Dynamical analysis of euclidean algorithms Benoˆ ıt Daireaux Dynamical analysis of euclidean algorithms Introduction Dynamical analysis of euclidean algorithms Benoˆ ıt Daireaux Application to accelerated GREYC, Universit´ e de Caen algorithms Joint works with Lo¨ ıck Lohte, V´ eronique Maume-Deschamps et Brigitte Analyse de Vall´ ee l’algorithme LSB Conclusion LORIA, Nancy march 23rd 2006 1 / 51
Dynamical analysis of euclidean Outline of the talk algorithms Benoˆ ıt Daireaux Introduction 1 Introduction Dynamical analysis of euclidean algorithms 2 Dynamical analysis of euclidean algorithms Application to accelerated algorithms 3 Application to accelerated algorithms Analyse de l’algorithme LSB Conclusion 4 Analyse de l’algorithme LSB 5 Conclusion 2 / 51
Dynamical analysis of euclidean Gcd computation algorithms Benoˆ ıt Daireaux Integer gcd computation: Introduction Integer gcd • gave rise to the eldest known algorithm... algorithms State of the art • most complex of basic arithmetic operations Dynamical analysis of • intensively used in many areas: cryptography, computer euclidean algorithms algebra, rational computations... Application to accelerated algorithms Analyse de l’algorithme Analyses of the algorithms: LSB Conclusion • worst-case analyses not very usefull • average-case analysis ∼ theoretical colement to experiments, helps to understand the mecanisms of the algorithms 3 / 51
Dynamical analysis of euclidean Many families of algorithms algorithms Benoˆ ıt Daireaux Introduction “Basics algorithms” (= sequence of divisions) Integer gcd algorithms • MSB algorithms(Most Significant Bits) State of the art Euclid and its variants: Centered, α -euclideans, Dynamical analysis of By-Excess... euclidean algorithms • LSB algorithms (Least Significant Bits) Application to accelerated • Mixed algorithms algorithms Binary, Plus-Minus et generalisations Analyse de l’algorithme LSB Conclusion Accelerated algorithms (= simulation of the divisions on a truncated part of the integers) • Lehmer-Euclide, Knuth-Sch¨ onhage, recursive LSB 4 / 51
Dynamical analysis of euclidean State of the art algorithms Benoˆ ıt Daireaux Introduction Average case analysis Integer gcd Digit cost Bit comp. algorithms Euclid [Dix70], [Hei69], [Va] [AV00], [Val00], State of the art MSB Variants [YK75],[Rie78], [Val03] [Val00] Dynamical α -euclideans [BDV02] [BDV02] analysis of LSB [DMDV05] [DMDV05] euclidean algorithms Mixed Binary [Bre76], [Va98a] [Va98a] Lehmer [DV04] [DV04] Application to Accelerated Knuth-Sch¨ onhage [DLMV06] [DLMV06] accelerated algorithms Analyse de Distributional analysis l’algorithme Digit cost Bit Comp. LSB Euclid [Hen94], [BV04], [Lho05], MSB Variants [BV04] Conclusion α -euclideans LSB Mixed Binary Accelerated Lehmer 5 / 51
Dynamical analysis of 1 Introduction euclidean algorithms Integer gcd algorithms Benoˆ ıt State of the art Daireaux Introduction 2 Dynamical analysis of euclidean algorithms Dynamical General principle analysis of euclidean An example: Euclid algorithm algorithms General principle An example: Euclid algorithm 3 Application to accelerated algorithms Application to The Knuth-Sch¨ onhage algorithm accelerated algorithms Interrupted algorithms Analyse de l’algorithme LSB 4 Analyse de l’algorithme LSB Conclusion Extension continue Produits de matrices al´ eatoires 5 Conclusion 6 / 51
Dynamical analysis of euclidean Dynamical analysis: general algorithms Benoˆ ıt principle Daireaux A dynamical analysis has 3 steps: Introduction Dynamical analysis of euclidean algorithms 1 Modelization into a dynamical system: General principle An example: Extension of divisions to a continuous world Euclid algorithm Application to accelerated algorithms 2 Study of the continuous model Analyse de Statistical properties, evolution of densities, operators l’algorithme LSB Conclusion 3 Return to the discrete model 7 / 51
Dynamical analysis of euclidean Dynamical analysis: general algorithms Benoˆ ıt principle Daireaux Algorithmes Systèmes Introduction euclidiens dynamiques Dynamical analysis of euclidean algorithms General principle An example: Euclid algorithm Application to Séries Opérateurs génératrices de transfert accelerated algorithms Analyse de l’algorithme LSB Conclusion Étude Étude analytique spectrale Extraction de coefficients Comportement probabiliste de l’algorithme 7 / 51
Dynamical analysis of euclidean Euclid algorithm algorithms Benoˆ ıt Let ( u , v ) be an input of the algorithm, u ≥ v . Daireaux • The algorithm performs the sequence of divisions Introduction Dynamical u 0 = u 1 q 1 + u 2 , u 1 = u 2 q 2 + u 3 , . . . u p − 1 = u p q p + 0 , analysis of euclidean � u i � algorithms General principle q i +1 = An example: u i +1 Euclid algorithm � 0 � Application to 1 accelerated • With Q = the algorithm computes the algorithms 1 q Analyse de sequence l’algorithme LSB Conclusion M i := Q 1 · Q 2 · Q 3 · · · Q i • in particular one has � u 1 � u i +1 � � = M i · u 0 u i 8 / 51
Dynamical analysis of euclidean Number of iterations of the Euclid algorithms Benoˆ ıt algorithm Daireaux Study of the number of iterations P ( u , v ) on the sets Introduction Dynamical Ω := { ( u , v ) , u > v ≥ 0 , pgcd( u , v ) = 1 } , analysis of euclidean algorithms Ω N := { ( u , v ) ∈ Ω , ℓ 2 ( u ) = N } General principle An example: Euclid algorithm Application to accelerated algorithms Analyse de l’algorithme LSB Conclusion 9 / 51
Dynamical analysis of euclidean Number of iterations of the Euclid algorithms Benoˆ ıt algorithm Daireaux Study of the number of iterations P ( u , v ) on the sets Introduction Dynamical Ω := { ( u , v ) , u > v ≥ 0 , pgcd( u , v ) = 1 } , analysis of euclidean algorithms Ω N := { ( u , v ) ∈ Ω , ℓ 2 ( u ) = N } General principle An example: Euclid algorithm Generating functions: Application to accelerated � P ( u , v ) algorithms • F ( s ) = Analyse de u s l’algorithme ( u , v ) ∈ Ω LSB Conclusion 9 / 51
Dynamical analysis of euclidean Number of iterations of the Euclid algorithms Benoˆ ıt algorithm Daireaux Study of the number of iterations P ( u , v ) on the sets Introduction Dynamical Ω := { ( u , v ) , u > v ≥ 0 , pgcd( u , v ) = 1 } , analysis of euclidean algorithms Ω N := { ( u , v ) ∈ Ω , ℓ 2 ( u ) = N } General principle An example: Euclid algorithm Generating functions: Application to accelerated � � f n algorithms • F ( s ) = f n = P ( u , v ) Analyse de n s l’algorithme n ≥ 1 ( u , v ) ∈ Ω LSB u = n Conclusion 9 / 51
Dynamical analysis of euclidean Number of iterations of the Euclid algorithms Benoˆ ıt algorithm Daireaux Study of the number of iterations P ( u , v ) on the sets Introduction Dynamical Ω := { ( u , v ) , u > v ≥ 0 , pgcd( u , v ) = 1 } , analysis of euclidean algorithms Ω N := { ( u , v ) ∈ Ω , ℓ 2 ( u ) = N } General principle An example: Euclid algorithm Generating functions: Application to accelerated � � f n algorithms • F ( s ) = f n = P ( u , v ) Analyse de n s l’algorithme n ≥ 1 ( u , v ) ∈ Ω LSB u = n Conclusion Average number of iterations on Ω N : 2 N � 1 • E N [ P ] = f k | Ω N | k =2 N − 1 9 / 51
Dynamical analysis of euclidean Th´ eor` eme Taub´ erien algorithms Benoˆ ıt Th´ eor` eme Soit F ( s ) une s´ erie de Dirichlet ` a coefficients Daireaux positifs telle que F ( s ) converge pour ℜ ( s ) > σ > 0 . Si Introduction ( i ) F ( s ) est analytique pour ℜ ( s ) = σ, s � = σ , et Dynamical ( ii ) pour γ ≥ 0 , F ( s ) s’´ ecrit analysis of euclidean algorithms A ( s ) General principle F ( s ) = ( s − σ ) γ +1 + C ( s ) , An example: Euclid algorithm Application to u A ( s ) et C ( s ) sont analytiques en s = σ et A ( σ ) � = 0 , o` accelerated algorithms Analyticité Analyse de l’algorithme LSB Conclusion σ Singularité 10 / 51
Dynamical analysis of euclidean Th´ eor` eme Taub´ erien algorithms Benoˆ ıt Th´ eor` eme Soit F ( s ) une s´ erie de Dirichlet ` a coefficients Daireaux positifs telle que F ( s ) converge pour ℜ ( s ) > σ > 0 . Si Introduction ( i ) F ( s ) est analytique pour ℜ ( s ) = σ, s � = σ , et Dynamical ( ii ) pour γ ≥ 0 , F ( s ) s’´ ecrit analysis of euclidean algorithms A ( s ) General principle F ( s ) = ( s − σ ) γ +1 + C ( s ) , An example: Euclid algorithm Application to u A ( s ) et C ( s ) sont analytiques en s = σ et A ( σ ) � = 0 , o` accelerated algorithms Alors, Analyse de l’algorithme 2 N LSB � f n = K γ,σ · 2 σ N · N γ · [1 + ǫ ( N ) ] , Conclusion n =2 N − 1 A ( σ ) σ Γ( γ + 1) (1 − 2 − σ )(2 log 2) γ , K γ,σ = lim N →∞ ǫ ( N ) = 0 . 10 / 51
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