1 Introduction
In August 2015 the National Security Agency (NSA) announced plans to upgrade security standards; the goal is to replace all deployed cryptographic protocols with quantum secure protocols. This transition requires a new security standard to be accepted by the National Institute of Standards and Technology (NIST). Proposals for quantum secure cryptosystems and protocols have been submitted for the standardization process. There are six main primitives currently proposed to be quantumsafe: (1) latticebased (2) codebased (3) isogenybased (4) multivariatebased (5) hashbased, and (6) groupbased cryptographic schemes.
One goal of cryptography, as it relates to complexity theory, is to analyze the complexity assumptions used as the basis for various cryptographic protocols and schemes. A central question is determining how to generate intractible instances of these problems upon which to implement an actual cryptographic scheme. The candidates for these instances must be platforms in which the hardness assumption is still reasonable. Determining if these groupbased cryptographic schemes are quantumsafe begins with determining the groups in which these hardness assumptions are invalid in the quantum setting.
In what follows we address the quantum complexity of the Hidden Subgroup Problem (HSP) to determine the groups in which the hardness assumption still stands. The Hidden Subgroup Problem (HSP) asks the following: given a description of a group and a function for some finite set is guaranteed to be strictly periodic, i.e. constant and distinct on left (resp. right) cosets of a subgroup , find a generating set for .
It is important to note that Simon’s problem of computing a XORmask, Shor’s algorithm for factoring and finding the discrete log, Boneh’s algorithm for finding a hidden linear function, and Kitaev’s algorithm for the abelian stabilizer problem are all special cases of HSP. Therefore, the HSP is directly related to problems such as breaking onetime pad, discrete logarithm problem, graph isomorphism problem (which is now known to be in qualipolynomial), latticebased problems, the problem for factoring for RSA.
The classical complexity of HSP is known [1]: Suppose that has a set of subgroups, such that . Then a classical computer must make queries to solve the HSP. The classical cases in which HSP is easy are the cases in which has only a polynomial number of subgroups, allowing bruteforce for the function on all subgroups.
We provide a survey of results regarding the complexity of quantum algorithms for solving HSP in various group platforms. We also provide information on the relationship between HSP and other computational problems. These results provide insight into potential platforms for quantum safe cryptography, when the underlying hard problem is reducible to HSP.
2 Groupbased Cryptography
Groupbased cryptography could be shown to be postquantum if the underlying security problem is NPcomplete or unsolvable; firstly, we need to analyze the problem’s equivalence to HSP, then analyze the applicability of Grover’s search problem. Cryptanalysis based on a reduction to solving HSP creates some obstacles as the groups under consideration below are mostly infinite and do not have an efficient algorithm for HSP. In the following cryptosystems a connection to HSP can assist in the analysis of security.
For example in [2] a practical cryptanalysis of WalnutDSA was proposed, a postquantum cryptosystem using the conjugacy search problem (CSP) over braid groups that was submitted to the NIST competition in 2017 [3]. It has been argued since the braid group does not contain any nontrivial finite subgroups, there does not seem to be any viable way to connect CSP with HSP. It has been shown there is no reduction connection between the CSP and HSP, [4, 5]. As for analysis via Grover’s algorithm [6], it has been mentioned that a majority of the time for signature verification in WalnutDSA is repeated EMultiplications.
There are alternative grouptheoretic problems and classes of groups which have been proposed for postquantum cryptography. For example, the first protocryptosystem based on groups was proposed by WagnerMagyarik in [7] for which the word choice problem was hard. Later on FloresKahrobaeiKoberda proposed rightangled Artin groups for various other cryptographic protocols [8], [9]. Eick and Kahrobaei proposed Polycyclic groups, using the Conjugacy Search Problem [10] for cryptography. Later on GryakKahrobaei wrote a survey and proposed other grouptheoretic problems for consideration using polycyclic groups [11]. KahrobaeiKoupparis [12] proposed a postquantum digital signature using polycyclic groups. KahrobaeiKhan proposed a publickey cryptosystem using polycyclic groups [13]. HabeebKahrobaeiKoupparisShpilrain proposed the use of a semigroup of matrices with a semidirect product structure [14].
Thompson groups have been considered by ShpilrainUshakov based on the Decomposition Search Problem [15]. Hyperbolic groups have been proposed by ChatterjiKahrobaeiLu using properties of subgroup distortion and the Geodesic Length Problem [16]. Free metabelian groups have been proposed based on the Subgroup Membership Search Problem by ShpilrainZapata [17]. KahrobaeiShpilirain proposed Free nilpotent pgroups for a semidirect product public key cryptosystem [18]. Linear groups were proposed by BaumslagFineXu [19]. Grigorchuk groups, have been proposed by [20]. Groups of matrices, for a Homomorphic Encryption scheme were proposed by GrigorievPonomarenko [21].
3 Relation of HSP to Other Computational Problems
Many computational problems are special cases of the HSP; an efficient algorithm for HSP over a certain group implies an efficient algorithm for some other computational problem. It is important to note that one method of determining an efficient quantum solution to a hard problem consists of reducing the problem to an instance of HSP over a group with a known efficient solution. This consists of determining the appropriate group , the subgroup and the strongly periodic function . For example, Simon’s problem can be viewed as an instantiation of HSP over with a subgroup of order . Duetsch’s algorithm solves a variant of HSP where is either (f is balanced) or (f is constant). Shor’s algorithm solves period finding as a special case of HSP, allowing for an efficient quantum algorithm for factoring and discrete log.
The graph automorphism (resp. isomorphism) problem can also be framed as an instance of HSP. To solve graph automorphism we consider HSP in the symmetric group on letters, , any function which hides the trivial subgroup is an automorphism. Analogously the graph isomorphism problem is an instance of HSP over the wreath product [22]. Also, solutions to HSP can solve the abelian stabilizer problem; when is acting on a finite set and where is the stabilizer of we have that can be defined such that is strongly periodic.
A solution to a particular instance of HSP is a solution to the hidden linear functions [23]; if is a permutation of and is such that , we have hiding . Additionally, selfshiftequivalent polynomials can be framed as an instance of HSP, in this case Grigoriev shows how to compute the hidden subgroup [24].
An efficient solution to HSP would imply an efficient solution to certain lattice problems. Specifically, the
Unique Shortest Vector Problem (USVP) is NPhard for
, and has a polynomial time classical solution when is large. The dihedral HSP, based on standardmethod (found below) can be used to solve poly()USVP [25]. HSP over the symmetric and dihedral groups are highly motivated open questions in postquantum groupbased cryptography.Another, related, computational problem is the Hidden Shift Problem, which has been proposed as a basis for postquantum cryptography in symmetric cryptosystems that are quantumCPA secure [26]. Other than the use of a generalization of Simon’s algorithm, and Kuperberg’s algorithm discussed above, very little is known about the Hidden Shift Problem. Clearly this problem is closely related to HSP as some solutions coincide. It is important to note that constructions baed on this Hidden Shift problem have also remained quantum secure.
4 Solution Methods
The standard method of solving HSP over performs the following steps. First, the algorithm queries the periodic function in superposition and discards the register which holds the output. This leaves the first register entangled in a hidden subgroup state, a superposition of coset representatives for some left traversal . Following this, the state can be sampled using postprocessing techniques to determine . In the following we have as the coset state. This approach reduces the problem to a problem of quantum mechanics: how to distinguish the members of an ensemble of quantum hidden subgroup states.
How do we measure the state? The problem of distinguishing these quantum states has some proposed solutions. Most namely, the often optimal solution entitled Pretty Good Measurement (PGM) can be used. An obstacle to performing PGM is the lack of an efficient QTF/CFT in the underlying group. For these instances we know of no efficient quantum algorithm for solving HSP.
5 Results
Finite Abelian and Finite NearAbelian.
The infamous quantum algorithms of Simon and Shor provide quantum solutions to HSP in the abelian cases where and respectively. Shor’s algorithm extends to the general abelian case as well, providing a polynomial time quantum algorithms with bounded error [27, 28, 29]
. The probability of success can be improved to
when is abelian of smooth order, i.e. if all prime factors of are at most [30].In the case that is nearly abelian, i.e. if the value where is the normalizer for is sufficiently large there are established computational bounds on HSP. The size of this intersection relative to the group is a measure of the abelianness of . Gavinsky [31] gave results to show that an efficient algorithm exists when .
The HSP can be solved in polynomial time by a quantum algorithm to find hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2subgroup of small index or with cyclic factor group [32]. Subexponential algorithms for HSP in any solvable group have been given by Friedl et al. [33].
When is a known finite abelian group with a subgroup , given blackbox access to the hiding function , we know that a quantum computer can uniquely and completely determine in time and query complexity. When is nearly abelian, or built from abelian parts, one can leverage this fact to obtain an efficient algorithm for HSP.
Finite, NonAbelian.
The finite nonabelian case of HSP is more elusive. Shor’s algorithm extends to any group when
is normal if quantum fourier transform (QFT) can be efficiently computed over the group
[34]. The algorithm also extends to when has few conjugates, requiring the quantum character transform (QCT) over the group algebra [35]. This variation is not applicable when has many conjugates, as in some of the following cases. Alternatively, when is normal in , a blackbox group, generators for can be found in time polynomial in the input size + [32] without requiring an efficient QFT over .Additionally, the quantum computation of the discrete log in semigroups [36] is an instance of HSP.When the group is the (discrete) Heisenberg group it is sufficient to be able to distinguish cyclic subgroups of order , . Thus, finding an arbitrary reduces to determining two parameters , given the coset state produced by the standard method using which hides . This can be efficiently computed with an overall success probability close to .
In a more difficult case we consider instances of HSP in the dihedral group of order , where the function hides of order . In this case , a hidden reflection, has many conjugates in and the QCT based solution is not applicable. Kuperberg stated that finding an arbitrary hidden subgroup of reduces to finding the slope of a hidden reflection and provides a quantum algorithm with both time and query complexity of , applicable to for all values of but achieving an even tighter complexity bound for specific smooth values of [37] . Kuperberg’s algorithm also provides a solution to the hidden shift problem in an arbitrary finitely generated abelian group . Regev improved upon the bounds of Kuperberg’s original algorithm providing a polynomial space variation to the original superpolynomial space algorithm, which still achieves subexponential complexity [38]. Regev showed that an efficient solution to the dihedral HSP implies a quantum solution to lattice problems [25].
When is a type of wreath product , Roetteler et al. [39] provide a positive result for finding an efficient solution to the nonabelian HSP within . This result is due to the existence of an efficient nonabelian QFT in . Wreath product groups are in turn a subset of semidirect product groups. When is (one of some groups that are) a semidirect product of abelian groups, alternative efficient algorithms have been proposed. The polycyclic HSP has been addressed for for fixed prime power [40], with and , certain affine groups [41], [42], with , where where and [43], and where has a special prime factorization [44].
In general, when is a group of finite order the HSP problem has quantum query complexity of as shown by Ettinger, Høyer and Knill [45]; for any group , queries provides sufficient statistical information to solve HSP. This result provides no guarantees on computational complexity. The real problem is determining how to implement the queries efficiently, as well as how to control the amount of postprocessing required by the algorithm. In the case of the dihedral group, an algorithm with the lower bound on query complexity has been constructed, but the postprocessing required is exponential. In many cases the inefficiency of a proposed quantum algorithm is primarily due to the inefficiency of the required quantum measurement or postprocessing within the group.
Infinite.
What seems to be an obstacle for infinite groups is that the quantum computer should assume the following state: The meaning of this is clear for finite groups. The abelian infinite HSP was clearly first considered with Shor’s algorithm, over . In [46], infinitedimensional HSP has been mentioned, particularly for infinite abelian groups . Additionally, HSP has been defined and considered for infinite abelian groups of the form for some finite group [47]. Other than the cases of , , , and combinations of these in which an efficient algorithm exists, the infinite and continuous HSP has not been addressed within the literature for the nonabelian case.
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