Mathematical Difficulties of Perturbative Functional Integrals

AbstractMontgomery’s and Barrett’s modular multiplication algorithms are widely used in modular exponentiation algorithms, e.g. to compute RSA or ECC operations. While Montgomery’s multiplication algorithm has been studied extensively in the literature and many side-channel attacks have been detected, to our best knowledge no thorough analysis exists for Barrett’s multiplication algorithm. This article closes this gap. For both Montgomery’s and Barrett’s multiplication algorithm, differences of the execution times are caused by conditional integer subtractions, so-called extra reductions. Barrett’s multiplication algorithm allows even two extra reductions, and this feature increases the mathematical difficulties significantly. We formulate and analyse a two-dimensional Markov process, from which we deduce relevant stochastic properties of Barrett’s multiplication algorithm within modular exponentiation algorithms. This allows to transfer the timing attacks and local timing attacks (where a second side-channel attack exhibits the execution times of the particular modular squarings and multiplications) on Montgomery’s multiplication algorithm to attacks on Barrett’s algorithm. However, there are also differences. Barrett’s multiplication algorithm requires additional attack substeps, and the attack efficiency is much more sensitive to variations of the parameters. We treat timing attacks on RSA with CRT, on RSA without CRT, and on Diffie–Hellman, as well as local timing attacks against these algorithms in the presence of basis blinding. Experiments confirm our theoretical results.

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Application of functional integrals to stochastic equations

Mathematical Models and Computer Simulations ◽

10.1134/s2070048217030024 ◽

2017 ◽

Vol 9 (3) ◽

pp. 339-348 ◽

Cited By ~ 2

Author(s):

E. A. Ayryan ◽

A. D. Egorov ◽

D. S. Kulyabov ◽

V. B. Malyutin ◽

L. A. Sevastyanov

Keyword(s):

Stochastic Equations ◽

Functional Integrals

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THE FUNCTIONAL INTEGRAL ON THE HALF-LINE

International Journal of Modern Physics A ◽

10.1142/s0217751x90001422 ◽

1990 ◽

Vol 05 (15) ◽

pp. 3029-3051 ◽

Cited By ~ 38

Author(s):

EDWARD FARHI ◽

SAM GUTMANN

Keyword(s):

Time Evolution ◽

Wide Class ◽

Green's Functions ◽

Green’S Functions ◽

Functional Integral ◽

Half Line ◽

Functional Integrals ◽

Quantum Hamiltonian ◽

Do So

A quantum Hamiltonian, defined on the half-line, will typically not lead to unitary time evolution unless the domain of the Hamiltonian is carefully specified. Different choices of the domain result in different Green’s functions. For a wide class of non-relativistic Hamiltonians we show how to define the functional integral on the half-line in a way which matches the various Green’s functions. To do so we analytically continue, in time, functional integrals constructed with real measures that give weight to paths on the half-line according to how much time they spend near the origin.

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