ and } \text{ so } \\ B^{\prime } &{} = &{} B/V = L_Q\left( \frac{1}{3},c_b-c_v\right) . \end{array} \end{aligned}$$
(28)
With these choices of B and V, it is possible to analyse the MNFS variant for Algorithm \(\mathcal {A}\) for three cases, namely, the medium prime case, the boundary case and the large characteristic case. Below we present the details of the boundary case. This presents a new asymptotic result.

Theorem 4

(MNFS-Boundary Case). Let k divide n, \(r\ge k\), \(t \ge 2\) and
$$\begin{aligned} p=L_Q\left( \frac{2}{3},c_p\right) \text{ where } c_p=\frac{1}{n}\left( \frac{\ln Q}{\ln \ln Q}\right) ^{1/3}. \end{aligned}$$
It is possible to ensure that the runtime of the MNFS algorithm is \(L_Q(1/3,2c_b)\) where
$$\begin{aligned} c_b= & {} \frac{4r+2}{6ktc_p} + \sqrt{\frac{r(3r+2)}{(3ktc_p)^2} + \frac{c_pk(t-1)}{3(r+1)}}. \end{aligned}$$
(29)

Proof

Note the following computations.
$$\begin{aligned} \Gamma _1= & {} ||\phi ||_{\infty }^{\mathrm{deg}(f)} = E^{2\mathrm{deg}(f)/t} = E^{(2d(r+1))/t} = E^{(2n(r+1))/(kt)} \\= & {} L_Q\left( \frac{2}{3},\frac{2(r+1)c_b}{ktc_p}\right) ; \\ \pi _1^{-1}= & {} L_Q\left( \frac{1}{3},\frac{2(r+1)}{3ktc_p}\right) ; \\ \Gamma _2= & {} ||\phi ||_{\infty }^{\mathrm{deg}(g)}\times ||g||_{\infty }^{\mathrm{deg}(\phi )} = E^{2\mathrm{deg}(g)/t}\times Q^{(t-1)/(d(r+1))} \\= & {} E^{(2rd)/t}\times Q^{(t-1)/(d(r+1))} = E^{(2rn)/(kt)}\times Q^{k(t-1)/(n(r+1))} \\= & {} L_Q\left( \frac{2}{3},\frac{2rc_b}{c_pkt} + \frac{kc_p(t-1)}{r+1} \right) ; \\ \pi _2^{-1}= & {} L_Q\left( \frac{1}{3},-c_v + \frac{1}{3(c_b-c_v)}\left( \frac{2rc_b}{c_pkt} + \frac{kc_p(t-1)}{r+1}\right) \right) ; \\ \pi ^{-1}= & {} L_Q\left( \frac{1}{3},\frac{2(r+1)}{3ktc_p}-c_v + \frac{1}{3(c_b-c_v)}\left( \frac{2rc_b}{c_pkt} + \frac{kc_p(t-1)}{r+1}\right) \right) ; \\ \end{aligned}$$
From the condition \(\pi ^{-1}=B\), we obtain the following equation.
$$\begin{aligned} c_b= & {} \frac{2(r+1)}{3ktc_p}-c_v + \frac{1}{3(c_b-c_v)}\left( \frac{2rc_b}{c_pkt} + \frac{kc_p(t-1)}{r+1}\right) . \end{aligned}$$
(30)
We wish to find \(c_v\) such that \(c_b\) is minimised subject to the constraint (30). Using the method of Lagrange multipliers, the partial derivative of (30) with respect to \(c_v\) gives
$$\begin{aligned} c_v = \frac{r+1}{3ktc_p}. \end{aligned}$$
Using this value of \(c_v\) in (30) provides the following quadratic in \(c_b\).
$$\begin{aligned} (3ktc_p)c_b^2-(4r+2)c_b + \frac{(r+1)^2}{3ktc_p}-\frac{(c_pk)^2t(t-1)}{r+1} = 0. \end{aligned}$$
Solving this and taking the positive square root, we obtain
$$\begin{aligned} c_b= & {} \frac{4r+2}{6ktc_p} + \sqrt{\frac{r(3r+2)}{(3ktc_p)^2} + \frac{c_pk(t-1)}{3(r+1)}}. \end{aligned}$$
(31)
Hence the overall complexity of MNFS for the boundary case is \(L_Q\left( \frac{1}{3} , 2 c_b \right) \).    \(\square \)

8.1 Further Analysis of the Boundary Case

Theorem 4 expresses \(2c_b\) as a function of \(c_p\), t, k and r. Let us write this as \(2c_b=\mathbf {C}(c_p,t,k,r)\). It turns out that fixing the values of (tkr) gives a set S(tkr) such that for \(c_p\in S(t,k,r)\), \(\mathbf {C}(c_p,t,k,r)\le \mathbf {C}(c_p,t^{\prime },k^{\prime },r^{\prime })\) for any \((t^{\prime },k^{\prime },r^{\prime })\ne (t,k,r)\). In other words, for a choice of (tkr), there is a set of values for \(c_p\) where the minimum complexity of MNFS-\(\mathcal {A}\) is attained. The set S(tkr) could be empty implying that the particular choice of (tkr) is sub-optimal.

For \(1.12 \le c_p\le 4.5\), the appropriate intervals are given in Table 4. Further, the interval (0, 1.12] is the union of S(t, 1, 1) for \(t\ge 3\). Note that the choice \((t,k,r)=(t,1,1)\) specialises MNFS-\(\mathcal {A}\) to MNFS-Conjugation. So, for \(c_p\in (0,1.12]\cup [1.45,3.15]\) the complexity of MNFS-\(\mathcal {A}\) is the same as that of MNFS-Conjugation.
Table 4.

Choices of (tkr) and the corresponding S(tkr).

(tkr)

S(tkr)

(t, 1, 1), \(t\ge 3\)

\(\bigcup _{t\ge 3} S(t,1,1)\approx (0,1.12]\)

(2, 3, 3)

\([(1/3)(4\sqrt{21} + 20)^{1/3},(\sqrt{78}/9 + 29/36)^{1/3}]\approx [1.12,1.21]\)

(2, 2, 2)

\([(\sqrt{78}/9 + 29/36)^{1/3},(1/2)(4\sqrt{11}+ 11)^{1/3}]\approx [1.21,1.45]\)

(2, 1, 1)

\([(1/2)(4\sqrt{11}+ 11)^{1/3},(2\sqrt{62} + 31/2)^{1/3}]\approx [1.45,3.15]\)

(2, 1, 2)

\([(2\sqrt{62} + 31/2)^{1/3},(8\sqrt{33} + 45)^{1/3}]\approx [3.15,4.5]\)

In Fig. 4, we have plotted \(2c_b\) given by Theorem 4 against \(c_p\) for some values of t, k and r where the minimum complexity of MNFS-\(\mathcal {A}\) is attained. The plot is labelled MNFS-\(\mathcal {A}\). The sets S(tkr) are clearly identifiable from the plot. The figure also shows a similar plot for NFS-\(\mathcal {A}\) which shows the complexity in the boundary case given by Theorem 1. For comparison, we have plotted the complexities of the GJL and the Conjugation methods from [5] and the MNFS-GJL and the MNFS-Conjugation methods from [24].

Based on the plots given in Fig. 4, we have the following observations.

  1. 1.

    Complexities of NFS-\(\mathcal {A}\) are never worse than the complexities of NFS-GJL and NFS-Conjugation. Similarly, complexities of MNFS-\(\mathcal {A}\) are never worse than the complexities of MNFS-GJL and MNFS-Conjugation.

     
  2. 2.

    For both the NFS-\(\mathcal {A}\) and the MNFS-\(\mathcal {A}\) methods, increasing the value of r provides new complexity trade-offs.

     
  3. 3.

    There is a value of \(c_p\) for which the minimum complexity is achieved. This corresponds to the MNFS-Conjugation. Let \(L_Q(1/3,\theta _0)\) be this complexity. The value of \(\theta _0\) is determined later.

     
  4. 4.

    Let the complexity of the MNFS-GJL be \(L_Q(1/3,\theta _1)\). The value of \(\theta _1\) was determined in [24]. The plot for MNFS-\(\mathcal {A}\) approaches the plot for MNFS-GJL from below.

     
  5. 5.

    For smaller values of \(c_p\), it is advantageous to choose \(t>2\) or \(k>1\). On the other hand, for larger values of \(c_p\), the minimum complexity is attained for \(t=2\) and \(k=1\).

     
Fig. 4.

Complexity plot for boundary case

From the plot, it can be seen that for larger values of \(c_p\), the minimum value of \(c_b\) is attained for \(t=2\) and \(k=1\). So, we decided to perform further analysis using these values of t and k.

8.2 Analysis for \(t=2\) and \(k=1\)

Fix \(t=2\) and \(k=1\) and let us denote \(\mathbf {C}(c_p,2,1,r)\) as simply \(\mathbf {C}(c_p,r)\). Then from Theorem 4 the complexity of MNFS-\(\mathcal {A}\) for \(p=L_Q(2/3,c_p)\) is \(L_Q(1/3,\mathbf {C}(c_p,r))\) where
$$\begin{aligned} \mathbf {C}(c_p,r)=2c_b= & {} 2\,\sqrt{\frac{c_{p}}{3 \, {\left( r + 1\right) }} + \frac{{\left( 3 \, r + 2\right) } r}{36 \, c_{p}^{2}}} + \frac{2 \, r + 1}{3 \, c_{p}}. \end{aligned}$$
(32)
Figure 4 shows that for each \(r\ge 1\), there is an interval \([\epsilon _0(r),\epsilon _1(r)]\) such that for \(c_p\in [\epsilon _0(r),\epsilon _1(r)]\), \(\mathbf {C}(c_p,r)<\mathbf {C}(c_p,r^{\prime })\) for \(r\ne r^{\prime }\). For \(r=1\), we have
$$\begin{aligned} \epsilon _0(1) = \frac{1}{2} \, {\left( 4 \, \sqrt{11} + 11\right) }^{\frac{1}{3}} \approx 1.45;&\epsilon _1(1) = \left( 2 \, \sqrt{62} + \frac{31}{2}\right) ^{\frac{1}{3}} \approx 3.15. \end{aligned}$$
For \(p=L_Q(2/3,c_p)\), the complexity of MNFS-\(\mathcal {A}\) is same as the complexity of MNFS-Conj. for \(c_p\) in [1.45, 3.15]; for \(c_p > 3.15\), the complexity of MNFS-\(\mathcal {A}\) is lower than the complexity of all prior methods. The following result shows that the minimum complexity attainable by MNFS-\(\mathcal {A}\) approaches the complexity of MNFS-GJL from below.

Theorem 5

For \(r\ge 1\), let \(C(r)=\min _{c_p>0}\mathbf {C}(c_p,r)\). Then
  1. 1.

    \(C(1)=\theta _0=\left( \frac{146}{261} \, \sqrt{22} + \frac{208}{87}\right) ^{1/3}\).

     
  2. 2.

    For \(r\ge 1\), C(r) is monotone increasing and bounded above.

     
  3. 3.

    The limiting upper bound of C(r) is \(\theta _1=\left( \frac{2\times (13\sqrt{13}+46) }{ 27} \right) ^{1/3}\).

     

Proof

Differentiating \(\mathbf {C}(c_p,r)\) with respect to \(c_p\) and equating to 0 gives
$$\begin{aligned} \frac{\frac{6}{r + 1} - \frac{{\left( 3 \, r + 2\right) } r}{c_{p}^{3}}}{18 \, \sqrt{\frac{c_{p}}{3 \, {\left( r + 1\right) }} + \frac{{\left( 3 \, r + 2\right) } r}{36 \, c_{p}^{2}}}} - \frac{2 \, r + 1}{3 \, c_{p}^{2}}=0 \end{aligned}$$
(33)
On simplifying we get,
$$\begin{aligned} \frac{6c_p^3 - (3r+2)r(r+1) }{ \sqrt{\Big (12c_p^3 +(r+1)(3r+2)r \Big )(r+1)}} - \frac{2 \, r + 1}{1}=0 \end{aligned}$$
(34)
Equation (34) is quadratic in \(c_p^3\). On solving we get the following value of \(c_p\).
$$\begin{aligned} \nonumber c_p= & {} \left( \frac{7}{6} \, r^{3} + 2 \, r^{2} + \frac{1}{6} \, \sqrt{13 \, r^{2} + 8 \, r + 1} {\left( 2 \, r^{2} + 3 \, r + 1\right) } + r + \frac{1}{6}\right) ^{1/3}\\= & {} \rho (r)\;\left( \text {say}\right) . \end{aligned}$$
(35)
Putting the value of \(c_p\) back in (32), we get the minimum value of C (in terms of r) as
$$\begin{aligned} C(r)=2\,\sqrt{\frac{\rho (r)}{3 \, {\left( r + 1\right) }} + \frac{{\left( 3 \, r + 2\right) } r}{36 \, \rho (r)^{2}}} + \frac{2 \, r + 1}{3 \, \rho (r)}. \end{aligned}$$
(36)
All the three sequences in the expression for C(r), viz, \(\frac{\rho (r)}{3 \, {\left( r + 1\right) }}\), \( \frac{{\left( 3 \, r + 2\right) } r}{36 \, \rho (r)^{2}}\) and \( \frac{2 \, r + 1}{3 \, \rho (r)}\) are monotonic increasing. This can be verified through computation (with a symbolic algebra package) as follows. Let \(s_r\) be any one of these sequences. Then computing \(s_{r+1}/s_r\) gives a ratio of polynomial expressions from which it is possible to directly argue that \(s_{r+1}/s_r\) is greater than one. We have done these computations but, do not present the details since they are uninteresting and quite messy. Since all the three sequences \(\frac{\rho (r)}{3 \, {\left( r + 1\right) }}\), \( \frac{{\left( 3 \, r + 2\right) } r}{36 \, \rho (r)^{2}}\) and \( \frac{2 \, r + 1}{3 \, \rho (r)}\) are monotonic increasing so is C(r).
Note that for \(r\ge 1\), \(\rho (r)> (7/6)^{1/3}r > 1.05r\). So, for \(r>1\),
$$\begin{aligned} \frac{(3r+2)r}{\rho (r)^2}= & {} 3\left( \frac{r}{\rho (r)}\right) ^2 + 2\frac{r}{\rho (r)^2}< 3\times \left( \frac{1}{1.05}\right) ^2+2\times \frac{1}{1.05}. \\ \frac{(2r+1)}{\rho (r)}= & {} 2\frac{r}{\rho (r)} + \frac{1}{\rho (r)} < 2\times \frac{1}{1.05} + \frac{1}{1.05}. \end{aligned}$$
This shows that the sequences \(\frac{(3r+2)r}{\rho (r)^2}\) and \(\frac{(2r+1)}{\rho (r)}\) are bounded above. For \(r>8\), we have \((3r+1)< (8r+1) < r^2\) and \((2r^2 + r + 1/6) < r^3/3\) which implies that for \(r>8\), \(\rho (r)< (7/6 + 1/6\times \sqrt{14}\times 3 +1/3)^{1/3} r < 1.5\,r\). Using \(\rho (r) < 1.5 r \text { for } r > 8\), it can be shown that the sequence \(\left( \frac{\rho (r)}{r+1}\right) _{r>8}\) is bounded above by 1.5. Since the three constituent sequences \(\frac{\rho (r)}{{\left( r + 1\right) }}\), \( \frac{{\left( 3 \, r + 2\right) } r}{\rho (r)^{2}}\) and \( \frac{2 \, r + 1}{ \rho (r)}\) are bounded above, it follows that C(r) is also bounded above. Being monotone increasing and bounded above C(r) is convergent. We claim that
$$\begin{aligned} \displaystyle \lim \limits _{r\rightarrow \infty } C(r)= \left( \frac{2\times (13\sqrt{13}+46) }{ 27} \right) ^{1/3}. \end{aligned}$$
The proof of the claim is the following. Using the expression for \(\rho (r)\) from (35) we have \(\displaystyle \lim \limits _{r\rightarrow \infty } \frac{\rho (r)}{r}= {\left( \frac{2}{6} \, \sqrt{13} + \frac{7}{6}\right) }^{\frac{1}{3}}\). Now,
$$\begin{aligned} C(r)=2\,\sqrt{\frac{\rho (r)/r}{3 \, {\left( 1 + 1/r\right) }} + \frac{{\left( 3 + 2/r\right) } }{36 \, \rho (r)^{2}/r^2}} + \frac{2 + 1/r}{3 \, \rho (r)/r}. \end{aligned}$$
(37)
Hence,
$$\begin{aligned} \lim \limits _{r\rightarrow \infty } C(r)= & {} 2\,\sqrt{ \frac{(2\sqrt{13}+7)^{1/3}}{3\times 6^{1/3}} + \frac{3\times 6^{2/3}}{36 \,(2\sqrt{13}+7)^{2/3}} } + \frac{2\times 6^{1/3}}{3 \,(2\sqrt{13}+7)^{1/3}}\\ \end{aligned}$$
After further simplification, we get
$$\begin{aligned} \lim \limits _{r\rightarrow \infty } C(r)= & {} \left( \frac{2\times (13\sqrt{13}+46) }{ 27} \right) ^{1/3}. \end{aligned}$$
The limit of C(r) as r goes to infinity is the value of \(\theta _1\) where \(L_Q(1/3,\theta _1)\) is the complexity of MNFS-GJL as determined in [24]. This shows that as r goes to infinity, the complexity of MNFS-\(\mathcal {A}\) approaches the complexity of MNFS-GJL from below.
We have already seen that C(r) is monotone increasing for \(r\ge 1\). So, the minimum value of C(r) is obtained for \(r=1\). After simplifying C(1), we get the minimum complexity of MNFS-\(\mathcal {A}\) to be
$$\begin{aligned} L_Q\left( 1/3, \left( \frac{146}{261} \, \sqrt{22} + \frac{208}{87}\right) ^{1/3} \right) =L\left( 1/3,1.7116 \right) . \end{aligned}$$
(38)
This minimum complexity is obtained at \(c_p=\rho (1)=\left( \sqrt{22}+\frac{13}{3}\right) ^{1/3}= 2.0819\).    \(\square \)

Note 1

As mentioned earlier, for \(r=k=1\), the new method of polynomial selection becomes the Conjugation method. So, the minimum complexity of MNFS-\(\mathcal {A}\) is the same as the minimum complexity for MNFS-Conjugation. Here we note that the value of the minimum complexity given by (38), is not same as the one reported by Pierrot in [24]. This is due to an error in the calculation in [24]2.

Complexity of NFS- \(\mathcal {A}\) : From Fig. 4, it can be seen that there is an interval for \(c_p\) for which the complexity of NFS-\(\mathcal {A}\) is better than both MNFS-Conjugation and MNFS-GJL. An analysis along the lines as done above can be carried out to formally show this. We skip the details since these are very similar to (actually a bit simpler than) the analysis done for MNFS-\(\mathcal {A}\). Here we simply mention the following two results:
  1. 1.

    For \(c_p\ge {\left( 2 \, \sqrt{89} + 20\right) }^{\frac{1}{3}}\approx 3.39\), the complexity of NFS-\(\mathcal {A}\) is better than that of MNFS-Conjugation.

     
  2. 2.

    For \(c_p\le \frac{1}{8} \, \sqrt{390} \sqrt{{\left( 5 \, \sqrt{13} - 18\right) } {\left( \frac{26}{27} \, \sqrt{13} + \frac{92}{27}\right) }^{\frac{1}{3}}} + \frac{45}{8} \, {\left( \frac{26}{27} \, \sqrt{13} + \frac{92}{27}\right) }^{\frac{2}{3}} \approx 20.91\), the complexity of NFS-\(\mathcal {A}\) is better than that of MNFS-GJL.

     
  3. 3.

    So, for \(c_p\in [3.39,20.91]\), the complexity of NFS-\(\mathcal {A}\) is better than the complexity of all previous method including the MNFS variants.

     
Current state-of-the-art: The complexity of MNFS-\(\mathcal {A}\) is lower than that of NFS-\(\mathcal {A}\). As mentioned earlier (before Table 4) the interval (0, 1.12] is the union of \(\cup _{t\ge 3}S(t,1,1)\). This fact combined with Theorem 5 and Table 4 show the following. For \(p=L_Q(2/3,c_p)\), when \(c_p\in (0,1.12]\cup [1.45,3.15]\), the complexity of MNFS-\(\mathcal {A}\) is the same as that of MNFS-Conjugation; for \(c_p\notin (0,1.12]\cup [1.45,3.15]\) and \(c_p>0\), the complexity of MNFS-\(\mathcal {A}\) is smaller than all previous methods. Hence, MNFS-\(\mathcal {A}\) should be considered to provide the current state-of-the-art asymptotic complexity in the boundary case.

8.3 Medium and Large Characteristic Cases

In a manner similar to that used to prove Theorem 4, it is possible to work out the complexities for the medium and large characteristic cases of the MNFS corresponding to the new polynomial selection method. To tackle the medium prime case, the value of t is taken to be \(t=c_tn\left( (\ln Q)(\ln \ln Q)\right) ^{-1/3}\) and to tackle the large prime case, the value of r is taken to be \(r=c_r/2\left( (\ln Q)(\ln \ln Q)\right) ^{1/3}\). This will provide a relation between \(c_b,c_v\) and r (for the medium prime case) or t (for the large prime case). The method of Lagrange multipliers is then used to find the minimum value of \(c_b\). We have carried out these computations and the complexities turn out to be the same as those obtained in [24] for the MNFS-GJL (for large characteristic) and the MNFS-Conjugation (for medium characteristic) methods. Hence, we do not present these details.

9 Conclusion

In this work, we have proposed a new method for polynomial selection for the NFS algorithm for fields \(\mathbb {F}_{p^n}\) with \(n>1\). Asymptotic analysis of the complexity has been carried out both for the classical NFS and the MNFS algorithms for polynomials obtained using the new method. For the boundary case with \(p=L_Q(2/3,c_p)\) for \(c_p\) outside a small set, the new method provides complexity which is lower than all previously known methods.

Footnotes

  1. 1.

    The value of \(\theta _0\) obtained in [24] is incorrect.

  2. 2.

    The error is the following. The solution for \(c_b\) to the quadratic \((18t^2c_p^2)c_b^2-(36tc_p)c_b+8-3t^2(t-1)c_p^3=0\) is \(c_b=1/(tc_p) + \sqrt{5/(9(c_pt)^2)+(c_p(t-1))/6}\) with the positive sign of the radical. In [24], the solution is erroneously taken to be \(1/(tc_p) + \sqrt{5/((9c_pt)^2)+(c_p(t-1))/6}\).

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Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.Applied Statistics UnitIndian Statistical InstituteKolkataIndia

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