From 1a7252b71a5ce7b89a5428c29da5ab92a56f1faa Mon Sep 17 00:00:00 2001 From: KlausC Date: Sun, 12 Nov 2023 21:56:55 +0100 Subject: [PATCH] use Conway polynomials for Galois fields --- CommutativeRings/Project.toml | 1 - Project.toml | 4 +- data/CPimport.txt | 35359 ++++++++++++++++++++++++++++++++ src/CommutativeRings.jl | 5 +- src/conway.jl | 181 + src/galoisfields.jl | 64 +- src/intfactorization.jl | 6 +- src/typevars.jl | 2 +- src/zzmod.jl | 4 +- test/galoisfields.jl | 11 +- test/specialseries.jl | 1 + test/univarpolynom.jl | 8 +- 12 files changed, 35598 insertions(+), 48 deletions(-) delete mode 100644 CommutativeRings/Project.toml create mode 100644 data/CPimport.txt create mode 100644 src/conway.jl diff --git a/CommutativeRings/Project.toml b/CommutativeRings/Project.toml deleted file mode 100644 index 81648c0..0000000 --- a/CommutativeRings/Project.toml +++ /dev/null @@ -1 +0,0 @@ -[deps] diff --git a/Project.toml b/Project.toml index 7fb4cd2..f54e373 100644 --- a/Project.toml +++ b/Project.toml @@ -3,7 +3,7 @@ uuid = "a6d4fa9c-9e0b-4795-89f3-f481b7b5e384" license = "MIT" authors = ["Klaus Crusius "] repo = "https://github.com/KlausC/CommutativeRings.jl" -version = "0.5.1" +version = "0.5.2" [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" @@ -12,7 +12,7 @@ Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" [compat] -Polynomials = "1,2,3" +Polynomials = "1,2,3,4" Primes = "0.5" julia = "1" diff --git a/data/CPimport.txt b/data/CPimport.txt new file mode 100644 index 0000000..9c775c1 --- /dev/null +++ b/data/CPimport.txt @@ -0,0 +1,35359 @@ +allConwayPolynomials = [ +[2,1,[1,1]], +[2,2,[1,1,1]], +[2,3,[1,1,0,1]], +[2,4,[1,1,0,0,1]], +[2,5,[1,0,1,0,0,1]], +[2,6,[1,1,0,1,1,0,1]], +[2,7,[1,1,0,0,0,0,0,1]], +[2,8,[1,0,1,1,1,0,0,0,1]], +[2,9,[1,0,0,0,1,0,0,0,0,1]], +[2,10,[1,1,1,1,0,1,1,0,0,0,1]], +[2,11,[1,0,1,0,0,0,0,0,0,0,0,1]], +[2,12,[1,1,0,1,0,1,1,1,0,0,0,0,1]], +[2,13,[1,1,0,1,1,0,0,0,0,0,0,0,0,1]], 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FractionRing, QuotientRing, Polynomial export ZZ, QQ, ZZmod, Frac, Quotient, UnivariatePolynomial, MultivariatePolynomial export GaloisField, FSeries -export PowerSeries, O, precision, absprecision +export SpecialPowerSeries, PowerSeries, O, precision, absprecision +export Conway export Hom, Ideal @@ -78,6 +79,8 @@ include("linearalgebra.jl") include("rationalcanonical.jl") include("powerseries.jl") include("specialseries.jl") +include("conway.jl") using .SpecialPowerSeries +using .Conway end # module diff --git a/src/conway.jl b/src/conway.jl new file mode 100644 index 0000000..904e295 --- /dev/null +++ b/src/conway.jl @@ -0,0 +1,181 @@ +module Conway + +export conway, quasi_conway + +using Primes +using ..CommutativeRings +using CommutativeRings: intpower, _isprimitive, coeffs + +const CONWAY_POLYNOMIALS = Dict{Tuple{Int,Int},Any}() + +function store_conway!(row::Vector) + p, n, c = row + key = (p, n) + @assert c[n+1] == 1 + c = c[1:n] + ix = findlast(!iszero, c) + if ix !== nothing && ix > 0 + resize!(c, ix) + end + d = CONWAY_POLYNOMIALS + d[key] = c +end + +function store_conway!(poly::P) where P<:UnivariatePolynomial{<:ZZmod} + c = value.(coeffs(poly)) + p = characteristic(P) + map!(x -> x >= 0 ? x : p + x, c, c) + n = deg(poly) + store_conway!([p, n, c]) +end + +""" + conway(p, n) + +Return the conway-polynomial of degree `n` over `ZZ/p`, as far as available. + +For `n in (1,2)` the polynomials are calculated and memoized in the data cache. +""" +function conway(p::Integer, n::Integer, X::Symbol = :x) + isprime(p) || throw(ArgumentError("p is not prime ($p)")) + n > 0 || throw(ArgumentError("n is not positive ($n)")) + + c = get(CONWAY_POLYNOMIALS, (p, n), missing) + if ismissing(c) && n <= 2 + poly = quasi_conway(p, n, X) + store_conway!(poly) + return poly + end + ismissing(c) && return c + B = (ZZ/p)[X] + B(c) + monom(B, n) +end + +""" + quasi_conway(p, m, X::Symbol, nr::Integer=1, factors=nothing) + +Return the first irreducible polynomial over `ZZ/p` of degree `m`. + +The polynomials are scanned in the canonical order for Conway polynomials. + +Variable symbol is `X`. If given, `factors` must be the prime factorization of `m`. +If `nr > 0` is given, the `nr+1`^st of found polynomials is returned. +""" +function quasi_conway(p::Integer, m::Integer, X::Symbol = :x, nr = 0, factors = nothing) + Z = ZZ / p + P = Z[X] + fact = factors === nothing ? factor(intpower(p, m) - 1) : factors + mm = prod(fact) + x = monom(P) + g = generator(Z) + s = (-1)^(m - 1) + nx = max(nr, 0) + # find the next irreducible, for which x is primitive (drop first nr-1) + for poly in Monic(P, m - 1) + gen = (poly(-x) * x - g) * s # observation for all calculated Conway polynomials + if isirreducible(gen) && _isprimitive((x, gen), mm, fact) + nx == 0 && return gen + nx -= 1 + end + end + text = "no irreducible polynomial of degree $m found with generator '$X' (nr = $nr)" + throw(ArgumentError(text)) +end + +""" + has_conway_property1(poly) + +Check if `poly` is monomial. +""" +function has_conway_property1(poly::T) where {P,S<:ZZmod{P},T<:UnivariatePolynomial{S}} + ismonic(poly) +end + +""" + has_conway_property2(poly) + +Check if an irreducible polynomial `poly` over `ZZ/p` of degree `n` has property 2 (is primitive). + +That means that the standard generator `x` in the quotient ring (field) `(ZZ/p) / poly` +is also generator in the multiplicative group of the ring. + +If `m` is the order of this group (`p^n - 1` if a field), it sufficient to check, if +`x ^ (m / s)` != 1 for all prime factors of `m`. (`y ^ m == 1` for all `y != 0`) + +The function returns `missing` if `p^n - 1` has no known prime factors. +""" +function has_conway_property2( + poly::T, + factors = nothing, +) where {P,S<:ZZmod{P},T<:UnivariatePolynomial{S}} + p = P + fact = factors === nothing ? Primes.factor(intpower(p, m) - 1) : factors + mm = prod(fact) + isirreducible(poly) || throw(ArgumentError("polynomial is not irreducible")) + x = monom(T) + _isprimitive((x, poly), mm, fact) +end + +""" + has_conway_property3(poly::(ZZ/p)[:x]}) + +Check if a polynomial `poly` over `ZZ/p` of degree `n` has property 3. + +That means, for all divisors `1 <= m < n` we have +`conway(p, m)(x^w) == 0 mod poly` with `w = (p^n - 1) / (p^m - 1)`. + +The definition is recursive and requires the knowledge of Conway polynomials of lesser degree. + +If such polynomials are unknown, `missing` is returned. +""" +function has_conway_property3(poly::T) where {P,S<:ZZmod{P},T<:UnivariatePolynomial{S}} + n = deg(poly) + p = P + Q = T / poly + for m in factors(n) + if m < n + nm = n ÷ m + q = big(p)^m + # w = (p^n - 1) / (p^m - 1); Z = x^w mod poly + X = monom(Q) + Y = X^q + Z = X * Y + for i = 2:nm-1 + Y = Y^q + Z *= Y + end + w = (q^nm - 1) ÷ (q - 1) + @assert X^w == Z + c = conway(p, m) + ismissing(c) && return missing + !iszero(c(Z)) && return false + end + end + return true +end + +function readparse(fn::AbstractString, action::Function) + open(fn, "r") do io + readparse(io, action) + end +end +function readparse(io::IO, action::Function) + while !eof(io) + line = readline(io) + if line[1] == '[' && line[end] == ',' + row = eval(Meta.parse(line[1:end-1])) + action(row) + end + end +end + +function __init__() + # file imported from + # `https://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/CPimport.txt.gz` + # at 2023-11-11 + file = joinpath(@__DIR__, "..", "data", "CPimport.txt") + empty!(CONWAY_POLYNOMIALS) + readparse(file, r -> store_conway!(r)) +end + +end # module diff --git a/src/galoisfields.jl b/src/galoisfields.jl index 846c486..a62b41f 100644 --- a/src/galoisfields.jl +++ b/src/galoisfields.jl @@ -25,11 +25,12 @@ function GF(n::Integer, k::Integer = 1; mod = nothing, nr = 0, maxord = 2^20) f = Primes.factor(n) length(f) == 1 || throw(ArgumentError("$n is not p^r with p prime and r >= 1")) p, r = f.pe[1] - _GF(p, r * k; mod, nr, maxord) + _GF(p, r * k, mod, nr, maxord) end -function _GF(p::Integer, r::Integer; nr::Integer = 0, mod = nothing, maxord = 2^20) - r == 1 || mod === nothing || throw(ArgumentError("given modulus requires prime base")) - mm = intpower(p, mod === nothing ? r : deg(mod)) - 1 +function _GF(p::Integer, r::Integer, mod, nr::Integer, maxord::Int) + modpol = mod isa UnivariatePolynomial + r == 1 || !modpol || throw(ArgumentError("given modulus requires prime base")) + mm = intpower(p, !modpol ? r : deg(mod)) - 1 fact = Primes.factor(mm) Q = GFImpl(p, r, fact; nr, mod) ord = order(Q) @@ -250,7 +251,8 @@ end Return the generator element of this implementation of Galois field. """ function generator(::Type{G}) where {Id,G<:GaloisField{Id,<:Integer}} - G(2, NOCHECK) + id = min(2, order(G) - 1) + G(id, NOCHECK) end function generator(::Type{G}) where {Id,G<:GaloisField{Id,<:Quotient}} G(monom(Quotient(G))) @@ -362,12 +364,12 @@ function tovalue(::Type{<:GaloisField{Id,V}}, num::Integer) where {Id,V<:Quotien end function tonumber(a::Quotient{<:UnivariatePolynomial}, p::Integer) - s = 0 u = a.val + s = zero(intpower(p, deg(u)+1)) for c in reverse(u.coeff) s = s * p + c.val end - s * p^ord(u) + s * intpower(p, ord(u)) end function toquotient(g::G) where {Id,T<:Integer,Q,G<:GaloisField{Id,T,Q}} @@ -382,7 +384,7 @@ end function toquotient( a::Integer, ::Type{Q}, -) where {Z,P<:UnivariatePolynomial{Z,:α},Q<:Quotient{P}} +) where {Z,P<:UnivariatePolynomial{Z},Q<:Quotient{P}} p = characteristic(Q) r = dimension(Q) ord = order(Q) @@ -429,34 +431,32 @@ function GFImpl( ) isprime(p) || throw(ArgumentError("base $p must be prime")) m > 0 || throw(ArgumentError("exponent m=$m must be positive")) - Z = ZZ / p - m == 1 && mod === nothing && return Z - P = Z[:α] - if mod === nothing - fact = factors === nothing ? Primes.factor(intpower(p, m) - 1) : factors - mm = prod(fact) - x = monom(P) - nx = max(nr, 0) - # find the next irreducible, for which x is primitive (drop first nr-1) - for gen in irreducibles(P, m) - if _isprimitive((x, gen), mm, fact) - nx == 0 && return P / gen - nx -= 1 - end + + m == 1 && mod === nothing && return ZZ / p + if mod === :conway + gen = Conway.conway(p, m, :γ) + if !ismissing(gen) + return typeof(gen) / gen end - throw( - ArgumentError( - "no irreducible polynomial of degree $m found with generator p(γ) = γ (nr = $nr)", - ), - ) - else + mod = nothing + end + + if isnothing(mod) + poly = Conway.quasi_conway(p, m, :α, nr, factors) + return typeof(poly) / poly + + elseif mod isa UnivariatePolynomial m == 1 || throw(ArgumentError("given mod requires prime base")) + Z = ZZ / p + P = Z[:α] # do not check if x is primitive here gen = P(Z.(mod.coeff), ord(mod)) if isirreducible(gen) return P / gen end - throw(ArgumentError("given polynomial $gen is not irreducible in $P")) + throw(ArgumentError("given polynomial $gen is not irreducible over $Z")) + else + throw(ArgumentError("modulus '$mod' not supported")) end end @@ -465,9 +465,9 @@ function Base.show(io::IO, g::G) where G<:GaloisField m = dimension(G) p = characteristic(G) cc = toquotient(g).val - print(io, '{', cc[m-1]) + print(io, '{', cc[m-1].val) for k = m-2:-1:0 - print(io, ':', cc[k]) + print(io, ':', cc[k].val) end print(io, '%', p, '}') end @@ -563,7 +563,7 @@ function *( end function monom(::Type{Q}, k::Integer, lc = 1) where {P<:UnivariatePolynomial,Q<:Quotient{P}} - Q(monom(P, k, lc)) + k < dimension(Q) ? Q(monom(P, k, lc)) : lc == 1 ? Q(monom(P))^k : Q(monom(P))^k * lc end """ diff --git a/src/intfactorization.jl b/src/intfactorization.jl index 077ac7b..1c002a6 100644 --- a/src/intfactorization.jl +++ b/src/intfactorization.jl @@ -370,7 +370,7 @@ calculate array of bounds `b` with `abs(v[i]) <= b[i+1] for i = 0:m`. Algorithm see TAoCP 2.Ed 4.6.2 Exercise 20. """ function coeffbounds(u::UnivariatePolynomial{ZZ{T},X}, m::Integer) where {T<:Integer,X} - I = widen(T) + W = widen(T) n = deg(u) 0 <= m <= n || throw(ArgumentError("required m ∈ [0,deg(u)] but $m ∉ [0,$n]")) accuracy = 100 # use fixed point decimal arithmetic with accuracy 0.01 for the norm @@ -379,7 +379,7 @@ function coeffbounds(u::UnivariatePolynomial{ZZ{T},X}, m::Integer) where {T<:Int iszero(u0) && throw(ArgumentError("required u(0) != 0")) rua = norm(value.(u.coeff)) * accuracy ua = Integer(ceil(rua)) - v = zeros(I, m + 1) + v = zeros(W, m + 1) if m > 0 v[m+1], v[1] = un, u0 else @@ -387,7 +387,7 @@ function coeffbounds(u::UnivariatePolynomial{ZZ{T},X}, m::Integer) where {T<:Int end u0 *= accuracy un *= accuracy - bk = I(1) + bk = W(1) for j = m-1:-1:1 bj = bk bk = bk * j ÷ (m - j) diff --git a/src/typevars.jl b/src/typevars.jl index c07de38..687ef2a 100644 --- a/src/typevars.jl +++ b/src/typevars.jl @@ -64,5 +64,5 @@ sintern(m::IdentSymbols) = m sintern(m::BigInt) = Symbol(m) sintern(a::IdentSymbols...) = Symbol(tuple(a...)) sintern(a::AbstractVector{<:IdentSymbols}) = Symbol(a) -sintern(a::Tuple{Vararg{<:IdentSymbols}}) = Symbol(a) +sintern(a::Tuple{Vararg{T}}) where T<:IdentSymbols = Symbol(a) sintern(a::Any) = Symbol(Base.hash(a)) diff --git a/src/zzmod.jl b/src/zzmod.jl index 96edbd1..ddb8e45 100644 --- a/src/zzmod.jl +++ b/src/zzmod.jl @@ -76,7 +76,7 @@ function value(a::ZZmod{X,T}) where {X,T} m = modulus(a) m1 = m >> 1 m2 = m - m1 - v <= m2 ? S(v) : S(v - m2) - S(m1) + v < m2 ? S(v) : S(v - m2) - S(m1) end # get type variable @@ -166,7 +166,7 @@ _unsigned(x::Integer) = unsigned(x) function Base.show(io::IO, a::ZZmod) v = a.val m = modulus(a) - if m >= 100 && v > m ÷ 2 + if m > 10 && v > m ÷ 2 print(io, '-', signed(m - v), '°') else print(io, signed(v), '°') diff --git a/test/galoisfields.jl b/test/galoisfields.jl index d3dda8b..d9b8e43 100644 --- a/test/galoisfields.jl +++ b/test/galoisfields.jl @@ -28,7 +28,7 @@ randmatrix(p::Integer, n::Integer) = rand(rng, 0:p-1, n, n) A = ma53(4, 3, 0) + ma53(4, 3, 1) + ma53(4, 3, 2) @test inv(A) * A == I - @test sprint(show, GFImpl(5, 3)([1, 2, 3])) == "3°*α^2 + 2°*α + 1° mod(α^3 + 3°*α + 2°)" + @test sprint(show, GFImpl(5, 3)([1, 2, 3])) == "3°*α^2 + 2°*α + 1° mod(α^3 + 3°*α + 3°)" end @testset "Galois Fields" begin @@ -104,7 +104,7 @@ end @test sprint(show, g1) !== nothing @test sprint(show, q1) !== nothing - @test endswith(sprint(show, generator(G)), r"{(0°:)*1°:0°%[1-9]}") + @test endswith(sprint(show, generator(G)), r"{(0:)*1:0%[1-9]}") @test G(one(ZZ / p)) == one(G) @test length(modulus(G).(G)) == length(G) @@ -116,7 +116,7 @@ end @test_throws ArgumentError GF(p, r, nr = 10000000) @test log(generator(G)^20) == 20 - @test det(normalmatrix(Q)) == 1 + @test det(normalmatrix(Q)) != 0 end @testset "Galois Field Implementation - Homomorphisms" begin @@ -211,4 +211,9 @@ end @test sort(collect(allzeros(p, vx))) == ofindex.(fa .- 1, Ref(G)) end +@testset "Conway" begin + @test GF(67, 18, mod=:conway)(ones(Int, 18)) isa GaloisField + @test GF(109987, 2, mod=:conway) <: GaloisField +end + end #module diff --git a/test/specialseries.jl b/test/specialseries.jl index 477d40a..d8eb936 100644 --- a/test/specialseries.jl +++ b/test/specialseries.jl @@ -3,6 +3,7 @@ module SpecialSeriesTest using CommutativeRings using CommutativeRings.SpecialPowerSeries using Test +using LinearAlgebra const Names = names(CommutativeRings.SpecialPowerSeries) ∩ names(Base.Math) const Funcs = eval.(Names) diff --git a/test/univarpolynom.jl b/test/univarpolynom.jl index a626e38..7de9141 100644 --- a/test/univarpolynom.jl +++ b/test/univarpolynom.jl @@ -302,14 +302,16 @@ end G = GF(7, 3) GX = G[:x] x = monom(GX) - p = x^3 + ofindex((7*2+1)*7+5, G)x + 1 - q = x^3 + 1 + ofindex(200,G) * x + p = x^3 + ofindex((7 * 2 + 1) * 7 + 4, G) * x + 1 + r = x + ofindex(28, G) + q = (x^2 + ofindex(201, G) * x + 1) * r @test isirreducible(p) @test factor(q) |> length == 3 @test deg(p^2) == 6 @test p * q == q * p @test p + q - p == q - @test q / (x + ofindex(28, G)) |> deg == 2 + @test q / r |> deg == 2 + @test_throws DomainError q / (r + 1) end @testset "primpart and content" begin