Alberto Zanoni - Academia.edu (original) (raw)

Papers by Alberto Zanoni

We consider the multiplication of long integers when one factor is much bigger than the other one... more We consider the multiplication of long integers when one factor is much bigger than the other one. We describe an iterative approach using Toom-Cook unbalanced methods, by evaluating the smaller factor just once. The particular case of Toom-2.5 is considered in full detail, and a further optimization depending on the parity of shortest operand evaluation in 1 is described. A comparison with GMP library is also presented.

Iterative Toom-Cook methods for very unbalanced long integer multiplication, Jul 25, 2010

We consider the multiplication of long integers when one factor is much bigger than the other one... more We consider the multiplication of long integers when one factor is much bigger than the other one. We describe an iterative approach using Toom-Cook unbalanced methods, by evaluating the smaller factor just once.
The particular case of Toom-2.5 is considered in full detail, and a further optimization depending on the parity of shortest operand evaluation in 1 is described. A comparison with GMP library is also presented.

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis comp... more In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known. We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms. We suggest some algorithms to deal with the case of overdetermined and unstable systems.

… --AES: 4th international conference, AES 2004, …, 2005

An Algebraic Interpretation of AES− 128 Ilia Toli and Alberto Zanoni Dipartimento di Matematica L... more An Algebraic Interpretation of AES− 128 Ilia Toli and Alberto Zanoni Dipartimento di Matematica Leonida Tonelli, Universita di Pisa, Via Buonarroti 2, 56127 Pisa, Italy {toli, zanoni}@ posso. dm. unipi. it Abstract. We analyze an algebraic representation of AES− 128 as an ...

Preprint

Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. Th... more Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A definitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. In this paper we present some particular examples of useful matrices and a method to generate automatically, by means of optimised exhaustive searches on a graph, the best sequence of basic operations to invert them.

This work deals with Karatsuba method for multivariate polynomials, not recursing on variables nu... more This work deals with Karatsuba method for multivariate polynomials, not recursing on variables number, but using an iterative scheme, with an eye to a better parallelism exploitation. Integers base 2 and 3 expansions are used in order to access the needed data. AMS Subject Classification: 11A05, 11A25, 11K65, 11Y70

Multiplication of univariate dense polynomials with long integer unbalanced (having different len... more Multiplication of univariate dense polynomials with long integer unbalanced (having different lengths) coefficients is considered. By reducing the problem to the product of bivariate polynomials with balanced coefficients, Toom–Cook approach is shown, pointing out some optimizations in order to reduce the computational cost. As a byproduct, univariate sparse Toom–Cook is also sketched. Lastly, some experimental results concerning performance comparisons are presented.

Karatsuba and Toom-Cook are well-known methods used to efficiently multiply univariate polynomial... more Karatsuba and Toom-Cook are well-known methods used to efficiently multiply univariate polynomials and long integers. For multivariate polynomials, asymptotically good approaches like Kronecker’s trick combined with FFT become truly effective only when the degree is above some threshold. In this paper we analyze Karatsuba and some of Toom-Cook methods for multivariate polynomials, considering density in a different way with respect to Kronecker, and present some algorithms for fast multivariate polynomial multiplication in practical cases, when degrees are not huge. A fast sparse polynomial multiplication algorithm is also proposed. 2000 Mathematics Subject Classification: 11A05, 11A25, 11K65, 11Y70

ACM SIGSAM Bulletin, 2000

This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert... more This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' Gröbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gröbner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Acm Sigsam Bulletin, 2000

This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert... more This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' Gröbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gröbner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02, 2002

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis comp... more In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known. We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms. We suggest some algorithms to deal with the case of overdetermined and unstable systems.

Computer Algebra in Scientific Computing, 2012

A new approach for the computation of long integer cube (third power) based on a splitting-in-two... more A new approach for the computation of long integer cube (third power) based on a splitting-in-two divide et impera approach and on a modified Toom-Cook-3 unbalanced method is presented, showing that the "classical" square-and-multiply algorithm is not (always) optimal. The new algorithm is used as a new basic tool to improve long integer exponentiation: different techniques combining binary and ternary exponent expansion are shown. Effective implementations by using the GMP library are tested, and performance comparisons are presented.

Lecture Notes in Computer Science, 2006

In Gröbner bases computation, as in other algorithms in commutative algebra, a general open quest... more In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial-and-error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm.

2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2011

In this paper the problem of univariate polynomial evaluation is considered. When both polynomial... more In this paper the problem of univariate polynomial evaluation is considered. When both polynomial coefficients and the evaluation "point" are integers, unbalanced multiplications (one factor having many more digits than the other one) in classical Ruffini-Horner rule do not let computations completely benefit of sub quadratic methods, like Karatsuba, Toom-Cook and Schonhage-Strassen's. We face this problem by applying an approach

Lecture Notes in Computer Science, 2005

We analyze an algebraic representation of AES−128 as an embedding in BES, due to Murphy and Robsh... more We analyze an algebraic representation of AES−128 as an embedding in BES, due to Murphy and Robshaw. We present two systems of equations S and K concerning encryption and key generation processes. After some simple but rather cumbersome substitutions, we should obtain two new systems C1 and C2. C1 has 16 very dense equations of degree up to 255 in each of its 16 variables. With a single pair (p, c), with p a cleartext and c its encryption, its roots give all possible keys that should encrypt p to c. C2 may be defined using 11 or more pairs (p, c), and has 16 times as many equations in 176 variables. K and most of S is invariant for all key choices.

IFIP International Federation for Information Processing

We analyze an algebraic representation of AES-128 as an embedding in BES, due to Murphy and Robsh... more We analyze an algebraic representation of AES-128 as an embedding in BES, due to Murphy and Robshaw. We present two systems of equations S and K concerning encryption and key generation processes. After some simple but rather cumbersome substitutions, we should obtain two new systems C1 and C2. C1 has 16 very dense equations of degree up to 255 in each of its 16 variables. With a single pair (p, c), with p a cleartext and c its encryption, its roots give all possible keys that should encrypt p to c. C2 may be defined using 11 or more pairs (p, c), and has 16 times as many equations in 176 variables. K and most of S is invariant for all key choices.

2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2009

... GMP represents long integers using arrays of so-called “limbs” (β-bits unsigned C integers – ... more ... GMP represents long integers using arrays of so-called “limbs” (β-bits unsigned C integers – a typical value for β is eg 32). ... VIII. CONCLUSION In this paper we presented a description andimplementation of Toom-Cook 8-way method for long integers multiplication and ...

We consider the multiplication of long integers when one factor is much bigger than the other one... more We consider the multiplication of long integers when one factor is much bigger than the other one. We describe an iterative approach using Toom-Cook unbalanced methods, by evaluating the smaller factor just once. The particular case of Toom-2.5 is considered in full detail, and a further optimization depending on the parity of shortest operand evaluation in 1 is described. A comparison with GMP library is also presented.

Iterative Toom-Cook methods for very unbalanced long integer multiplication, Jul 25, 2010

We consider the multiplication of long integers when one factor is much bigger than the other one... more We consider the multiplication of long integers when one factor is much bigger than the other one. We describe an iterative approach using Toom-Cook unbalanced methods, by evaluating the smaller factor just once.
The particular case of Toom-2.5 is considered in full detail, and a further optimization depending on the parity of shortest operand evaluation in 1 is described. A comparison with GMP library is also presented.

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis comp... more In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known. We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms. We suggest some algorithms to deal with the case of overdetermined and unstable systems.

… --AES: 4th international conference, AES 2004, …, 2005

An Algebraic Interpretation of AES− 128 Ilia Toli and Alberto Zanoni Dipartimento di Matematica L... more An Algebraic Interpretation of AES− 128 Ilia Toli and Alberto Zanoni Dipartimento di Matematica Leonida Tonelli, Universita di Pisa, Via Buonarroti 2, 56127 Pisa, Italy {toli, zanoni}@ posso. dm. unipi. it Abstract. We analyze an algebraic representation of AES− 128 as an ...

Preprint

Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. Th... more Karatsuba and Toom-Cook are well-known methods used to multiply efficiently two long integers. There have been different proposal about the interpolating values used to determine the matrix to be inverted and the sequence of operations to invert it. A definitive word about which is the optimal matrix (values) and the (number of) basic operations to invert it seems still not to have been said. In this paper we present some particular examples of useful matrices and a method to generate automatically, by means of optimised exhaustive searches on a graph, the best sequence of basic operations to invert them.

This work deals with Karatsuba method for multivariate polynomials, not recursing on variables nu... more This work deals with Karatsuba method for multivariate polynomials, not recursing on variables number, but using an iterative scheme, with an eye to a better parallelism exploitation. Integers base 2 and 3 expansions are used in order to access the needed data. AMS Subject Classification: 11A05, 11A25, 11K65, 11Y70

Multiplication of univariate dense polynomials with long integer unbalanced (having different len... more Multiplication of univariate dense polynomials with long integer unbalanced (having different lengths) coefficients is considered. By reducing the problem to the product of bivariate polynomials with balanced coefficients, Toom–Cook approach is shown, pointing out some optimizations in order to reduce the computational cost. As a byproduct, univariate sparse Toom–Cook is also sketched. Lastly, some experimental results concerning performance comparisons are presented.

Karatsuba and Toom-Cook are well-known methods used to efficiently multiply univariate polynomial... more Karatsuba and Toom-Cook are well-known methods used to efficiently multiply univariate polynomials and long integers. For multivariate polynomials, asymptotically good approaches like Kronecker’s trick combined with FFT become truly effective only when the degree is above some threshold. In this paper we analyze Karatsuba and some of Toom-Cook methods for multivariate polynomials, considering density in a different way with respect to Kronecker, and present some algorithms for fast multivariate polynomial multiplication in practical cases, when degrees are not huge. A fast sparse polynomial multiplication algorithm is also proposed. 2000 Mathematics Subject Classification: 11A05, 11A25, 11K65, 11Y70

ACM SIGSAM Bulletin, 2000

This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert... more This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' Gröbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gröbner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Acm Sigsam Bulletin, 2000

This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert... more This paper shows how to solve homogeneous polynomial systems that contain parameters. The Hilbert function is used to check that the specialization of a 'generic' Gröbner basis of the parametric homogeneous polynomial system (computed in a polynomial ring containing the parameters and the unknowns as variables) is a Gröbner basis of the specialized homogeneous polynomial system. A preliminary implementation of these algorithms in PoSSoLib is also reported.

Proceedings of the 2002 international symposium on Symbolic and algebraic computation - ISSAC '02, 2002

In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis comp... more In this paper we consider the problem of the use of approximate arithmetics in Gröbner basis computation. This is useful to reduce the cost of integer arithmetic, but is especially necessary for overdetermined systems whose coefficients are only approximately known. We report on some numerical experiments, that show that the intrinsic instability of the problem is high but not such as to make the problem unmanageable, and that there is space to improve the numerical stability of the algorithms. We suggest some algorithms to deal with the case of overdetermined and unstable systems.

Computer Algebra in Scientific Computing, 2012

A new approach for the computation of long integer cube (third power) based on a splitting-in-two... more A new approach for the computation of long integer cube (third power) based on a splitting-in-two divide et impera approach and on a modified Toom-Cook-3 unbalanced method is presented, showing that the "classical" square-and-multiply algorithm is not (always) optimal. The new algorithm is used as a new basic tool to improve long integer exponentiation: different techniques combining binary and ternary exponent expansion are shown. Effective implementations by using the GMP library are tested, and performance comparisons are presented.

Lecture Notes in Computer Science, 2006

In Gröbner bases computation, as in other algorithms in commutative algebra, a general open quest... more In Gröbner bases computation, as in other algorithms in commutative algebra, a general open question is how to guide the calculations coping with numerical coefficients and/or not exact input data. It often happens that, due to error accumulation and/or insufficient working precision, the obtained result is not one expects from a theoretical derivation. The resulting basis may have more or less polynomials, a different number of solution, roots with different multiplicity, another Hilbert function, and so on. Augmenting precision we may overcome algorithmic errors, but one does not know in advance how much this precision should be, and a trial-and-error approach is often the only way to follow. Coping with initial errors is an even more difficult task. In this experimental work we propose the combined use of syzygies and interval arithmetic to decide what to do at each critical point of the algorithm.

2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2011

In this paper the problem of univariate polynomial evaluation is considered. When both polynomial... more In this paper the problem of univariate polynomial evaluation is considered. When both polynomial coefficients and the evaluation "point" are integers, unbalanced multiplications (one factor having many more digits than the other one) in classical Ruffini-Horner rule do not let computations completely benefit of sub quadratic methods, like Karatsuba, Toom-Cook and Schonhage-Strassen's. We face this problem by applying an approach

Lecture Notes in Computer Science, 2005

We analyze an algebraic representation of AES−128 as an embedding in BES, due to Murphy and Robsh... more We analyze an algebraic representation of AES−128 as an embedding in BES, due to Murphy and Robshaw. We present two systems of equations S and K concerning encryption and key generation processes. After some simple but rather cumbersome substitutions, we should obtain two new systems C1 and C2. C1 has 16 very dense equations of degree up to 255 in each of its 16 variables. With a single pair (p, c), with p a cleartext and c its encryption, its roots give all possible keys that should encrypt p to c. C2 may be defined using 11 or more pairs (p, c), and has 16 times as many equations in 176 variables. K and most of S is invariant for all key choices.

IFIP International Federation for Information Processing

We analyze an algebraic representation of AES-128 as an embedding in BES, due to Murphy and Robsh... more We analyze an algebraic representation of AES-128 as an embedding in BES, due to Murphy and Robshaw. We present two systems of equations S and K concerning encryption and key generation processes. After some simple but rather cumbersome substitutions, we should obtain two new systems C1 and C2. C1 has 16 very dense equations of degree up to 255 in each of its 16 variables. With a single pair (p, c), with p a cleartext and c its encryption, its roots give all possible keys that should encrypt p to c. C2 may be defined using 11 or more pairs (p, c), and has 16 times as many equations in 176 variables. K and most of S is invariant for all key choices.

2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, 2009

... GMP represents long integers using arrays of so-called “limbs” (β-bits unsigned C integers – ... more ... GMP represents long integers using arrays of so-called “limbs” (β-bits unsigned C integers – a typical value for β is eg 32). ... VIII. CONCLUSION In this paper we presented a description andimplementation of Toom-Cook 8-way method for long integers multiplication and ...