����� � ���� � ��� �� ���� ���� ��� ����������� � � � �������� � � � � � � �������������������������� ������������������������ ������ ���� ����� �� Fast computation of minimal interpolation bases § , † , ‡ Vincent Neiger ´ Claude-Pierre Jeannerod § Eric Schost † Gilles Villard § § AriC, LIP, ´ Ecole Normale Sup´ erieure de Lyon, France † University of Waterloo, Ontario, Canada ‡ Partially supported by the mobility grants Explo’ra doc from R´ egion Rhˆ one-Alpes / Globalink Research Award - Inria from Mitacs & Inria / Programme Avenir Lyon Saint-´ Etienne JNCF, Cluny, France, November 2, 2015 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 1 / 24
Problem Outline Problem 1 Application to decoding algorithms 2 Fast algorithm for small weights 3 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 2 / 24
Problem Polynomial approximation Hermite-Pad´ e approximation Input: f = ( f 1 , . . . , f m ) polynomials over K , order σ Find p = ( p 1 , . . . , p m ) polynomials such that � p 1 f 1 + · · · + p m f m = 0 mod X σ minimal deg( p ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 3 / 24
Problem Polynomial approximation Hermite-Pad´ e approximation Input: f = ( f 1 , . . . , f m ) polynomials over K , order σ Find p = ( p 1 , . . . , p m ) polynomials such that � p 1 f 1 + · · · + p m f m = 0 mod X σ minimal deg( p ) M-Pad´ e approximation (without multiplicities) Input: f = ( f 1 , . . . , f m ) polynomials over K , points x 1 , . . . , x σ Find p = ( p 1 , . . . , p m ) polynomials such that � p 1 ( x j ) f 1 ( x j ) + · · · + p m ( x j ) f m ( x j ) = 0 for all j minimal deg( p ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 3 / 24
Problem Same problem. . . Define p · f in K 1 × σ Hermite-Pad´ e p · f = [coefficients of p 1 f 1 + · · · + p m f m of degree < σ ] M-Pad´ e p · f = [evaluations of p 1 f 1 + · · · + p m f m at points x 1 , . . . , x σ ] � Unified framework [Beckermann - Labahn, 2000] p interpolant for f : p · f = 0 minimal deg( p ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 4 / 24
Problem Same problem. . . Define p · f in K 1 × σ Hermite-Pad´ e p · f = [coefficients of p 1 f 1 + · · · + p m f m of degree < σ ] M-Pad´ e p · f = [evaluations of p 1 f 1 + · · · + p m f m at points x 1 , . . . , x σ ] � Unified framework [Beckermann - Labahn, 2000] p interpolant for f : p · f = 0 minimal deg( p ) − → minimal deg w ( p ) Degree weights w = ( w 1 , . . . , w m ) deg w ( p ) = max(deg( p j ) + w j ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 4 / 24
Problem . . . different algorithms? Hermite-Pad´ e Algorithm ⋆ : small weights Cost Output O ( m (2) σ 2 ) [Van Barel - Bultheel, 1991] ( ⋆ ) Basis [Beckermann - Labahn, 1994] O ˜( m ω σ ) Basis O ˜( m ω − 1 σ ) [Zhou - Labahn, 2012] ⋆ Basis General problem Algorithm ⋆ : small weights Cost Output O ( m (2) σ 2 ) [Beckermann - Labahn, 1997/2000] ( ⋆ ) Basis O ( m (2) σ 2 ) [K¨ otter; as in McEliece, 2003] ( ⋆ ) Basis Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 5 / 24
Problem This talk Algorithm solves the general problem for small weights cost bound O ˜( m ω − 1 σ ) outputs a minimal basis Focusing on M-Pad´ e approximation without multiplicities Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 6 / 24
Application to decoding algorithms Outline Problem 1 Application to decoding algorithms 2 Fast algorithm for small weights 3 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 7 / 24
Application to decoding algorithms List-decoding: Sudan algorithm given σ points { ( x 1 , y 1 ) , . . . , ( x σ , y σ ) } f solution: deg f � k and f ( x i ) = y i for � σ − e points [Sudan, 1997] Compute degree constraints m and b Interpolation step compute Q ( X , Y ) = Q 0 + Q 1 Y + · · · + Q m Y m such that Q 0 , . . . , Q m have small weighted degree: deg Q j < b − jk Q ( x i , y i ) = 0 for all points Root-finding step the solutions f are among the Y -roots of Q ( X , Y ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 8 / 24
Application to decoding algorithms Interpolation steps in related contexts [Guruswami - Sudan, 1999] List-decoding of Reed-Solomon codes, further extends the error-correction bound Compute Q ( X , Y ) = Q 0 + Q 1 Y + · · · + Q m Y m such that Q 0 , . . . , Q m have small weighted degree Q ( x i , y i ) = 0 with multiplicity µ for all points Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 9 / 24
Application to decoding algorithms Interpolation steps in related contexts [K¨ otter - Vardy, 2003] Soft-decision decoding of Reed-Solomon codes x 1 , . . . , x n are not pairwise distinct Compute Q ( X , Y ) = Q 0 + Q 1 Y + · · · + Q m Y m such that Q 0 , . . . , Q m have small weighted degree Q ( x i , y i ) = 0 with multiplicity µ i for all points Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 9 / 24
Application to decoding algorithms Interpolation steps in related contexts [Guruswami - Rudra, 2006] List-decoding of folded Reed-Solomon codes: extends the error-correction bound up to the information-theoretic limit [Devet - Goldberg - Heninger, 2012] Optimally robust Private Information Retrieval ( j 1 ,..., j s ) ∈ Γ Q j 1 ,..., j s Y j 1 1 · · · Y j s Compute Q ( X , Y 1 , . . . , Y s ) = � s such that the Q ( j 1 ,..., j s ) have small weighted degree Q ( x i , y i 1 , . . . , y is ) = 0 with multiplicity µ for all points Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 9 / 24
Fast algorithm for small weights Outline Problem 1 Application to decoding algorithms 2 Fast algorithm for small weights 3 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 10 / 24
Fast algorithm for small weights M-Pad´ e M-Pad´ e approximation Input: f = ( f 1 , . . . , f m ) polynomials, points x 1 , . . . , x σ , weights w Find p = ( p 1 , . . . , p m ) polynomials such that p interpolant for f : p · f = 0 p has minimal weighted-degree deg w ( p ) where p · f = [evaluations of p 1 f 1 + · · · + p m f m at points x 1 , . . . , x σ ] and deg w ( p ) = max(deg( p j ) + w j ) Iterative algorithm: cost quadratic in σ returns a basis of interpolants Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 11 / 24
Fast algorithm for small weights Iterative algorithm [Beckermann-Labahn / K¨ otter] � − p 1 − � . = Identity in K [ X ] m × m . 1. P = . − p m − 2. For i from 1 to σ : ( p 1 · f )( x i ) . . = ( P · f )( x i ) a. Compute evaluations . ( p m · f )( x i ) b. Choose pivot π with smallest w π such that ( p π · f )( x i ) � = 0 Update pivot weight w π = w π + 1 c. Eliminate: For j � = π do p j = p j − ( p j · f )( x i ) ( p π · f )( x i ) p π /* ∀ j � = π, ( p j · f )( x i ) = 0 */ p π = ( X − x i ) p π /* ( p π · f )( x i ) = 0 */ After i iterations: P basis of small interpolants for ( x 1 , . . . , x i ) Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 12 / 24
Fast algorithm for small weights M-Pad´ e Iterative algorithm σ = 8 m = 4 w = [0 , 2 , 4 , 6] , base field F 97 Parameters: Input: (24 , 31 , 15 , 32 , 83 , 27 , 20 , 59) and f = (1 , R , R 2 , R 3 ) Iteration: i = 1 Point: 24 , 31 , 15 , 32 , 83 , 27 , 20 , 59 Weights [0 2 4 6] 1 0 0 0 0 1 0 0 Basis 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 80 73 73 35 66 46 91 64 Values 95 91 91 61 88 79 36 22 34 47 47 1 85 45 75 50 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 13 / 24
Fast algorithm for small weights M-Pad´ e Iterative algorithm σ = 8 m = 4 w = [0 , 2 , 4 , 6] , base field F 97 Parameters: Input: (24 , 31 , 15 , 32 , 83 , 27 , 20 , 59) and f = (1 , R , R 2 , R 3 ) Iteration: i = 1 Point: 24 , 31 , 15 , 32 , 83 , 27 , 20 , 59 Weights [0 2 4 6] 1 0 0 0 0 1 0 0 Basis 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 80 73 73 35 66 46 91 64 Values 95 91 91 61 88 79 36 22 34 47 47 1 85 45 75 50 Vincent Neiger (ENS de Lyon) Computing minimal interpolation bases Cluny, Nov 2015 13 / 24
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