Editing Uncloneable Encryption (section)

==Protocol Description==

Pre-shared key <math>(k, e, c_1, b)</math> is established.

===Encoding:===

# Divide the <math>n</math>-bit message into <math>r</math> groups of <math>s</math> bits.
# Define a polynomial <math>f</math> of degree <math>r</math> whose first <math>r</math> coefficients are the registers <math>m_0, m_1,\ldots, m_{r-1}</math> of the <math>n</math>-bit message.
# The constant term, <math>m_r</math>, is chosen such that <math>f(k) = 0</math>.
# XOR the string <math>(m_0, \ldots , m_r)</math> with <math>e</math>, producing a new classical string <math>y</math> of length <math>n+s</math> bits.
# Consider the particular coset of the classical error-correcting code <math>C_1</math> given by the syndrome <math>c_1</math>.
# Pick the string <math>z</math> at random so that its coset of <math>C_1/C_2^{\perp}</math> in the coset <math>c_1</math> of <math>C_1</math> corresponds to <math>y</math>
# Transmit <math>N</math> qubits after operations based on b such that:
## When the <math>i^{th}</math> bit of <math>b</math> is <math>0</math>, transmit the <math>i^{th}</math> bit of <math>z</math> in the computational basis.
## When the <math>i^{th}</math> bit of <math>b</math> is <math>1</math>, transmit the <math>i^{th}</math> bit of <math>z</math> in the Hadamard basis.

===Decoding:===
# Upon receiving the <math>N</math> qubits, measure in the computational basis if the <math>i^{th}</math> bit of <math>b</math> is <math>0</math>, else in the Hadamard basis the <math>i^{th}</math> bit of <math>b</math> is <math>1</math>, to get <math>z</math>.
# Calculate the parity checks of the classical code <math>C_1^{\perp}</math>. If they are not equal to the string <math>c_1</math>, there are errors in the state, which can be corrected using the standard decoding map.
# Evaluate the parity checks of <math>C_2/C_1^{\perp}</math>, producing a <math>n + s</math>-bit string <math>y</math>.
# XOR <math>y</math> with <math>e</math>, producing a new string <math>(m_0, \ldots , m_r)</math>.
# Consider the <math>(m_0, \ldots, m_r)</math> as the coefficients of a polynomial <math>f</math>. Accept only if <math>f(k) = 0</math>.

<div style='text-align: right;'>''*contributed by Natansh Mathur''</div>