CryptoBook
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  • Fundamentals
    • Mathematical Notation
    • Division and Greatest common divisor
      • Euclidean Algorithm
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        • Fermat's Little Theorem in Detail
        • Euler's Theorem in Detail
      • Quadratic Residues
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      • Another take on groups
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    • Untitled
  • Lattices
    • Introduction
    • LLL reduction
      • Gram-Schmidt Orthogonalization
      • Lagrange's algorithm
      • LLL reduction
    • Lattice reduction
      • Minkowski reduced
      • HKZ reduced
      • LLL reduced
    • Applications
      • Coppersmith algorithm
      • Extensions of Coppersmith algorithm
    • Hard lattice problems
    • Lattices of interest
    • Cryptographic lattice problems
      • Short integer solutions (SIS)
      • Learning with errors (LWE)
      • Ring-LWE
      • NTRU
    • Interactive fun
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  • Asymmetric Cryptography
  • RSA
    • Proof of correctness
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    • Low Private Component Attacks
      • Wiener's Attack
      • Boneh-Durfee Attack
    • Common Modulus Attack
    • Recovering the Modulus
  • Diffie-Hellman
    • MITM
  • Elliptic Curve Cryptography
  • Symmetric Cryptography
    • Encryption
    • The One Time Pad
    • AES
      • Rijndael Finite Field
      • Round Transformations
  • Hashes
    • Introduction / overview
    • The Birthday paradox / attack
  • Isogeny Based Cryptography
    • Introduction to Isogeny Cryptography
    • Isogenies
    • Isogeny and Ramanujan Graphs
  • Appendices
    • Sets and Functions
    • Probability Theory
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  1. Lattices
  2. Lattice reduction

HKZ reduced

Definition

Let πi\pi_iπi​as the projection to the orthogonal complement of {bj}j=1i−1\left\{b_j\right\}_{j=1}^{i-1}{bj​}j=1i−1​.Then the basis is HKZ-reduced if it is size-reduced and ∣∣bi∗∣∣=λ1(πi(L))||b_i^*||=\lambda_1\left(\pi_i(L)\right)∣∣bi∗​∣∣=λ1​(πi​(L)). This definition gives us a relatively simple way to compute a HKZ-reduced basis by iteratively finding the shortest vector in orthogonal projections.

Bounds

4i+3≤(∣∣bi∣∣λi(L))2≤i+34\frac4{i+3}\leq\left(\frac{||b_i||}{\lambda_i(L)}\right)^2\leq\frac{i+3}4i+34​≤(λi​(L)∣∣bi​∣∣​)2≤4i+3​
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Last updated 4 years ago

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