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Lattice

CVP_embedding(lattice_basis, v, M=1)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/embeddings.py 
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def CVP_embedding(lattice_basis: SquareMatrix, v: Vector, M = 1) -> SquareMatrix:
    r'''

    Args:

    Returns:

    '''
    assert v.shape[0] == lattice_basis.shape[0]
    n = lattice_basis.shape[0]


    return np.block([[lattice_basis, np.zeros((n, 1), dtype=lattice_basis.dtype)], [v, np.array([M])]])

bai_galbraith_embedding(A, b, q)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/embeddings.py 
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def bai_galbraith_embedding(A: MatrixInt, b: VectorInt, q: int) -> SquareMatrixInt:
    r'''

    Args:

    Returns:

    '''
    #B = np.block([np.identity(m, int), A, -b.reshape(-1,1)])
    pass

dual_q_ary_basis(A, q)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/embeddings.py 
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def dual_q_ary_basis(A: MatrixInt, q: int) -> MatrixModInt:
    r'''

    Args:

    Returns:

    '''
    m, n = A.shape
    Y = matrix.mod_left_kernel(A,q)
    return np.block([[Y], [ np.zeros((n, m - n), dtype=np.int64), q * np.identity(n, int)]])

q_ary_basis(A, q)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/embeddings.py 
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def q_ary_basis(A: MatrixInt, q: int) -> MatrixModInt:
    r'''

    Args:

    Returns:

    '''
    m, n = A.shape
    B = np.block([[A.T], [q * np.identity(m, int)]])
    H, *_ = matrix.HNF(B)
    return H[:m, :m]

subset_sum_lattice(sequence, S)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/embeddings.py 
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def subset_sum_lattice(sequence, S):
    r'''

    Args:

    Returns:

    '''
    n = len(sequence)
    M = np.identity(n + 1, dtype=float) * 2
    M[-1] = 1
    M[:-1, -1] = np.array(sequence, dtype=float)
    M[-1, -1] = S

babai_cvp(arbitrary_vector, lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def babai_cvp(arbitrary_vector: Vector, lattice_basis: SquareMatrix) -> VectorInt:
    r'''

    Args:

    Returns:

    '''
    return np.rint(arbitrary_vector @ np.linalg.inv(lattice_basis)).astype(int)

dual_basis(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def dual_basis(lattice_basis: SquareMatrix) -> SquareMatrix:
    r'''

    Args:

    Returns:

    '''
    return np.linalg.inv(lattice_basis.T)

gaussian_expected_shortest_length(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def gaussian_expected_shortest_length(lattice_basis: SquareMatrix) -> float:
    r'''

    Args:

    Returns:

    '''
    n = rank(lattice_basis)
    return np.sqrt(n / (2 * np.pi * np.e)) * (volume(lattice_basis) ** (1/n))

hadamard_ratio(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def hadamard_ratio(lattice_basis: SquareMatrix) -> float:
    r'''

    Args:

    Returns:

    '''
    return (volume(lattice_basis) / np.linalg.norm(lattice_basis, axis=1).prod()) ** (1/rank(lattice_basis))

rank(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def rank(lattice_basis: SquareMatrix) -> int:
    r'''

    Args:

    Returns:

    '''
    return lattice_basis.shape[0]

transition_matrix(from_basis, to_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def transition_matrix(from_basis: SquareMatrix, to_basis: SquareMatrix) -> SquareMatrixInt:
    r'''

    Args:

    Returns:

    '''
    return np.rint(to_basis @ np.linalg.inv(from_basis)).astype(int)

volume(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/fullrank.py 
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@enforce_type_check
def volume(lattice_basis: SquareMatrix) -> float:
    r'''

    Args:

    Returns:

    '''
    return abs(np.linalg.det(lattice_basis))

GLR_2dim(lattice_basis)

Gaussian Lattice reduction in dimension 2

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def GLR_2dim(lattice_basis: SquareMatrix) -> SquareMatrixFloat:
    '''
    Gaussian Lattice reduction in dimension 2

    Args:

    Returns:

    '''
    if lattice_basis.shape != (2,2):
        raise ValueError(f"Lattice has to have rank 2 for gaussian reduction")

    w1 = lattice_basis[0]
    w2 = lattice_basis[1]

    v1 = w1.astype(float)
    v2 = w2.astype(float)
    if np.linalg.norm(v1) > np.linalg.norm(v2):
        v1, v2 = v2, v1

    while np.linalg.norm(v2) > np.linalg.norm(v1):
        m = round(np.dot(v1, v2) / np.dot(v1, v1))
        if m == 0:
            return v1, v2
        v2 = v2 - m * v1
        if np.linalg.norm(v1) > np.linalg.norm(v2):
            v1, v2 = v2, v1

    return np.array([v1, v2])

GSO(B)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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@enforce_type_check
def GSO(B: Matrix) -> Tuple[MatrixFloat, SquareMatrixFloat]:
    r'''

    Args:

    Returns:

    '''
    m, n = B.shape
    proj_coeff = lambda q, b: np.dot(b, q) / np.dot(q, q)
    B_star = B.astype(float)
    U = np.identity(m)

    for j in range(1, m):
        b = B_star[j].copy()
        for i in range(j):
            U[i,j] = proj_coeff(B_star[i], b)
            B_star[j] -= U[i][j] * B_star[i]

    # B = U.T @ B_star
    return B_star, U

LLL(lattice_basis, delta=0.75)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def LLL(lattice_basis: SquareMatrix, delta: float = 0.75) -> SquareMatrixFloat:
    r'''

    Args:

    Returns:

    '''
    n = lattice_basis.shape[0]
    B = lattice_basis.astype(float)
    while True:
        Bstar, _ = GSO(B)
        # Reduction Step
        for i in range(1, n):
            for j in range(i-1, -1, -1):
                cij = round(np.dot(B[i], Bstar[j]) / np.dot(Bstar[j], Bstar[j]))
                B[i] = B[i] - cij * B[j]
        # Swap step
        exists = False
        for i in range(n - 1):
            u = np.dot(B[i + 1], Bstar[i]) / np.dot(Bstar[i], Bstar[i])
            r = u * Bstar[i] + Bstar[i + 1]
            if delta * np.dot(Bstar[i], Bstar[i]) > np.dot(r, r):
                B[[i, i + 1]] = B[[i + 1, i]]
                exists = True
                break
        if not exists:
            break
    return B

babai_nearest_plane(lattice_basis, w)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def babai_nearest_plane(lattice_basis: SquareMatrix, w: VectorFloat):
    r'''

    Args:

    Returns:

    '''
    n = lattice_basis.shape[0]
    B = LLL(lattice_basis, 0.75)
    b = w.astype(float)
    for j in range(n - 1, -1, -1):
        Bstar, _ = GSO(B)
        cj = round(np.dot(b, Bstar[j]) / np.dot(Bstar[j], Bstar[j]))
        b = b - cj * B[j]
    return w - b

is_LLL_reduced(lattice_basis, delta)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def is_LLL_reduced(lattice_basis: Matrix, delta: float):
    r'''

    Args:

    Returns:

    '''
    return is_size_reduced(lattice_basis) and lovasz_condition(lattice_basis, delta)

is_size_reduced(lattice_basis)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def is_size_reduced(lattice_basis: Matrix) -> bool:
    r'''

    Args:

    Returns:

    '''
    _, U = GSO(lattice_basis)
    return np.all(np.abs(U[np.fromfunction(lambda i, j: i < j, U.shape).nonzero()]) <= 0.5)

lovasz_condition(lattice_basis, delta)

Args:

Returns:

Source code in src/lbpqc/primitives/lattice/reductions.py 
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def lovasz_condition(lattice_basis: Matrix, delta: float) -> bool:
    r'''

    Args:

    Returns:

    '''
    norm2 = lambda x: np.sum(x * x, axis=1)
    G, U = GSO(lattice_basis)
    lhs = delta * norm2(G[:-1])
    rhs = norm2(G[1:] + np.diag(U, 1)[:, np.newaxis] * G[:-1])
    return np.all(lhs <= rhs)