Faster relaxed multiplication by Joris van der Hoeven CNRS, Laboratoire d'informatique École polytechnique 91128 Palaiseau Cedex France Email: joris@texmacs.org November 3, 2023

Abstract

In previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up to a given order . The fastest currently known algorithm works over an effective base field with sufficiently many -th roots of unity and has algebraic time complexity . In this paper, we will generalize this algorithm to the cases when is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a -module and comes with an additional algorithm for partial division by integers. In particular, we may take to be any effective field. We will also present an asymptotically faster algorithm for relaxed multiplication of -adic numbers.

Keywords: power series, multiplication, on-line algorithm, FFT, computer algebra

A.M.S. subject classification: 68W30, 30B10, 68W25, 33F05, 11Y55, 42-04

1Introduction

1.1Relaxed resolution of recursive equations

Let be an effective (possibly non-commutative) ring; i.e., we assume data structures for representing the elements of and algorithms for performing the ring operations , and . The aim of algebraic complexity theory is to study the cost of basic or more complex algebraic operations over (such as the multiplication of polynomials or matrices) in terms of the number of operations in .

The algebraic complexity usually does not coincide with the bit complexity, which also takes into account the potential growth of the actual coefficients in . Nevertheless, understanding the algebraic complexity usually constitutes a first useful step towards understanding the bit complexity. Of course, in the special case when is a finite field, both complexities coincide up to a constant factor.

One of the most central operations is polynomial multiplication. We will denote by the number of operations required to multiply two polynomials of degrees in . If admits primitive -th roots of unity for all , then we have using FFT multiplication, which is based on the fast Fourier transform [12]. In general, it has been shown [28, 10] that . The complexities of most other operations (division, Taylor shift, extended g.c.d., multipoint evaluation, interpolation, etc.) can be expressed in terms of . Often, the cost of such other operations is simply , where stands for ; see [2, 6, 15] for some classical results along these lines.

The complexity of polynomial multiplication is fundamental for studying the cost of operations on formal power series in up to a given truncation order . Clearly, it is possible to perform the multiplication up to order in time : it suffices to multiply the truncated power series at order and truncate the result. Using Newton's method, and assuming that , it is also possible to compute , , etc. up to order in time . More generally, it has been shown in [9, 17, 29, 23] that the power series solutions of algebraic differential equations with coefficients in can be computed up to order in time . However, in this case, the “” hides a non-trivial constant factor which depends on the expression size of the equation that one wants to solve.

The relaxed approach for computations with formal power series makes it possible to solve equations in quasi-optimal time with respect to the sparse expression size of the equations. The idea is to consider power series as streams of coefficients and to require that all operations are performed “without delay” on these streams. For instance, for a multiplication of two power series , we require that is computed as soon as are known. Any algorithm which has this property will be called a relaxed or on-line algorithm for multiplication.

Given a relaxed algorithm for multiplication, it is possible to let the later coefficients of the input depend on the known coefficients of the output. For instance, given a power series with , we may compute using the formula

(1)

provided that . Indeed, extraction of the coefficient of in and yields

and only depends on . More generally, we define an equation of the form

(2)

to be recursive, if only depends on . Replacing by , we notice that the same terminology applies to systems of equations. In the case of an implicit equation, special rewriting techniques can be implied in order to transform the input equation into a recursive equation [24, 22, 5].

Let denote the cost of performing a relaxed multiplication up to order . If is an expression which involves multiplications and other “linear time” operations (additions, integrations, etc.), then it follows that (2) can be solved up to order in time . If we had , then this would yield an optimal algorithm for solving (2) in the sense that the computation of the solution would essentially require the same time as its verification.

1.2Known algorithms for relaxed multiplication

The naive algorithm for computing , based on the formula

is clearly relaxed. Unfortunately, FFT multiplication is not relaxed, since are computed simultaneously as a function of , in this case.

In [16, 17] it was remarked that Karatsuba's algorithm [25] for multiplying polynomials can be rewritten in a relaxed manner. Karatsuba multiplication and its relaxed version thus require exactly the same number of operations. In [16, 17], an additional fast relaxed algorithm was presented with time complexity

(3)

We were recently made aware of the fact that a similar algorithm was first published in [13]. However, this early paper was presented in a different context of on-line (relaxed) multiplication of integers (instead of power series), and without the application to the resolution of recursive equations (which is quite crucial from our perspective).

An interesting question remained: can the bound (3) be lowered further, be it by a constant factor? In [18], it was first noticed that an approximate factor of two can be gained if one of the multiplicands is known beforehand. For instance, if we want to compute for a known series with , then the coefficients of are already known in the product in (1), so only one of the inputs depends on the output. An algorithm for the computation of is said to be semi-relaxed, if is written to the output as soon as are known, but all coefficients of are known beforehand. We will denote by the complexity of semi-relaxed multiplication. We recall from [21] (see also Section 3) that relaxed multiplication reduces to semi-relaxed multiplication:

It has recently been pointed out [26] that the factor that was gained in the case of semi-relaxed multiplication can also be gained in the general case.

The first reduction of (3) by a non-constant factor was published in [21], and uses the technique of FFT blocking (which has also been used for the multiplication of multivariate polynomials and power series in [17, Section 6.3], and for speeding up Newton iterations in [3, 23]). Under the assumption that admits primitive -th roots of unity for all (or at least for all with ), we showed that

(4)

The function has slower growth than any strictly positive power of . It is convenient to write whenever for all . In particular, it follows that

In Section 3, we will recall the main ideas from [21] which lead to the complexity bound (4).

In fact, in Section 4, we will see that the complexity bound from [21] can be further reduced to

Since such complexity bounds involve rather complex expressions, it will be convenient to use the following abbreviations:

Clearly, .

1.3Improved complexity bounds

We recall that the characteristic of a ring is the integer such that the canonical ring homomorphism has kernel . If is torsion-free as a -module (i.e. for any and ), then we will say that admits an effective partial division by integers if there exists an algorithm which takes and on input and which returns the unique with on output. We will count a unit cost for such divisions. The main result of this paper is:

Theorem 1. Assume that one of the following two holds:

Then we have

(5)

We notice that the theorem holds in particular if is an effective field. In Section 8, we will also consider the relaxed multiplication of -adic numbers, with and . If we denote by the bit complexity of multiplying two -bit integers, then [13] essentially provided an algorithm of bit complexity for the relaxed multiplication of -adic numbers at order . Various algorithms and benchmarks for more general were presented in [4]. It is also well known [33, 11, 28, 14] that . Let denote the bit complexity of the relaxed multiplication of two -adic numbers modulo . In Section 8, we will prove the following new result:

Theorem 2. Let with . Then we have

uniformly in and .

For comparison, the best previously known bound for relaxed multiplication in was

We thus improved the previous bound by a factor at least, up to sublogarithmic terms.

The main idea which allows for the present generalizations is quite straightforward. In our original algorithm from [21], the presence of sufficiently many primitive -th roots of unity in gives rise to a quasi-optimal evaluation-interpolation strategy for the multiplication of polynomials. More precisely, given two polynomials of degrees , their FFT-multiplication only requires evaluation and interpolation points, and both the evaluation and interpolation steps can be performed efficiently, using only operations. Now it has recently been shown [8] that quasi-optimal evaluation-interpolation strategies still exist if we evaluate and interpolate at points in geometric progressions instead of roots of unity. This result is the key to our new complexity bounds, although further technical details have to be dealt with in order to make things work for various types of effective rings . We also notice that the main novelty of [8] concerns the interpolation step. Fast evaluation at geometric progressions was possible before using the so called chirp transform [27, 7]. For effective rings of positive characteristic, this would actually have been sufficient for the proving the bound (5).

Our paper is structured as follows. Since the algorithms of [8] were presented in the case when is an effective field, Section 2 starts with their generalization to more general effective rings . These generalizations are purely formal and contain no essentially new ideas. In Section 3, we give a short survey of the algorithm from [21], but we recommend reading the original paper for full technical details. In Section 4, we sharpen the complexity analysis for the algorithm from [21]. In Section 5, we prove Theorem 1 in the case when has characteristic zero. In Section 6, we turn our attention to the case when has prime characteristic . If the characteristic is sufficiently large, then we may find sufficiently large geometric progressions in in order to generalize the results from Section 5. Otherwise, we have to work over for some sufficiently large . In Section 7, we complete the proof of Theorem 1; the case when the characteristic is a prime power is a refinement of the result from Section 6. The remaining case is done via Chinese remaindering. In Section 8, we will prove Theorem 2.

Acknowledgments. I am grateful to the referees for their detailed comments and suggestions. A special thanks goes to the referee who made a crucial suggestion for the improved complexity analysis in Section 4.

2Multipoint evaluation and interpolation

Let be a (commutative) effective integral domain and let be its quotient field. Assume that has characteristic zero. Let be an effective torsion-free -module and . Elements of and are fractions with (resp. ) and , and the operations on such fractions are as usual:

For , we also have . It follows that is an effetive field and an effective -vector space. Moreover, all field operations in (and all vector space operations in ) can be performed using only operations in (resp. or ).

We will say that admits an effective partial division, if for every and , we can compute the unique with . In that case, and we will count any division of the above kind as one operation in . Similarly, given a fixed , we say that admits an effective partial division by , we can compute the unique with . Given , we define

Given and , we will denote by the number of operations in and which are needed in order to compute the product .

Lemma 3. Let be an effective integral domain and an effective torsion-free -module. There exists a constant , such that the following holds: for any , and such that are pairwise distinct, and such that admits effective divisions by and , we have:

Proof. In the case when is a field and , this result was first proven in [8]. More precisely, the conversions can be done using the algorithms NewtonEvalGeom, NewtonInterpGeom, NewtonToMonomialGeom and MonomialToNewtonGeom in that paper. Examining these algorithms, we observe that general elements in are only multiplied with elements in and divided by elements of the set . In particular, the algorithms can still be applied in the more general case when is a field and a vector space.

If is only an effective integral domain and an effective torsion-free -module with an effective partial division, then we define the effective field and the effective vector space as above, and we may still apply the generalized algorithms for multipoint evaluation and interpolation in . In particular, both multipoint evaluation and interpolation can still be done using operations in and , whence operations in and . If we know that the end-results of these algorithms are really in the subspace of (or in the submodule of ), then we use the partial division in to replace their representations in (or ) by representations in (or ).

3Survey of blockwise relaxed multiplication

Let be an effective (possibly non-commutative) ring and recall that

Given a power series and , we will also use the notations

The fast relaxed algorithms from [21] are all based on two main changes of representation: “blocking” and the fast Fourier transform. Let us briefly recall these transformations and how to use them for the design of fast algorithms for relaxed multiplication.

Blocking and unblocking.

Given a block size , the first operation of blocking rewrites a power series as a series in with coefficients in

Given , we may then compute using

where and

Discrete Fourier transforms.

Assume now that , that , and that admits a primitive -th root of unity . Then the discrete Fourier transform provides us with an isomorphism

and it is classical [12] that both and can be computed using operations in . The operations and extend naturally to via

Given , this allows us to compute using the formula

(6)

where is a pointwise product in . The first coefficients of can be computed using at most operations in .

Relaxed multiplication.

In formula (6) the -th coefficient of the right hand side may depend on the -th coefficients of and . In order to make (6) suitable for relaxed multiplication, we have to treat the first coefficients of and separately. Indeed, the formula

allows for the relaxed computation of at order using at most

operations in . Similarly, the formula

			(7)

allows for the semi-relaxed computation of at order using at most

(8)

operations in . For a given expansion order , one may take , and use the above formula in a recursive manner. This yields [21, Theorem 11]

Remark 4. Since the block size is chosen as a function of , the above method really describes a relaxed algorithm for computing the product up to an order which is specified in advance. In fact, such an algorithm automatically yields a genuine relaxed algorithm with the same complexity (up to a constant factor), by doubling the order each time when needed.

Reduction to semi-relaxed multiplication.

In the above discussion, we both provided bounds for and . In fact, there exists a straightforward reduction of relaxed multiplication to semi-relaxed multiplication. First of all, the relaxed multiplication of two power series up to order clearly reduces to the relaxed multiplication of the two polynomials and up to order . Now the formula

shows that a relaxed product of two polynomials and of degrees reduces to a relaxed product of half the size, two semi-relaxed products , , and one non-relaxed product . Under the assumptions that and are increasing, a routine calculation thus yields

Multiple block sizes.

Instead of using a single block size , one may use several block sizes. Applying this technique, we proved in [21, Theorem 12] that

			(9)

4Improved complexity analysis

One of the referees suggested to take instead of in (8). As a matter of fact, it is even better to take . In this section, we will show that this leads to the bound

which further improves on (9). We will prove a slightly more general result, which is useful for the analysis of algorithms which satisfy complexity bounds similar to (8).

Lemma 5. Let be an increasing function with and let

where . Then there exist constants and such that for all , , and , we have

Proof. Notice that and for sufficiently large . We have

For a suitable constant and all sufficiently large , it follows that

(10)

We also have

whence

For all sufficiently large , it follows that

Consequently,

(11)

Adding up (10) and (11), we obtain

for all sufficiently large .

Theorem 6. Let be an increasing function with . Assume that

(12)

for all sufficiently large , where

Then

Proof. We define functions by

and let , and be as in Lemma 5. Without loss of generality, we may pick sufficiently large such that and such that (12) holds for . Let be such that and denote and . For certain , we have and . Now (12) implies

Let and . Let us prove by induction that for all . This is clear for . Assume now that and for all . Then, with the above notations, implies and , whence

as desired. For all , we have thus shown that .

5Relaxed multiplication in characteristic zero

Let us now consider the less favourable case when is an effective ring which does not necessarily contain primitive -th roots of unity for arbitrarily high . In this section, we will first consider the case when is torsion-free as a -module and also admits a partial algorithm for division by integers.

Given a block size and (say ), we will replace the discrete Fourier transform at a -th primitive root of unity by multipoint evaluation at . More precisely, we define

and the inverse transform . By Lemma 3, these transforms can both be computed using operations in . In a similar way as for and , we extend and to power series in .

Theorem 7. Let both be an effective ring and an effective torsion-free -module with an effective partial division by elements in . Then

Proof. It suffices to prove the complexity bound for semi-relaxed multiplication. Instead of (7), we now compute using

			(13)

The bound (8) then has to be replaced by

(14)

Plugging in the bound from [10] and applying Theorem 6, the result follows.

6Relaxed multiplication in prime characteristic

Let now be an effective ring of prime characteristic . For expansion orders , the ring does not necessarily contain distinct points in geometric progression. Therefore, in order to apply Lemma 3, we will first replace by a suitable extension, in which we can find sufficiently large geometric progressions.

Given , let be even with . Let be such that the finite field is isomorphic to . Then the ring

has dimension over as a vector space, so we have a natural -linear bijection

The ring is an effective ring and one addition or subtraction in corresponds to additions or subtractions in . Similarly, one multiplication in can be done using operations in .

In order to multiply two series up to order , the idea is now to rewrite and as series in with . If we want to compute the relaxed product, then we also have to treat the first coefficients apart, as we did before for the blocking strategy. More precisely, we will compute the semi-relaxed product using the formula

where we extended to in the natural way:

From the complexity point of view, we get

(15)

Since contains a copy of , it also contains at least points in geometric progression. For the multiplication up to order of two series with coefficients in , we may thus use the blocking strategy combined with multipoint evaluation and interpolation.

Theorem 8. Let be an effective ring of prime characteristic . Then

Proof. With the notations from above, we may find a primitive -th root of unity in . We may thus use formula (13) for the semi-relaxed multiplication of two series in up to order . In a similar way as in the proof of Theorem 7, we thus get

Using classical fast relaxed multiplication [16, 17, 13], we also have

whence (15) simplifies to

(16)

Since and , the result follows.

Remark 9. As long as , then in (16), so the bound further reduces into

Remark 10. In our complexity analysis, we have not taken into account the computation of the polynomial with . Using a randomized algorithm, such a polynomial can be computed in time ; see [15, Corollary 14.44]. If , then this is really a precomputation of negligible cost .

If we insist on computing in a deterministic way, then one may use [30, Theorem 3.2], which provides us with an algorithm of time complexity .

Similarly, there both exist randomized and deterministic algorithms [32, 31] for the efficient computation of primitive -th roots of unity in . In particular, thanks to [31, Theorem 1] such a primitive root of unity can always be computed in time , when using a naive algorithm for factoring .

7Relaxed multiplication in positive characteristic

Let us now show that the technique from the previous section actually extends to the case when is an arbitrary effective ring of positive characteristic. We first show that the algorithm still applies when the characteristic of is a prime power. We then conclude by showing how to apply Chinese remaindering in our setting.

Theorem 11. Let be an effective ring of prime power characteristic . Then

Proof. Taking , let of degree be as in the previous section and pick a monic polynomial of degree in such that the reduction of modulo yields . Then we get a natural commutative diagram

where stands for reduction modulo . In particular, we have an epimorphism

with .

Now let be an element in of order . Then any lift of with has order at least . Moreover, and are all invertible. Consequently, and do not lie in , whence they are invertible as well. It follows that we may still apply multipoint evaluation and interpolation in at the sequence , whence Theorem 8 generalizes to the present case.

Remark 12. For a fixed prime number , we notice that the complexity bound is uniform in the following sense: there exists a constant such that for all effective rings of characteristic with , we have . Indeed, the choice of only depends on and , and any operation in or in the case corresponds to exactly one lifted operation in or in the general case.

Remark 13. Similarly as in Remark 9, the hypothesis leads to the improved bound

This bound is uniform in a similar way as in Remark 12.

Theorem 14. Let be an effective ring of non-zero characteristic . Then

Proof. We will prove the theorem by induction on the number of prime divisors of . If is a prime power, then we are done. So assume that , where and are relatively prime, and let be such that

Then we may consider the rings

These rings are effective, when representing their elements by elements of and transporting the operations from . Of course, the representation of an element of (or ) is not unique, since we may replace it by for any (or ). But this is not a problem, since our definition of effective ring did not require unique representability or the existence of an equality test.

Now let and let be their projections in , for . Consider the relaxed products , for . These products are represented by relaxed series via , for . By the induction hypotheses, we may compute and at order using operations in . The linear combination can still be expanded up to order with the same complexity. We claim that . Indeed,

Summing both relations, our claim follows.

Remark 15. A uniform bound interms of can be given along similar lines as in Remark 12. This time, such a bound depends linearly on the number of prime factors of .

8Relaxed multiplication of -adic numbers

Let be an integer, not necessarily a prime number, and denote . We will regard -adic numbers as series with , and such that the basic ring operations , and require an additional carry treatment.

In order to multiply two relaxed -adic numbers , we may rewrite them as series , multiply these series , and recover the product from the result. Of course, the coefficients of may exceed , so some relaxed carry handling is required in order to recover from . We refer to [4, Section 2.7] for details. In particular, we prove there that can be computed up to order using ring operations in of bit size .

Given , let , and consider two power series . We will denote by (resp. ) the bit complexity of multiplying and up to order using a relaxed (resp. semi-relaxed) algorithm.

Lemma 16. We have

Proof. Let be a prime number with . Such a prime can be computed in time using the polynomial time polynomial primality test from [1], together with the Bertrand-Chebychev theorem.

Let and consider such that . Let be the reductions of modulo . Then may be reconstructed up to order from the product . We thus get

By Theorem 11 and Remarks 12 and 13, while using the fact that , we have

and this bound is uniform in . Since , the result follows.

For the above strategy to be efficient, it is important that . This can be achieved by combining it with the technique of -adic blocking. More precisely, given a -adic block size , then any -adic number in can naturally be considered as a -adic number in , and vice versa. Assuming that numbers in are written in base , the conversion is trivial if is a power of two. Otherwise, the conversion involves base conversions and we refer to [4, Section 4] for more details. In particular, the conversions in both directions up to order can be done in time .

Let (resp. ) the complexity of relaxed (resp. semi-relaxed) multiplication in up to order .

Theorem 17. Setting , we have

Proof. Let , so that

Using the strategy of -adic blocking, a semi-relaxed product in may then be reduced to one semi-relaxed product in and one relaxed multiplication with an integer in . In other words,

where stands for the cost of semi-relaxed multiplication of two -adic numbers in up to order . By [4, Proposition 4], we have

By Lemma 16, we also have

Notice that

which completes the proof of the theorem.

9Final remarks

For the moment, we have not implemented any of the new algorithms in practice. Nevertheless, our old implementation of the algorithm from [21] allowed us to gain some insight on the practical usefulness of blockwise relaxed multiplication. Let us briefly discuss the potential impact of the new results for practical purposes.

Characteristic zero.

In the case of integer coefficients, it is best to re-encode the integers as polynomials in for a prime number which fits into a machine word and such that admits many -th roots of unity (it is also possible to take several primes and use Chinese remaindering). After that, one may again use the old algorithm from [21]. Also, integer coefficients usually grow in size with , so one really should see the power series as a bivariate power series in with a triangular support. One may then want to use TFT-style multiplication [19, 20] in order to gain another constant factor.

Finite fields.

For large finite fields, it is easy to find large geometric progressions, so the algorithms of this paper can be applied without the need to consider field extensions. Moreover, for finite fields of the form with sufficiently large and , it is possible to choose , thereby speeding up evaluation and interpolation. For small finite fields of the form , it is generally necessary to make the initial investment of working in a larger ring with sufficiently large geometric progressions. Of course, instead of the ring extensions considered in Section 6, we may directly use field extensions of the form with .

Semi-relaxed multiplication.

In principle, a factor 2 can be gained in the semi-relaxed (resp. general) case using the technique from [18] (resp. [26]). Unfortunately, the middle product (resp. the FFT-trick used in [26]) is not always easy to implement. For instance, if we rely on Kronecker substitution for multiplications in , then we will need to implement an ad hoc analogue for the middle product. Since we did not use a fast algorithm for middle products for our benchmarks in [21, Section 5], the timings for the semi-relaxed product in were only about instead of better than the timings for the fully relaxed product. Nevertheless, modulo increased implementation efforts, we stress that a gain should be achievable.

Cache friendliness.

Bilinear maps.

In order to keep the presentation reasonably simple, we have focussed on the case when is an effective ring. In fact, a more general setting for relaxed multiplication is to consider a bilinear mapping , where , and are effective -modules, and extend it into a mapping by . Under suitable hypothesis, the algorithms in this paper generalize to this setting.

Skew series.

The relaxed approach can also be generalized to the case when the coefficients of the power series are operators which commute with monomials in a non-trivial way. More precisely, assume that we have an effective ring homomorphism such that for all . For instance, one may take with , so that . Given a commutation rule of this kind, we define a skew multiplication on by

If denotes the cost of multiplying two polynomials and in with , then the classical fast relaxed multiplication algorithm from [17, 13] generalizes and still admits the time complexity . However, the blockwise algorithm from this paper does not generalize to this setting, at least not in a straightforward way.

Bibliography