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On the quantum complexity of the continuous hidden subgroup problem
Paper: https://eprint.iacr.org/2019/716.pdf
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- Introduction
- Problem statement
- Quantum algorithm
- Error analysis
On the quantum complexity of the continuous hidden subgroup problem - - PowerPoint PPT Presentation
On the quantum complexity of the continuous hidden subgroup problem - - PowerPoint PPT Presentation
On the quantum complexity of the continuous hidden subgroup problem Koen de Boer, Lo Ducas, Serge Fehr 1 / 20 Paper: https://eprint.iacr.org/2019/716.pdf Overview Introduction Problem statement Quantum algorithm Error analysis
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Introduction
- Generalizes the Hidden Subgroup Problem to
- Computes unit groups of number fjelds
- Used to prove a (quantum) hardness-gap between
Ideal-SVP and SVP [CGS14,CDPR16,BS16]
- Possible conseq. in crypto based on lattices with
algebraic structure [CDW17]
(2014)
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Introduction
Shortcomings
- No exclusion of intractable instances
- Polynomial in which variable?
- Only high-level reasoning in the ext. abstract
- Up to now, 5 years later, no full version published
Extended abstract (2014)
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Problem statement (informal)
- Given a ‘nice’ periodic function, fjnd its period.
- More psychedelic example in 2d:
- Insight: In higher dimensions the period is
encoded by a lattice
Period
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Problem statement (formal)
- Given black-box access to a function
that satisfjes the following: (i) is periodic w.r.t. some lattice (ii) is Lipschitz-continuous (iii) is seperable, i.e., not too constant.
- Find: A -approximate basis of the lattice
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Our contributions
- Statement with all dependencies on
parameters
- Rigorous proof of this statement
- Simplifying the quantum algorithm of
Eisenträger et al.
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High level approach
- Sample approx. dual lattice points
using a quantum algorithm
- From enough of such , recover an approx.
dual basis
- From recover an approx. primal basis
This talk
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Some important thoughts
- The notion of the dual lattice
- Defjne
- Every nice -periodic function can be written
with the Fourier decomposition of
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One more important thought
- The convolution theorem
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Global idea
- Create the Gaussian
superposition
- Query in superposition
- Apply the Fourier Transform
- Measure
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Global idea
- Create the Gaussian
superposition
- Query in superposition
- Apply the Fourier Transform
- Measure
- Q computers only have finitely many qubits
- We need to discretize and ‘window’ the wave
- Fourier transf. becomes Finite Fourier Transf.
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Quantum algorithm
Ideal case: Fourier transform restricted to grid Actual case: Finite Fourier transform
?
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Quantum algorithm
Ideal case: Fourier transform restricted to grid Actual case: Finite Fourier transform
?
- How ‘fine’ must the grid be?
- How is it related with parameters a, r, ε, τ ?
- How fast is the convergence?
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Fourier transforms
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Fourier transforms
- These are pointwise errors
- We want the error in the L2 -distance
We want the sum of those to be small
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Actual Analysis
- Grid → Unit cube: the Yudin-Jackson theorem
- Unit cube → real space: the Poisson Summation Formula
About optimal trigonometric approximations About the interplay between Fourier transforms the
- perations ‘restriction’
and ‘periodization’ on functions
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Main theorem
we need This high complexity is mostly due to numerical instability of generating a dual basis and inverting this basis to obtain a primal basis.
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Open questions
- Complexity unit group or class group computation?
- Complexity of Principal Ideal Problem?
- Are there assumptions on the oracle function
making the complexity better?
- Using BKZ to improve the numerical stability of
recovering the primal basis?
- Using sublattices or symmetries of lattices to
improve complexity
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