# Martin Ekerå's research while affiliated with The Swedish Armed Forces and other places

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## Publications (11)

We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order $r$ in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any $r$, without re-running the quantum part or increas...

We show that given the order of a single element selected uniformly at random from $${\mathbb {Z}}_N^*$$ Z N ∗ , we can with very high probability, and for any integer N , efficiently find the complete factorization of N in polynomial time. This implies that a single run of the quantum part of Shor’s factoring algorithm is usually sufficient. All p...

We significantly reduce the cost of factoring integers and computing discrete logarithms in finite fields on a quantum computer by combining techniques from Shor 1994, Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Ekerå-Håstad 2017, Ekerå 2017, Ekerå 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction using pl...

We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's alg...

We revisit the quantum algorithm for computing short discrete logarithms that was recently introduced by Ekerå and Håstad. By carefully analyzing the probability distribution induced by the algorithm, we show its success probability to be higher than previously reported. Inspired by our improved understanding of the distribution, we propose an impr...

We show that given the order of a single element selected uniformly at random from $\mathbb Z_N^*$, we can with very high probability, and for any integer $N$, efficiently find the complete factorization of $N$ in polynomial time. This implies that a single run of the quantum part of Shor's factoring algorithm is usually sufficient. All prime facto...

We significantly reduce the cost of factoring integers and computing discrete logarithms over finite fields on a quantum computer by combining techniques from Griffiths-Niu 1996, Zalka 2006, Fowler 2012, Eker{\aa}-H{\aa}stad 2017, Eker{\aa} 2017, Eker{\aa} 2018, Gidney-Fowler 2019, Gidney 2019. We estimate the approximate cost of our construction u...

We heuristically demonstrate that Shor's algorithm for computing general discrete logarithms, modified to allow the semi-classical Fourier transform to be used with control qubit recycling, achieves a success probability of approximately 60% to 70% in a single run. By slightly increasing the number of group operations that are evaluated quantumly,...

In this paper we generalize the quantum algorithm for computing short discrete logarithms previously introduced by Eker{\aa} so as to allow for various tradeoffs between the number of times that the algorithm need be executed on the one hand, and the complexity of the algorithm and the requirements it imposes on the quantum computer on the other ha...

We review, implement and evaluate a recent method for breaking one time pad ciphers when the same pad is reused. The attack can also be applied to a number of other ciphers when they are used incorrectly. Some possible enhancements to the algorithm are discussed, including how to exploit multiple reuses of the same pad, what happens when the the en...

We review, implement and evaluate a recent method for breaking one time pad ciphers when the same pad is reused. The attack can also be applied to a number of other ciphers when they are used incorrectly. Some possible enhancements to the algorithm are discussed, including how to exploit multiple reuses of the same pad, what happens when the the en...

## Citations

... As a corollary, we use the reduction from the IFP to the OFP in [7] to prove a lower bound on the probability of completely factoring any integer N efficiently in a single run of the order-finding algorithm. Compared to the bound in [7], our lower bound also accounts for the probability of the order-finding algorithm failing to recover r. ...

... Year Time complexity Space complexity Shor [58] 1994 O(n 3 ) O(n) Proos et al. [71] 2003 O(n 2 ) -Ekera et al. [72] 2019 -O(n 2 ) With the efficiency of quantum computers, the security provided by cryptosystems, which are based on IFP and discrete logarithmic problems (DLP), seems to be short-lived. Speaking of IFP, RSA and Paillier encryption are vulnerable against Shor's algorithm. ...

... In 2001, it was shown how to successfully factor 15 via Shor's algorithm on a 7-qubit nuclear magnetic resonance (NMR) quantum computer [41]. In fact, few years ago, it was argued in Ref. [42] that it would take about 8 hours to factor 2048-bit RSA integers using 20 million noisy qubits! We can summarize the algorithm in following steps: ...

Reference: Notes on Quantum Computation and Information

... Here, Euclidean lattice simulations on classical computers help quantify the scheme-dependent systematic errors [7,11,19,[73][74][75]. We can draw another analogy to the case of prime factorization where Ekerå and Håstad's modifications [76][77][78][79] of Shor's algorithms [41] used classical processing to reduce qubits and quantum arithmetic while increasing the success rate. In the same way, lattice calculations have a number of steps that can potentially be offloaded to classical resources. ...

... Here, Euclidean lattice simulations on classical computers help quantify the scheme-dependent systematic errors [7,11,19,[73][74][75]. We can draw another analogy to the case of prime factorization where Ekerå and Håstad's modifications [76][77][78][79] of Shor's algorithms [41] used classical processing to reduce qubits and quantum arithmetic while increasing the success rate. In the same way, lattice calculations have a number of steps that can potentially be offloaded to classical resources. ...