69 lines
2.3 KiB
Python
69 lines
2.3 KiB
Python
# Constantine
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# Copyright (c) 2018-2019 Status Research & Development GmbH
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# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
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# Licensed and distributed under either of
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# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
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# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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# ############################################################
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#
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# Quadratic and Cubic Non-Residue
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#
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# ############################################################
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#
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# This script checks the compatibility of a field modulus
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# with given tower extensions
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# ############################################################
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# 1st try
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# # Create the field of x ∈ [0, p-1]
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# K.<p> = NumberField(x - 1)
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#
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# # Tower Fp² with Fp[u] / (u² + 1) <=> u = 𝑖
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# L.<im> = K.extension(x^2 + 1)
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#
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# TODO how to make the following work?
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# # Tower Fp^6 with Fp²[v] / (v³ - (u + 1))
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# M.<xi> = L.extension(x^3 - (im + 1))
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# ############################################################
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# 2nd try
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# # Create the field of u ∈ [0, p-1]
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# p = Integer('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47')
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# Fp = GF(p)
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# Elem.<u> = Fp[]
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# print("p mod 4 = ", p % 4)
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#
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# # Tower Fp² with Fp[u] / (u² + 1) <=> u = 𝑖
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# Fp2.<im> = Fp.extension(u^2 + 1)
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# Elem2.<v> = Fp2[]
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#
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# # Tower Fp^6 with Fp²[v] / (v³ - (u + 1))
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# Fp6.<xi> = Fp.extension(v^3 - (im + 1))
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# Elem6.<w> = Fp6[]
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# ############################################################
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# 3rd try
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# K.<xi, im, p> = NumberField([x^3 - I - 1, x^2 + 1, x - 1])
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# ############################################################
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# Let's at least verify Fp6
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print('Verifying non-residues')
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modulus = Integer('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47')
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Fp.<p> = NumberField(x - 1)
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r1 = Fp(-1).residue_symbol(Fp.ideal(modulus),2)
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print('Fp² = Fp[sqrt(-1)]: ' + str(r1))
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Fp2.<im> = Fp.extension(x^2 + 1)
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xi = Fp2(1+im)
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r2 = xi.residue_symbol(Fp2.ideal(modulus),3)
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# ValueError: The residue symbol to that power is not defined for the number field
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# ^ AFAIK that means that Fp2 doesn't contain the 3rd root of unity
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# so we are clear
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print('Fp6 = Fp²[cubicRoot(1+I)]: ' + str(r2))
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