the dynamic approach to gröbner basis computationimplementing f4 cone evolution? method of judging...

The dynamicapproach to

Gröbner basiscomputation

John Perry

Motivation,technicalbackgroundFirst World problems

McEliece

Gröbner bases

DynamicalgorithmsQuestion #1

Question #2

Question #3

ExperimentalresultsImplementing F4

Cone evolution?

Method of judging

Remarks on aDynamic F5

Conclusions,futuredirections

The dynamic approach to Gröbner basiscomputation

John Perry

University of Southern Mississippi

2019



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Outline

1 Motivation, technical background

2 Dynamic algorithms

3 Experimental results

4 Conclusions, future directions



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§1







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§1.1

First World problems



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Nim in Action

I am a Consumer.

I find meaning in life by Consuming.Right now I want to Consume a book.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Nim in Action

I am a Consumer.I find meaning in life by Consuming.

Right now I want to Consume a book.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Nim in Action

I am a Consumer.I find meaning in life by Consuming.Right now I want to Consume a book.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



AWESOME: Amazon has it in stock!



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Checkout

What’s this thing?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Checkout

For those who haven’t noticed it before:



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



What’s a lock doing there?

You had best be glad it’s there.

Excellent, unaltered image downloaded from The Economist. Please don’t sue. Fair use principles at work.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Public-key cryptography

1 Amazon broadcasts• encryption protocol E• encryption key e (public)

2 Buyer broadcasts c= E (m, e)3 Only Amazon

• knows decryption key d4 Eavesdropper knows

• c• encryption key• encryption and decryption protocols

. . . but cannot decrypt



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





2 Buyer broadcasts c= E (m, e)

3 Only Amazon• knows decryption key d

4 Eavesdropper knows• c• encryption key• encryption and decryption protocols




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging






• knows decryption key d

4 Eavesdropper knows• c• encryption key• encryption and decryption protocols




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging






• knows decryption key d4 Eavesdropper knows

• c• encryption key• encryption and decryption protocols




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



How is this secure?

• D, E inverse functions

• computing d “difficult” even knowing D, E, e

• factoring integers• discrete logarithm• elliptic curve arithmetic



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



How is this secure?

• D, E inverse functions• computing d “difficult” even knowing D, E, e

• factoring integers• discrete logarithm• elliptic curve arithmetic



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example

RSA Challenge: Want $200,000?

Factor this 617-digit number.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA

Big-time fact!

Crucial link in cryptography is Rivest (1976)ShamirAdelman

Big-time fact!

Cracking RSA communication as “easy” as

6= 2× 3

Big-time fact!

If that doesn’t bother you, it should.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA Challenge

Quantum challenge

1994 Shor: “quantum algorithm” to factor integers inpolynomial time

2001 IBM factors 15= 3× 52012 group factors 143= 11× 132014 oh, look, we also factored 56153= 233× 241

(“without the awareness of the authors”)2017 NIST: proposals for post-quantum cryptography



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA Challenge

Quantum challenge


2001 IBM factors 15= 3× 5

2012 group factors 143= 11× 132014 oh, look, we also factored 56153= 233× 241




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA Challenge

Quantum challenge


2001 IBM factors 15= 3× 52012 group factors 143= 11× 13

2014 oh, look, we also factored 56153= 233× 241(“without the awareness of the authors”)

2017 NIST: proposals for post-quantum cryptography



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA Challenge

Quantum challenge



(“without the awareness of the authors”)

2017 NIST: proposals for post-quantum cryptography



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



RSA Challenge

Quantum challenge






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§1.2

McEliece



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



McEliece

• Alice chooses• G (linear code correcting t errors)• S, P (nonsingular, permutation)

• Alice broadcasts t, bG= SGP• Bob broadcasts c=mbG+ z (message + t errors)• Only Alice knows

• G (can identify, remove z)• S,P (hence S−1, P−1)

• Eve knows• c, t, bG• encryption and decryption methods

. . .but cannot decrypt



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



McEliece


• Alice broadcasts t, bG= SGP

• Bob broadcasts c=mbG+ z (message + t errors)• Only Alice knows






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



McEliece


• Alice broadcasts t, bG= SGP• Bob broadcasts c=mbG+ z (message + t errors)

• Only Alice knows• G (can identify, remove z)• S,P (hence S−1, P−1)





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



McEliece


• Alice broadcasts t, bG= SGP• Bob broadcasts c=mbG+ z (message + t errors)• Only Alice knows






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Good idea?

Big-time fact!

“Classic McEliece” susceptible to attacks? Unknown, but NISTaccepted it for post-Quantum standard (2019).

Big-time fact!

Why do you never hear about it?

• encryption, decryption faster than RSA, but. . .

• key very large (bG)

Proposal: “McEliece Variants”

(2009) Choose G w/particular form



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Bad idea!

Big-time fact!

McEliece Variants cracked in < 1s! (Faugère et al., 2010)

How?1 System polynomial equations2 Structure simplify, rewrite3 Gröbner basis4 Keep “smallest” polys

Question

Gröbner basis?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§1.3

Gröbner bases



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Analogies

linear

non-linear

structure vector space

polynomial ring

multipliers field elements

monomials

workspace subspace

ideal

presentation spanning set

basis

goodbasis

Gröbner basis

presentation

transformation Gauss-Jordan

Buchberger



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Analogies

linear non-linear

structure vector space polynomial ring

multipliers field elements monomials

workspace subspace ideal

presentation spanning set basis

goodbasis Gröbner basis

presentation

transformation Gauss-Jordan Buchberger



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example: linear

2x+ 3y= 4x+ 2y= 3

=⇒2x+ 3y= 4

− 2x+ 4y= 6−y=−2

row reduction



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example: linear

2x+ 3y= 4x+ 2y= 3 =⇒

2x+ 3y= 4− 2x+ 4y= 6

−y=−2row reduction



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example: non-linear

x2+ y2 = 4xy= 1

=⇒x2y+ y3 = 4y

− x2y = xy3 = 4y− x

S-poly reduction



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example: non-linear

x2+ y2 = 4xy= 1 =⇒

x2y+ y3 = 4y− x2y = x

y3 = 4y− xS-poly reduction



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Analogy goes deeper

vector space bases, Gröbner bases both answer. . .• existence of solutions• dimension of solutions• explicit description of solutions• which vectors are in subspace

. . . in similar ways!



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



F4/F5 [Lazard, 1983], [Faugère, 1999]

Gaussian reduction¡ Buchberger’s algorithm

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...

1 1 − 4 xg

1 − 4 yg

1 1 −4 g. . . . . .

...

1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...

1 1 − 4 xg

1 − 4 yg

1 1 −4 g. . . . . .

...1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...

1 1 − 4 xg

1 1 − 4 yg

1 1 −4 g. . . . . .

...1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 − 4 xg

1 1 − 4 yg

1 1 −4 g. . . . . .

...1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 − 4 xg

0 1 1 − 4 yg

1 1 −4 g. . . . . .

...1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 − 4 xg

1 − 4 yg

1 1 −4 g. . . . . .

...1 − 1 xf

1 − 1 yf

1 −1 f

�

xy− 1 , x2+ y2− 4 , y3+ x− 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Wait. What about Buchberger?

Typically the first one taught• first algorithm (1965)• easier to implement



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why F4/F5, then?

1 Have a mystique

2 Matrix view: easier exposition (IMHO)3 Buchberger ( F4

• e.g., select 1 pair of rows of matrix4 I’ll use it later in the talk



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why F4/F5, then?

1 Have a mystique2 Matrix view: easier exposition (IMHO)

3 Buchberger ( F4• e.g., select 1 pair of rows of matrix

4 I’ll use it later in the talk



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why F4/F5, then?

1 Have a mystique2 Matrix view: easier exposition (IMHO)3 Buchberger ( F4

• e.g., select 1 pair of rows of matrix

4 I’ll use it later in the talk



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why F4/F5, then?

1 Have a mystique2 Matrix view: easier exposition (IMHO)3 Buchberger ( F4

• e.g., select 1 pair of rows of matrix4 I’ll use it later in the talk



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



F4? F5? What’s the difference?

F4 GB algorithm using matrix arithmetic

F5

GB algorithm w/matrix arithmetic and row labels

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



F4? F5? What’s the difference?

F4 GB algorithm using matrix arithmeticF5 GB algorithm w/matrix arithmetic and row labels

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§2







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§2.1

Question #1



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Optimization in linear algebra

Triangularize?

1 1 1 1 11 1 1 11 1 11 11

−→

1 1 1 1 11 1 1 1

1 1 11 1

1

Exchange columns

Can we do this with Gröbner bases?

• change ordering during algorithm• dynamic variant of static algorithm• adapt, not abolish



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why?

1 Why not?2 Might be “practical”

• faster?

• maybe. . .• . . . but not if overhead is too large

[even if not faster:]

• fewer basis polys?

• apply Gröbner bases• smaller basis: faster application• tradeoff could be worth it!



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why?

1 Why not?

2 Might be “practical”

• faster?







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why?


• faster?







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why?


• faster?• maybe. . .• . . . but not if overhead is too large






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why?


• faster?• maybe. . .• . . . but not if overhead is too large


• fewer basis polys?• apply Gröbner bases• smaller basis: faster application• tradeoff could be worth it!



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Practical?

User POV: no CAS does this.

Provider POV: Gröbner property depends on ordering.

• monomials¡ columns• swap columns? out of order!• out of order? not Gröbner!



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



For example

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



For example

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



For example

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



For example

x3 y2 xy2 y3 x2 xy x2y x y 1... . . . . . .

...1 1 −4 xg

1 1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f

�

x2y+ x− 4y,y2+ x2− 4,x2y− 1

not GB under any order



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Past work

Dynamic Buchberger implemented, improved, explored[Caboara, 1993] AlPI, CoCoA (lost)

[Golubitsky, unpublished] Maple[Caboara & Perry, 2014] Sage

[Hashemi & Talaashrafi, 2016] Sage[Perry, 2017] C++

*[Gritzmann & Sturmfels, 1993] on theory, not implementation



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1

Can we implement a Dynamic F4 algorithm?

(One person told me no)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1

Can we implement a Dynamic F4 algorithm?(One person told me no)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #11/2




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§2.2

Question #2



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Term orderings

Weighted degree ordering: ω = (ω1, . . . ,ωn)

Example

Dot product ofω, exponentsx3+ y4+ x2z

Break ties w/additional vectors:

Theorem ([Robbiano, 1986])Every admissible ordering a nonsingular matrix over R.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Term orderings


Example

Dot product ofω, exponentsx3+ y4+ x2z(1,0,0)





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Term orderings


Example






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Term orderings


Example

Dot product ofω, exponentsx3+ y4+ x2z





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Effect

x1+ x2+ · · ·+ x7,x1x2+ x2x3+ · · ·+ x7x1,

x1x2x3+ x2x3x4+ · · ·+ x7x1x2,...

x1x2x3x4x5x6x7− h7

11

...

!

� · · · 1 1 1· · · 1 1· · · 1

�

wdeg

#-spolys 5463 2199 404#polys 977 443 107

time† (sec) 14401 11.1 5.0

†wdeg uses (2526, 1461, 2625, 2639, 1, 2702, 2703, 1714)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Compute ordering, subject to constraints?

xα11 · · ·x

αnn > xβ1

1 · · ·xβnn

α ·ω > β ·ω

∑

(αi−βi)ωi > 0



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Compute ordering, subject to constraints?

n

�∑

(αi−βi)ωi > 0

xβ ∈ Supp{g}

o

g∈G

inexact: simplex (GLPK)exact: double description method (PPL, Skeleton)

• speed/size tradeoff?• geometry of solutions?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Complexity issues?

Question

With thousands, even millions of monomials being ordered,won’t this overwhelm the system?

x5+ x4y2+wz3+w3z+wx4+w2yz2+ x2

new poly −→ new monomialsmany monomials −→ many constraintsmany constraints −→ too much time

Question

How can we reduce the number of constraints?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Many elements “strung together”

· · · > x3 > x2 > x > 1

Divisibility criterion [Caboara, 1993]t | u? no need to consider t.

(limited, esp. in homogeneous systems)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Many elements “caught between”

xy 6= lm�

x2+ xy+ y2�

Question

Under what condition(s) is u 6= lm (t+ u+ v)?

New criterion

u2 | tv

(Necessary, but not sufficient)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Many elements “fenced outside”

f = x+ y

g= x2+ y2

x= lm (f ) ⇐⇒ y2 6= lm (g)

How can we detect this?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Geometric view

linear constraints −→ polyhedral cone

τ

σ

µ

2d 3d

DefinitionSkeleton: extreme vectors & adjacency relationship



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Usefulness

Big-time fact!

Skeleton identifies minimal set of potential lm’sconsistent with previous choices

(Proved from Corner Point Theorem)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example

f = x+ y

g= x2+ y2

1 Choose x= lm (x+ y)2 x> y? skeleton {(1,0) , (1,1)}3 x2 > y2 for both

∴ y2 6= lm�

x2+ y2�



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example

f = x+ y

g= x2+ y2

1 Choose x= lm (x+ y)

2 x> y? skeleton {(1,0) , (1,1)}3 x2 > y2 for both

∴ y2 6= lm�

x2+ y2�



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example

f = x+ y

g= x2+ y2

1 Choose x= lm (x+ y)2 x> y? skeleton {(1,0) , (1,1)}

3 x2 > y2 for both

∴ y2 6= lm�

x2+ y2�



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Example

f = x+ y

g= x2+ y2

1 Choose x= lm (x+ y)2 x> y? skeleton {(1,0) , (1,1)}3 x2 > y2 for both

∴ y2 6= lm�

x2+ y2�



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Challenge

CAS’s require integer orderings• simplex w/float fast• simplex w/int slower than molasses in winter

• narrow cone? large integer entries(CAS also don’t like large integer products)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Tool

Double description method• Exact• Iterative: well-suited for refinement!

• add constraints as needed• detect, discard redundant constraints

• Well-studied• Motzkin et al., 1953• Fukuda and Prodon, 1996• [Zolotykh, 2012] (← implemented this)

• Worst case not great (exponential complexity)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Use the entire skeleton. . .

• Refinement adds constraints• Explosion in # of vectors?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




• Refinement adds constraints

• Explosion in # of vectors?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




• Refinement adds constraints• Explosion in # of vectors?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



. . . or an approximation?

compute only max, minωi

disallow some needed, allow only needed

• O (n) space• O (n) “easy” calls to simplex: change objective function



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Practical results

#constraints size of basis (pols ×mons)div only app skel dyn grevlex

Caboara 1 29 20 35× 137 239× 478Caboara 2 17 26 21× 42 553× 1106Cyclic-5 61 31 11× 68 20× 85Cyclic-6 250 56 20× 129 45× 199Cyclic-7 >1000 63 63× 1049 209× 1134

Cyclic-5 hom 60 33 11× 98 38× 197Cyclic-6 hom 303 39 33× 476 99× 580Cyclic-7 hom >1000 105 222× 5181 443× 3395

• Able to compute dyn GB we could not do before!• Reduction consumes most time (usually)• Katsura-n: similar size basis as static, fewer constraints

[Caboara & Perry, 2014]



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #2

What can we say about the tradeoffof an approximate skeleton v. an exact one?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§2.3

Question #3



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



How to judge orderings?

Definition (Heuristic on leading terms)

H : Tn× 2Tn−→ S

S: scores, linearly orderedTn: monomials

Aims• educated: rationale for mapping• efficient: minimal penalty



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Degree heuristic

DefinitionChoose monomial of smallest “degree”

efficient? yeseducated?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Degree heuristic



We often prefer lower-degree terms. . .

deg�

x3y2�

= 5 degy4 = 4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Degree heuristic



. . . though not always. . . also, independent of ordering?

deg (x) = 1 degy3 = 3



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Hilbert heuristic

Definition (HF (d))

minimize Hilbert data (dimension, number of residues)

Educated?

• invariant of homogeneous ideal

• approaching from above• proposed by smart people[Gritzmann & Sturmfels, 1993,Caboara, 1993]

Efficient?

• well-studied [Bigatti, 1997, Roune, 2010]

• HP coeffs can grow very large



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Hilbert heuristic

Definition (HF (d))


Educated?• invariant of homogeneous ideal


Efficient?

• well-studied [Bigatti, 1997, Roune, 2010]




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Hilbert heuristic

Definition (HF (d))


Educated?• invariant of homogeneous ideal


Efficient?• well-studied [Bigatti, 1997, Roune, 2010]




John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Critical pairs heuristic

Definition (β)

minimize number of critical pairs after adding monomial

Educated?

• want to minimize• related to invariant of homogeneous ideal

(Betti number)• proposed by smart person (Eder)

Efficient?

• well-studied [Gebauer & Möller, 1988]• minimal overhead



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




Definition (β)


Educated?• want to minimize• related to invariant of homogeneous ideal


Efficient?

• well-studied [Gebauer & Möller, 1988]• minimal overhead



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




Definition (β)


Educated?• want to minimize• related to invariant of homogeneous ideal


Efficient?• well-studied [Gebauer & Möller, 1988]• minimal overhead



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #3

Which of these heuristics performs well in practice:Degree, Hilbert, Betti, or Random?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§3







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Review our questions

• Can we implement a Dynamic F4 algorithm?• What can we say about the tradeoff of an approximate

skeleton v. an exact one?• Which of these heuristics performs well in practice: Degree,

Hilbert, Betti, or Random?

Yes Lots Hilbert (best) & Betti (maybe)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging







Yes

Lots Hilbert (best) & Betti (maybe)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging







Yes Lots

Hilbert (best) & Betti (maybe)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging







Yes Lots Hilbert (best) & Betti (maybe)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§3.1

Implementing F4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1


(1) Can it be done?(2) Can it be done efficiently?(3) Can it be done in an existing computer algebra system?

(1) Yes (2) I’d say so (3) Yes, but not without a lot of work*

*Except Sage (I think)

http://www.sagemath.org/



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1


(1) Can it be done?

(2) Can it be done efficiently?(3) Can it be done in an existing computer algebra system?






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1


(1) Can it be done?(2) Can it be done efficiently?

(3) Can it be done in an existing computer algebra system?






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1








John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1



(1) Yes

(2) I’d say so (3) Yes, but not without a lot of work*





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1



(1) Yes (2) I’d say so

(3) Yes, but not without a lot of work*





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #1








John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Dynamic

Buchberger Algorithm

choose initial ordering

repeat• choose 2 unconsidered polys• reduce s-poly

• non-zero poly?

• reconsider ordering• re-order polys

until all pairs reduce to zero



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Dynamic Buchberger Algorithm

choose initial orderingrepeat• choose 2 unconsidered polys• reduce s-poly• non-zero poly?

• reconsider ordering• re-order polys

until all pairs reduce to zero



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Dynamic

F4 Algorithm

choose initial ordering

repeat• choose m unconsidered rows• perform Gauss-Jordan

• non-zero rows?

• reconsider ordering• adjust matrix

until all remaining rows reduce to zero rows



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Dynamic F4 Algorithm

choose initial orderingrepeat• choose m unconsidered rows• perform Gauss-Jordan• non-zero rows?

• reconsider ordering• adjust matrix

until all remaining rows reduce to zero rows



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Implementation?

DynGB, software for exploring dynamic algorithms to computea Gröbner basis

Still in development

Work-in-progress (as of Spring 2019)• works, but still sort of slow

• Dynamic Buchberger is remarkably slow• Dynamic F4 is pretty good

• learned a lot of things along the way• half-tempted to junk it and start from scratch• eventual plan is to fold dynamic code into existing CAS

(e.g., Eder’s F4 code)

https://github.com/johnperry-math/DynGB

https://github.com/ederc/gb



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Implementation?





• learned a lot of things along the way

• half-tempted to junk it and start from scratch• eventual plan is to fold dynamic code into existing CAS






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Implementation?





• learned a lot of things along the way• half-tempted to junk it and start from scratch

• eventual plan is to fold dynamic code into existing CAS(e.g., Eder’s F4 code)





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Implementation?





• learned a lot of things along the way• half-tempted to junk it and start from scratch• eventual plan is to fold dynamic code into existing CAS






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Reinventing the wheel?

Why not modify a CAS?

It’s not easy.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




Why not modify a CAS? It’s not easy.

• decades-old code and/or non-existent documentation

• organized, optimized, tuned for static computation• 67% penalty for same work (sometimes more)[ i.e., grevlex v. wgrevlex(1,. . . ,1) ]• pulling one thread unravels the whole cloth



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





• decades-old code and/or non-existent documentation• organized, optimized, tuned for static computation

• 67% penalty for same work (sometimes more)[ i.e., grevlex v. wgrevlex(1,. . . ,1) ]• pulling one thread unravels the whole cloth



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





“Dive in and destroy; we’ll sort it out later.”

I spent 2 years trying to modify 2 existing CAS’s.I have not given up, but I needed a reference implementation.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Sample performance

Cyclic-7h

Buch DBuch F4 DF4

time (sec) 17.6 9.5 4.0 1.0s-polys 2199 396 2199 409

size of basis 443 106 443 108highest degree 20 14 20 15

Mid-2012 MacBook Pro, 2.5 GHz Intel Core i5,16 GB RAM, MacOS 10.13 High Sierra



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Sample performance

Cyclic-8h

Buch DBuch F4 DF4

time 775 1519 148 28s-polys 7026 2048 7025 1991

size of basis 1182 415 1182 404highest degree 30 18 30 17

Mid-2012 MacBook Pro, 2.5 GHz Intel Core i5,16 GB RAM, MacOS 10.13 High Sierra



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§3.2

Cone evolution?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #2

What can we say about the tradeoffof an approximate skeleton v. an exact one?

Similar results in [Hashemi & Talaashrafi, 2016](independent work)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Cone evolution

Size of skeleton (vectors) v. number of refinements

not too frightening:

top-bot (from left): eco8, Es 1, Es 2, kotsireas



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Cone evolution


downright pleasant:

top-bot (from left): Cyc7, Cyc6, Cyc5



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Cone evolution


also reassuring:

Cyclic-5h Cyclic-6h Cyclic-7h



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Cone evolution


not too disconcerting:

Cab Es1 (rand binom) Cab Es2 (int prog binom)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Exact or inexact?

Exact solver typically smaller & faster. . .

Cyc-7h, Z32003 |basis| #s time (sec)inexact 401 1373 25.2exact 107 404 4.99static 443 2199 10.8



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Exact or inexact?

. . . but not always. . .

Cyc-6h, Z32003 |basis| #s time (sec)inexact 30 81 0.15exact 38 110 0.09static 99 389 0.15



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Exact or inexact?

. . . and in some cases there’s no choice!

4× 4 |basis| #s time (sec)inexact 7 6 .418exact * * *static 44 202 .396



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Not all the news is great

“4× 4” system [Yan, 1998]• ~500 −→~60,000 vectors in 2 refinements• exact approaches choke / terminate after 2 minutes• approximate approaches compute GB in half a second!

• GLPK (simplex)



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§3.3

Method of judging



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Question #3

Which of these heuristics performs well in practice:Degree, Hilbert, Betti, or Random?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

Caboara 1

| basis | #s time (sec)static 239 1188 .562betti 154 711 .281

degree 95 396 .183hilbert 34 137 .089

random (×3)63 266 .141346 1685 1.63138 625 .273



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

Caboara 2

| basis | #s time (sec)static 414 6781 42.96betti 6 681 .451

degree 6 33 .011hilbert 7 30 .113

random (×3)6 63 .0266 63 .0266 30 .018



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

4× 4


degree 7 7 .401hilbert 7 7 .404

random (×3)47 233 8.6267 379 86.933 144 5.40

*GLPK only



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

Cyc-7

| basis | #s time (sec)static 209 2099 10.7betti 81 2634 43.0

degree 54 2099 50.5hilbert 54 2147 54.8

random (×3)35 2227 51.146 2721 43.752 2214 50.4



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

Cyc-7h


degree 106 399 5.05hilbert 107 404 5.05

random (×3)170 842 27.7155 756 25.8214 907 13.0



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

Buch85


degree 11 22 .150hilbert 11 27 .156

random (×3)11 20 .22711 25 .17214 30 .163



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

butcher8

| basis | #s time (sec)static 29 435 .441betti 10 386 1.27

degree * * *hilbert 29 366 .367

random (×3)* * *8 260 .5198 167 .253

*terminated after several seconds



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

eco8


degree * * *hilbert 12 362 1.54

random (×3)8 513 4.708 247 .7968 382 2.41

*terminated after several seconds



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

kotsireas


degree 6 731 11.0hilbert 27 311 .809

random (×3)* * *6 737 11.1* * *



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

katsura8


degree 15 1588 51.8hilbert 133 822 13.3

random (×3)15 1776 72.5* * *30 1266 66.7



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

katsura8h


degree 130 790 15.8hilbert 249 1745 38.2

random (×3)183 1191 25.0207 1369 26.5329 2332 96.3



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

noon5


degree 15 577 5.88hilbert 53 314 .542

random (×3)15 577 5.9115 578 5.8215 577 5.90



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Benchmark systems

s9_1


degree 8 14 .017hilbert 14 28 .021

random (×3)8 13 .0188 13 .0208 13 .019



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Heuristic performance

Dynamic sometimes longer?• weighted degree• longer intermediate polys• fewer pairs, smaller basis; faster computation

No heuristic has advantage?!?

Even random choice competes?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Heuristic performance

Dynamic sometimes longer?• weighted degree• longer intermediate polys• fewer pairs, smaller basis; faster computation

No heuristic has advantage?!? Even random choice competes?



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why is random sometimes better?

• finitely many orderings (equiv. classes, really)

• sometimes “very” finite!

Example (Cyclic-4)

96 possible orderings via gfan4–8 GB size16 size 416 size 5

33% probability of choosing “small” basis!

For most systems tried, random is terrible



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging




• finitely many orderings (equiv. classes, really)• sometimes “very” finite!

Example (Cyclic-4)






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





Example (Cyclic-4)

96 possible orderings via gfan

4–8 GB size16 size 416 size 5





John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging





Example (Cyclic-4)






John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§3.4

Remarks on a Dynamic F5



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Quick review

F4 GB algorithm using matrix arithmetic

F5

GB algorithm w/matrix arithmetic and row labels

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Quick review

F4 GB algorithm using matrix arithmeticF5 GB algorithm w/matrix arithmetic and row labels

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



More specifically

F5 GB algorithm w/matrix arithmetic and row labelsreduce row by lower rows only

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why the restriction?

row labels −→ signatures −→ “how row was generated”

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f

[yg] ?← [yg]− [xf]



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Why the restriction?

row labels −→ signatures −→ “how row was generated”

x3 x2y xy2 y3 x2 xy y2 x y 1... . . . . . .

...1 1 −4 xg

0 1 1 −4 yg

1 1 −4 g. . . . . .

...1 −1 xf

1 −1 yf

1 −1 f

[yg] ?← [yg]− [xf]



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Observations

signature depends on ordering

change order: change signature? more delicatere-evaluate order during reduction? more opportunity



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Observations


change order: change signature? more delicate

re-evaluate order during reduction? more opportunity



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Observations


change order: change signature? more delicatere-evaluate order during reduction? more opportunity



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



§4







John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Summary

• Dynamic algorithms seek out “good” orderings• can be faster, smaller. . . but no guarantee• metrics need further investigation

• implementation shows promise (IMHO)• Dynamic F5



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Thank you!

Massimo Caboara

Università di Pisa

Christian Eder

Centre for Computer Algebra

William Stein

Николай Юрьевич Золотых

Казанский Федеральний Университет

University of Southern Mississippi

Галина Валеревна Заморий



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Bibliography: General

Bigatti, “Computation of Hilbert-Poincaré Series.” JPAA 119 (1997),237–253.

Buchberger, An algorithm for finding the basis elements of the residue classring of a zero dimensional polynomial ideal. PhD Thesis (1965), Universityof Innsbruck. English translation in JSC 41 (2006) 475–511.

Gebauer and Möller, “On an Installation of Buchberger’s Algorithm,”JSC 6 (1988) 275–286.

Lazard, “Gröbner bases, Gaussian elimination, and resolution of systemsof algebraic equations.” EUROCAL ’83, Springer LNCS 162, 146–156.

Roune, “A Slice Algorithm for Corners and Hilbert-Poincaré Series ofMonomial Ideals.” ISSAC 2010, AMC Press, 115–122.

Yan, “The Geobucket Data Structure for Polynomials.” JSC 25 (1998)295–293.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Bibliography: Dynamic algorithm

Caboara, “A Dynamic Algorithm for Gröbner basis computation.”ISSAC ’93, ACM Press, 275–283.

Caboara and Perry, “Reducing the size and number of linear programs ina dynamic Gröbner basis algorithm.” AAECC 25 (2014) 99–117.

O. Golubitsky, “Converging term order sequences and the dynamicBuchberger algorithm.” Unpublished preprint received via privatecommunication.

Gritzmann and Sturmfels, “Minkowski Addition of Polytopes:Computational Complexity and Applications to Gröbner Bases.” SIAM J.Disc. Math 6 (1993) 246–269.

Hashemi and Talaashrafi, “A Note on Dynamic Gröbner BasesComputation.” CASC 2016, Springer, 276–288.

Mora and Robbiano, “The Gröbner fan of an ideal.” JSC 6 (1988) 183–208.

Perry, “Exploring the Dynamic Buchberger Algorithm.” ISSAC 2017,ACM Press, 365–372.

Robbiano, “On the theory of graded structures,” JSC 2 (1986) 139–170.

Zolotykh, “New Modification of the Double Description Method forConstructing the Skeleton of a Polyhedral Cone.” ComputationalMathematics and Mathematical Physics 52 (2012) 153–163.



John Perry


McEliece

Gröbner bases


Question #2

Question #3


Cone evolution?

Method of judging



Bibliography: Faugère

Faugère, “A new efficient algorithm for computing Gröbner bases (F4).”Journal of Pure and Applied Algebra 139 (1999) 61–88.

Faugère, “A new efficient algorithm for computing Gröbner baseswithout reduction to zero (F5).” ISSAC ’02, ACM Press, 75–82.

Faugère and Joux, “Algebraic cryptanalysis of Hidden Field Equation(HFE) cryptosystems using Gröbner bases.” Advances in Cryptology(2003) 44–60.

the dynamic approach to gröbner basis computationimplementing f4 cone evolution? method of judging...

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