29 research outputs found
Improved reversible and quantum circuits for Karatsuba-based integer multiplication
Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor\u27s algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba\u27s recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n^1.585) to O(n^1.427). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees
Quantum Circuits for Toom-Cook Multiplication
In this paper, we report efficient quantum circuits for integer
multiplication using Toom-Cook algorithm. By analysing the recursive tree
structure of the algorithm, we obtained a bound on the count of Toffoli gates
and qubits. These bounds are further improved by employing reversible pebble
games through uncomputing the intermediate results. The asymptotic bounds for
different performance metrics of the proposed quantum circuit are superior to
the prior implementations of multiplier circuits using schoolbook and Karatsuba
algorithms
Space-efficient quantum multiplication of polynomials for binary finite fields with sub-quadratic Toffoli gate count
Multiplication is an essential step in a lot of calculations. In this paper
we look at multiplication of 2 binary polynomials of degree at most ,
modulo an irreducible polynomial of degree with input and output
qubits, without ancillary qubits, assuming no errors. With straightforward
schoolbook methods this would result in a quadratic number of Toffoli gates and
a linear number of CNOT gates. This paper introduces a new algorithm that uses
the same space, but by utilizing space-efficient variants of Karatsuba
multiplication methods it requires only Toffoli gates at the
cost of a higher CNOT gate count: theoretically up to but in examples
the CNOT gate count looks a lot better.Comment: 15 pages, 5 figure
Another Concrete Quantum Cryptanalysis of Binary Elliptic Curves
This paper presents concrete quantum cryptanalysis for binary elliptic
curves for a time-efficient implementation perspective (i.e., reducing the circuit
depth), complementing the previous research by Banegas et al., that focuses on the
space-efficiency perspective (i.e., reducing the circuit width). To achieve the depth
optimization, we propose an improvement to the existing circuit implementation of
the Karatsuba multiplier and FLT-based inversion, then construct and analyze the
resource in Qiskit quantum computer simulator. The proposed multiplier architecture,
improving the quantum Karatsuba multiplier by Van Hoof et al., reduces the
depth and yields lower number of CNOT gates that bounds to O(nlog2(3)) while
maintaining a similar number of Toffoli gates and qubits. Furthermore, our improved
FLT-based inversion reduces CNOT count and overall depth, with a tradeoff
of higher qubit size. Finally, we employ the proposed multiplier and FLT-based inversion
for performing quantum cryptanalysis of binary point addition as well as the
complete Shor’s algorithm for elliptic curve discrete logarithm problem (ECDLP).
As a result, apart from depth reduction, we are also able to reduce up to 90% of the
Toffoli gates required in a single-step point addition compared to prior work, leading
to significant improvements and give a new insights on quantum cryptanalysis for a
depth-optimized implementation
Concrete quantum cryptanalysis of binary elliptic curves
This paper analyzes and optimizes quantum circuits for computing discrete logarithms on binary elliptic curves, including reversible circuits for fixed-base-point scalar multiplication and the full stack of relevant subroutines. The main optimization target is the size of the quantum computer, i.e., the number of logical qubits required, as this appears to be the main obstacle to implementing Shor’s polynomial-time discrete-logarithm algorithm. The secondary optimization target is the number of logical Toffoli gates. For an elliptic curve over a field of 2n elements, this paper reduces the number of qubits to 7n + ⌊log2 (n)⌋ + 9. At the same time this paper reduces the number of Toffoli gates to 48n3 + 8nlog2(3)+1 + 352n2 log2 (n) + 512n2 + O(nlog2(3)) with double-and-add scalar multiplication, and a logarithmic factor smaller with fixed-window scalar multiplication. The number of CNOT gates is also O(n3). Exact gate counts are given for various sizes of elliptic curves currently used for cryptography