Lattices Part II — Dual Lattices , Fourier Transform , Smoothing Parameter , Public Key Encryption
نویسنده
چکیده
Fourier transform Consider the interval [0, λ] and suppose that we identify the point λ with 0 (i.e., think of it as a Torus and work modulo λ). Another way to think about this is as the basic cell of the lattice λZ, whose dual is the lattice (1/λ)Z. A periodic function on this torus has to period length of the form λ/n for an integer n. Thus, the Fourier transform of a function on this torus involves representing it as a sum of functions of the form x 7→ e−2πinx/λ. More generally, the Fourier transform of a function f on P(L) represents f as the sum of functions of the form x 7→ e2πi〈x,y〉 where y is an element in L∗. That is, we have the following theorem:
منابع مشابه
Lattices Part II — Dual Lattices , Fourier Transform , Smoothing Parameter , Public Key
Fourier transform Consider the interval [0, λ] and suppose that we identify the point λ with 0 (i.e., think of it as a Torus and work modulo λ). Another way to think about this is as the basic cell of the lattice λZ, whose dual is the lattice (1/λ)Z. A periodic function on this torus has to period length of the form λ/n for an integer n. Thus, the Fourier transform of a function on this torus i...
متن کاملTowards Efficient Arithmetic for Lattice-Based Cryptography on Reconfigurable Hardware
In recent years lattice-based cryptography has emerged as quantum secure and theoretically elegant alternative to classical cryptographic schemes (like ECC or RSA). In addition to that, lattices are a versatile tool and play an important role in the development of efficient fully or somewhat homomorphic encryption (SHE/FHE) schemes. In practice, ideal lattices defined in the polynomial ring Zp[...
متن کاملEEH: AGGH-like public key cryptosystem over the eisenstein integers using polynomial representations
GGH class of public-key cryptosystems relies on computational problems based on the closest vector problem (CVP) in lattices for their security. The subject of lattice based cryptography is very active and there have recently been new ideas that revolutionized the field. We present EEH, a GGH-Like public key cryptosystem based on the Eisenstein integers Z [ζ3] where ζ3 is a primitive...
متن کاملEfficient Public Key Encryption Based on Ideal Lattices
We describe public key encryption schemes with security provably based on the worst case hardness of the approximate Shortest Vector Problem in some structured lattices, called ideal lattices. Under the assumption that the latter is exponentially hard to solve even with a quantum computer, we achieve CPA-security against subexponential attacks, with (quasi-)optimal asymptotic performance: if n ...
متن کاملE cient Public Key Encryption Based on Ideal Lattices ( Extended Abstract )
We describe public key encryption schemes with security provably based on the worst case hardness of the approximate Shortest Vector Problem in some structured lattices, called ideal lattices. Under the assumption that the latter is exponentially hard to solve even with a quantum computer, we achieve CPA-security against subexponential attacks, with (quasi-)optimal asymptotic performance: if n ...
متن کامل