The Structured Totient Preimage Problem: Reconstruction, Collisions, and Cryptographic Implications

The Structured Totient Preimage Problem: Reconstruction, Collisions, and Cryptographic Implications Luis Adrián Lizama-Pérez

arXiv:2608.19191v1 [cs.CR] 19 Aug 2026

Universidad Autónoma Metropolitana, Unidad Lerma, Mexico l.lizama@correo.ler.uam.mx ORCID: 0000-0001-5109-2927

Abstract We define and study the Structured Totient Preimage (STP) problem as a restricted reconstruction relation with a direct cryptographic motivation. Let p1 , . . . , pk be distinct primes of the same bit length and reveal only x=

k Y

(pi − 1).

i=1

Given (x, λ, k), STP asks for any set of k distinct λ-bit primes satisfying this product. The relation is efficiently verifiable, but its reconstruction complexity is not known. We establish three concrete results. First, for factored x we derive the exact number of ordered exponent allocations and a bound showing that direct reconstruction is polynomial for fixed k when Ω(x) = O(log λ); this rules out that regime as a basis for a strong hardness claim. Second, we give exhaustive algorithms for reconstruction and collision analysis. Third, we exhaustively evaluate 28 parameter pairs, with 2 ≤ k ≤ 5, up to λ = 16 for pairs and 4,588,935 prime sets in the largest census. The data quantify non-injectivity through collision participation, maximum multiplicity, and conditional ambiguity in bits. These results isolate STP from general inverse-totient computation and motivate a Structured Totient Preimage Assumption for explicitly growing parameter families. Under such an assumption, STP becomes a candidate preimage-resistant relation whose implications for commitments, proofs of knowledge of multiplicative witnesses, and authentication can be stated precisely. The paper establishes the computational foundation and parameter constraints for those constructions; it does not claim a security reduction or post-quantum hardness.

Keywords: Euler’s totient function; computational number theory; exact enumeration; shifted primes; multiplicative reconstruction; collision statistics; preimage resistance; cryptographic implications. 2020 Mathematics Subject Classification: 11Y16 (primary); 11A25, 94A60 (secondary).

1

Introduction

Euler’s totient function is easy to evaluate from a prime factorization. If n = φ(n) =

Y αj −1

qj

αj j qj , then

Q

(qj − 1).

j

Its inverse behavior is more complicated: some integers are not totients, the function is not injective, and the number of inverse images varies widely [3, 7, 5, 6]. Consequently, recognizing a totient, finding one inverse, and listing every inverse are different tasks. 1

This paper isolates a narrower relation. Distinct primes pi are sampled from one dyadic interval, while multiplication removes the boundaries between the shifted factors pi − 1. Reconstruction then means recovering any partition of the multiplicative resources into blocks di for which di + 1 are distinct primes of the prescribed size. This restriction is not a minor variant of asking for every solution of φ(n) = x: it fixes the number, size, distinctness, and square-free structure of the hidden primes. The motivation is both computational and cryptographic. Prospective commitment, witnessverification, and authentication mechanisms may use a valid prime set as a secret multiplicative witness and its shifted product as public data. Before such protocols can be assessed, the underlying relation must be defined independently: its witnesses, collisions, input models, elementary algorithms, and plausible parameter regimes must be known. Establishing that foundation is the purpose of this paper. Four research questions organize the work: RQ1. What is the precise search relation induced by products of shifted equal-bit primes? RQ2. What can be proved about reconstruction when the factorization of the public product is supplied? RQ3. How frequent and how large are collisions over exactly enumerable parameter ranges? RQ4. Which results support, constrain, or rule out the use of the relation as a cryptographic hardness candidate? The principal contributions are: 1. the formulation of STP as a standalone, efficiently verifiable search relation, with factored and unfactored input models; 2. an exact exponent-allocation count, a reconstruction procedure, and a proved parameter regime in which factored-input candidate enumeration is polynomial; 3. an exhaustive collision census for 28 parameter pairs, including explicit unique and non-unique reconstructions; 4. an entropy-based ambiguity statistic that measures how much of a uniformly sampled witness remains undetermined after x is revealed; and 5. a candidate hardness assumption and a precise boundary between the results proved here and the separate cryptographic constructions they are intended to support. The contribution is therefore not a new general inverse-totient algorithm. It is the isolation, exact finite analysis, and cryptographic delimitation of a restricted multiplicative witness relation. No unconditional NP-hardness, average-case hardness, security reduction, or post-quantum claim is made. To the best of our knowledge, prior inverse-totient work has not isolated this joint equal-bit, distinct-prime, fixed-cardinality restriction or reported the corresponding fiber collision census.

2

The structured totient preimage problem

Table 1 collects the notation used throughout the paper. In particular, the hidden primes pi , the shifted factors di = pi − 1, and the prime factors qj of the public value x are different objects.

2

Table 1: Notation used throughout the manuscript. Symbol

Meaning

Symbol

Meaning

k

Pλ

Bit length of every hidden prime Set of exactly λ-bit primes

P = {p1 , . . . , pk }

Hidden prime-set witness

di = pi − 1

x

Public shifted product

qj j

aji

Tλ,k

fλ,k

Share of exponent ej assigned to di Shifted-product map

µλ,k (x)

Nλ

Number of λ-bit primes

Vλ,k

Ak (x)

Number of ordered exponent allocations Random witness and its public image

Ω(x)

Number of distinct hidden primes Set of admissible k-element witnesses Shifted factor associated with pi Prime-power factors in the factorization of x Number of admissible prime sets Number of witnesses in the fiber over x Number of represented public values Prime factors of x, counted with multiplicity Ambiguity, collision participation, and mean fiber size

λ

P, X

2.1

Sλ,k

e

∆λ,k , Cλ,k , µλ,k

Relation and input models

For an integer λ ≥ 2, define the set of exactly λ-bit primes Pλ = {p prime : 2λ−1 ≤ p < 2λ }. For 1 ≤ k ≤ |Pλ |, let !

Sλ,k = {P ⊆ Pλ : |P | = k},

Tλ,k = |Sλ,k | =

|Pλ | . k

Define the shifted-product map fλ,k (P ) =

Y

(p − 1)

p∈P

and the multiplicity of a represented value x by µλ,k (x) = |{P ∈ Sλ,k : fλ,k (P ) = x}|. Definition 1 (Structured totient preimage). For given x, λ, and k, a structured preimage is a set P ∈ Sλ,k such that fλ,k (P ) = x. The search task is to return any such set under the promise that one exists. The enumeration task is to return all such sets. These tasks are relations: if µλ,k (x) > 1, recovering the set that originally generated x is not required and in general cannot be inferred from x alone. Two input models must be distinguished. Factored input. Q e The complete factorization x = rj=1 qj j is supplied. Unfactored input. Only x, λ, and k are supplied. A factor-first method must also factor x. 3

Every structured preimage P = {p1 , . . . , pk } determines the square-free integer n = i pi and satisfies φ(n) = x. The converse does not hold for an arbitrary inverse totient because inverse images may contain prime powers, primes of different sizes, or a different number of prime factors. Q

2.2

A different public–witness split

The familiar RSA relation publishes n = pq and keeps its factorization hidden; φ(n) is normally not public. STP uses a different split. The public value is 

x = φ

 Y

p =

p∈P

Y

(p − 1),

p∈P

while the witness is a prime set P satisfying explicit cardinality and bit-length restrictions. In the stronger factored-input model, even the prime factorization of x may be public; the remaining task is to reconstruct valid boundaries among its prime-power resources. The witness relation is efficiently verifiable. Given P , a verifier checks its cardinality, distinctness, bit lengths, primality, and shifted product. This combination of easy verification and potentially difficult reconstruction is the feature needed by the motivating cryptographic constructions. It is a candidate feature, not yet a hardness theorem.

2.3

Candidate hardness statement

Let k = k(λ) be a polynomially bounded growing function satisfying 2 ≤ k(λ) ≤ |Pλ |. On input 1λ , let GenSTP sample P uniformly from Sλ,k(λ) and output (x, λ, k(λ)), where x = fλ,k(λ) (P ). The following experiment records the exact assumption that a cryptographic construction based on STP would need. Candidate Assumption 1 (Structured Totient Preimage Assumption (STPA)). For every probabilistic polynomial-time classical algorithm A, 

Pr

−1 ′ P ∈ fλ,k(λ) (x) :

$

P ← − Sλ,k(λ) ,



x = fλ,k(λ) (P ),

P ′ ← A(1λ , k(λ), x)

= negl(λ),

where the probability is over the choices of P and the internal randomness of A. The factored STPA gives the algorithm the complete factorization of x in canonical form as additional input. It is the stronger candidate assumption and is the version required by constructions that intentionally disclose this factorization. The results below do not prove either assumption. They do, however, identify a factored-input regime with only polynomially many candidates; that regime cannot support factored STPA and must be excluded from cryptographic parameter selection. Table 2 separates established results from intended downstream use.

4

Table 2: Logical scope of the manuscript. Component

Status

Role

STP relation and input models Verification and allocation count Collision and ambiguity data

Defined here Proved here Exact counts; rounded entropy Candidate assumptions Separate constructions

Public statement and secret witness Correctness and parameter screening Quantifies non-injectivity

STPA and factored STPA Commitment, proof-of-knowledge, and authentication protocols

Possible preimage-resistant foundation Require their own security games and reductions

3

Main analytical and computational results

3.1

Forward collision census

For fixed (λ, k), the collision census is computed as follows: 1. generate every prime in Pλ with a sieve; 2. enumerate every unordered k-subset P ; 3. compute x = fλ,k (P ); and 4. increment the multiplicity stored for x. The result is the complete finite map x 7→ µλ,k (x). It is not a sample. If Nλ = |Pλ | and Vλ,k
denotes the number of represented values, the implementation performs Nkλ tuple evaluations and stores Vλ,k integer keys. Its direct time cost is therefore Nλ O k k

!!

arithmetic multiplications, in addition to the sieve, and its dictionary storage is O(Vλ,k ).

3.2

Reconstruction from a factorization

Suppose that x=

r Y ej

qj

j=1

is supplied in factored form. An ordered factorization x = d1 · · · dk is determined by nonnegative integers aji satisfying ej = aj1 + · · · + ajk ,

di =

r Y aji

qj .

j=1

The candidate primes are pi = di + 1. Proposition 1 (Exact allocation count). Before bit-length, primality, ordering, and distinctness tests, the number of ordered exponent allocations is Ak (x) =

r Y

!

ej + k − 1 . k−1

j=1

+k−1 Proof. For a fixed prime base qj , the exponent ej has ejk−1 weak allocations among k positions. Allocations for different prime bases are independent, so their counts multiply.

5

This is a count of candidates before rejection, not a running-time lower bound. Requiring d1 < · · · < dk , applying the dyadic bounds before primality testing, and abandoning partial products that are already too large can reduce the actual search. Proposition 2 (A simple allocation bound). Let Ω(x) = multiplicity. Then Ak (x) ≤ k Ω(x) .

Pr

j=1 ej count prime factors with

Proof. For one exponent e, every weak allocation into k positions is realized by at least one e+k−1
e assignment of e labelled copies to those positions. There are k such assignments, so k−1 ≤ k e . Multiplication over the prime bases gives the claim. Corollary 1 (A tractable factored regime). For fixed k, if Ω(x) = O(log λ), factored-input STP search is solvable in time polynomial in λ by direct allocation and testing. Proof. If Ω(x) ≤ c log λ, then Ak (x) ≤ k c log λ = λc log k , which is polynomial in λ for fixed k. Moreover, every valid instance satisfies x < 2kλ , so its candidates have O(kλ) = O(λ) bits in this regime. Forming and testing every allocation therefore takes polynomial time; deterministic primality testing is polynomial-time [1]. This corollary is also a cryptographic constraint: a factored-input construction cannot base its security on STP in this fixed-k regime. The reference reconstruction program implements this specialized direct procedure: generate all weak exponent allocations, form the di , sort to remove permutations, and retain a tuple only when every di + 1 is a distinct λ-bit prime. For k = 2, an equivalent procedure enumerates divisors d | x and tests the pair d + 1, x/d + 1. Observation 1 (Factor exposure). If λ ≥ 3, k ≥ 2, and a structured preimage is returned, then p − 1 is a proper nontrivial divisor of x for every returned prime p. This observation does not reduce the factorization of an arbitrary integer to structured reconstruction. It only records that a successful reconstruction on a valid instance reveals a factor of that particular shifted product.

4

Finite ambiguity measures

Collision counts distinguish represented values, but they do not by themselves describe how a uniformly generated input is affected. Let the random set P be uniform on Sλ,k and let X = fλ,k (P). Conditional on X = x, every set in the fiber is equally likely. Consequently, ∆λ,k := H(P | X) =

1 X Tλ,k

µλ,k (x) log2 µλ,k (x),

(1)

x

where the sum is over represented values. This is the standard conditional entropy identity for a deterministic map and a uniform input [9]. It is reported here only as a finite ambiguity measure; it is not a security parameter. Two additional exact summaries are used: Cλ,k = µλ,k =

1 Tλ,k x:µ Tλ,k , Vλ,k

X

µλ,k (x),

fraction of input sets in a non-singleton fiber,

λ,k (x)≥2

mean fiber size over represented values.

6

These quantities answer different questions. The maximum multiplicity records the largest observed fiber, Cλ,k measures how often a uniformly sampled input is ambiguous, and ∆λ,k weights each fiber by the number of bits needed to identify an input within it. All multiplicities and counts reported below are exact. Decimal evaluations of ∆λ,k and derived ratios are computed from those exact multiplicities and rounded to the displayed precision.

5

Exact experiments

5.1

Reproducibility and range

The program uses only the Python standard library. It enumerates combinations in lexicographic order, uses arbitrary-precision integer products, and stores exact multiplicities. Running the script rewrites both the complete CSV file and the LaTeX table used below. Repeated execution produced byte-identical data and table files; entropy values are deterministic numerical evaluations from the exact fibers and are rounded when displayed. The census contains 28 parameter pairs: k λ 2 4, . . . , 16 3 6, . . . , 12 4 6, . . . , 10 5 7, . . . , 9. The largest row contains 4,588,935 unordered prime pairs. The complete CSV includes the number of primes, input sets, represented values, collision values, colliding input sets, mean fiber size, ambiguity, and maximum multiplicity for every row. Table 3: Selected collision statistics. All counts are exact. A tuple is counted as colliding when its value x is shared by at least one other unordered tuple. The ambiguity column gives H(P | X) in bits for a uniformly sampled tuple, rounded to five decimal places.

k

λ

#Pλ

Tuples

Values

Colliding (%)

Ambiguity

Max. mult.

2 2 2 2 2 2 3 3 3 3 4 4 5 5

6 8 10 12 14 16 6 8 10 12 8 10 8 9

7 23 75 255 872 3,030 7 23 75 255 23 75 23 43

21 253 2,775 32,385 379,756 4,588,935 35 1,771 67,525 2,731,135 8,855 1,215,450 33,649 962,598

21 250 2,684 31,266 367,181 4,433,899 35 1,710 60,555 2,423,283 8,272 967,849 30,175 787,667

0.000 2.372 6.450 6.664 6.363 6.435 0.000 6.776 18.713 19.577 12.660 33.767 19.344 30.545

0.00000 0.02372 0.06640 0.07113 0.06833 0.07027 0.00000 0.06974 0.22382 0.25654 0.13551 0.48524 0.21669 0.42589

1 2 3 5 6 8 1 3 9 16 3 24 6 17

5.2

Observed finite-range behavior

Three patterns are visible in Table 3.

7

First, structured collisions are not exceptional artifacts. For example, the fraction of colliding pairs is between 5.90% and 6.66% for 12 ≤ λ ≤ 16. Over the same five rows, the conditional ambiguity is between 0.06275 and 0.07113 bits. These close finite values are empirical observations, not an assertion of convergence. Second, increasing k can produce substantially larger fibers in the tested ranges. At (λ, k) = (10, 4), 33.767% of the input sets belong to a non-singleton fiber and the largest fiber contains 24 sets. At (9, 5) the corresponding values are 30.545% and 17. Comparisons across different k should remain finite-range comparisons because the number of available primes and the number of subsets change simultaneously. Third, the rates are not monotone at small parameters. For example, the colliding fraction for k = 3 drops from 12.238% at λ = 7 to 6.776% at λ = 8, then increases again. This behavior is one reason not to infer an asymptotic law from a short table.

5.3

Explicit reconstructions

The following examples are independently checked by the reconstruction program. Unique pair. For x = 1440 = 25 32 5 and (λ, k) = (6, 2), the 36 ordered exponent allocations produce exactly one unordered structured preimage: {37, 41},

(37 − 1)(41 − 1) = 36 · 40 = 1440.

Collision. For x = 23400 = 23 32 52 13 and (λ, k) = (8, 2), the 72 ordered allocations produce two structured preimages: (131 − 1)(181 − 1) = 130 · 180 = 23400, (151 − 1)(157 − 1) = 150 · 156 = 23400. Thus a solver can reconstruct a valid pair without identifying which pair was used to generate the target. Permutation is not multiplicity.

For P = {37, 41, 43},

x = 36 · 40 · 42 = 60480. The six orderings describe one set, not six preimages. Exact enumeration finds this set unique for (λ, k) = (6, 3).

6

Relation to inverse-totient complexity

The general inverse-totient problem has a deeper literature than the restricted relation considered here. When the factorization of the target is supplied, Contini, Croot, and Shparlinski give a deterministic algorithm that constructs the full inverse set in time polynomial in the number of inverse-search paths, the divisor count, and the input length; they also obtain polynomial time for almost all targets. Separately, under a strong quantitative prime-pair conjecture, they prove NP-completeness for deciding whether a finite set of integers contains a totient [4]. Alekseyev subsequently gives a general dynamic-programming framework for computing inverses, counts, power sums, and extrema of multiplicative functions, with Euler’s totient as a principal example [2]. The exponent-allocation procedure in this paper should therefore be read as a transparent specialization to square-free, equal-bit, fixed-cardinality preimages, not as a replacement for general inverse-totient algorithms. The isolation of the STP relation, the restricted collision census, and its ambiguity measures are the present paper’s focus. These general results provide 8

important context, but they do not directly establish hardness for a single target constrained to k distinct equal-bit primes. There is also a natural resemblance to Product Partition, which is strongly NP-complete when the input items are indivisible [8]. In a factored structured-preimage instance, however, prime exponents can be split among candidate factors, and every completed factor must be one less than a prime. The resemblance therefore does not by itself give a reduction. The computational conclusions of this paper require neither result. The finite census is exact, the reconstruction algorithm is exhaustive for a supplied factorization, and the candidate count follows directly from weak compositions. Whether the restricted search relation is hard in a growing parameter regime remains open.

7

Cryptographic consequences and boundaries

7.1

Candidate preimage-resistant relation

The map P 7→ fλ,k (P ) is efficiently computable, and membership in a fiber is efficiently verifiable. If STPA holds for a specified growing parameter family, this map induces a candidate preimage-resistant relation: the goal of an adversary is to find any valid preimage, not necessarily the sampled one. This distinction is essential because the experiments prove that structured collisions occur. Non-injectivity is compatible with a preimage-resistant relation, but it changes how downstream protocols must define correctness and security.

7.2

Consequences for the motivating constructions

The present results have direct implications for the protocols that motivated STP. Commitment mechanisms. Whenever µλ,k (x) > 1, the rule “commit with x and open with any P ” is not perfectly binding: more than one valid opening exists. This fact alone does not rule out computational binding, which would require a separate game and an analysis of whether an efficient committer can find two openings. A viable construction must either enforce a uniqueness condition, bind auxiliary information to the sampled witness, or prove an appropriate computational-binding property. The collision census therefore identifies a design constraint, not merely a numerical curiosity. Proofs of knowledge. For polynomially bounded k(λ), STP supplies the explicit NP relation RSTP = {((x, λ, k), P ) : P ∈ Sλ,k , fλ,k (P ) = x}. It can serve as the statement–witness layer of a proof or verification procedure. A concrete proof system must still specify completeness, knowledge soundness, zero knowledge, and efficiency. Authentication primitives. A public value x can index a secret structured witness P , while a challenge– response protocol proves possession without directly releasing it. Freshness, replay resistance, witness leakage, and reduction to STPA belong to the separate protocol analysis. These applications explain why the relation is established before the protocols are presented. The mathematical object and its parameter limits should not depend on one particular construction, while every later construction can cite the same relation and state exactly which version of STPA it assumes. 9

7.3

Factored inputs and the post-quantum boundary

The two input models lead to different security interpretations. In the unfactored model, ordinary factorization may contribute substantially to the observed difficulty. In the factored model, only the allocation-and-primality layer remains. Corollary 1 proves that this residual layer is easy in an explicit fixed-k regime, so cryptographic parameters must avoid that regime. Quantum factoring removes the first layer [10]. Consequently, a post-quantum claim could rest only on factored STPA in a growing parameter family, together with a separate quantum attack analysis. Neither requirement is established here. This boundary prevents the computational evidence from being misread as a post-quantum security result.

7.4

Limitations and next tests

The census is exact but finite and therefore proves no asymptotic collision law. Direct subset enumeration scales combinatorially and cannot reach cryptographic bit lengths. The data establish neither average-case nor worst-case hardness, and no complete protocol or security game is included. The next mathematical-computational tests are now specific: study growing k(λ), measure exponent-allocation tree width under GenSTPλ,k , compare branch-and-bound and meet-inthe-middle reconstruction, and search for reductions or attacks against factored STPA. The downstream protocol papers can then use the surviving parameter families and supply their own security definitions.

8

Conclusion

This paper introduced STP as a standalone computational relation: given the product of shifted, distinct, equal-bit primes, recover any compatible prime set. The relation is narrower than general inverse-totient computation and separates two possible sources of difficulty: factoring the public product and reconstructing admissible shifted-prime boundaries. The main results are identifiable and complementary. The exact allocation formula characterizes the direct factored-input search space. Its upper bound proves a polynomial reconstruction regime and thereby excludes weak cryptographic parameter choices. Exhaustive enumeration over 28 parameter pairs establishes that the map is non-injective and quantifies its fibers by collision participation, maximum multiplicity, and conditional ambiguity. The supplied algorithms and complete data make every finite claim reproducible. These results do not prove STPA, but they give it a precise object, generator, success condition, and parameter boundary. They also show why collisions matter operationally: a naive commitment is not perfectly binding, while proof-of-knowledge and authentication constructions must treat STP as a relation rather than as an injective function. The contribution is therefore the computational foundation needed before those protocols are developed and evaluated separately. Establishing or refuting factored STPA for growing parameters is the decisive next step.

Data and code availability The source package accompanying this article contains both Python programs, the complete CSV data, the generated LaTeX table, and compilation instructions. The programs use only the Python standard library.

10

Acknowledgment The author thanks Dr. Luis Hernández Encinas for comments on an earlier version of this work.

Conflict of interest The author declares no conflict of interest.

References [1] Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. PRIMES is in P. Annals of Mathematics, 160(2):781–793, 2004. doi:10.4007/annals.2004.160.781. [2] Max A. Alekseyev. Computing the inverses, their power sums, and extrema for Euler’s totient and other multiplicative functions. Journal of Integer Sequences, 19:Article 16.5.2, 2016. https://cs.uwaterloo.ca/journals/JIS/VOL19/Alekseyev/alek5.html. [3] Robert D. Carmichael. On Euler’s ϕ-function. Bulletin of the American Mathematical Society, 13(5):241–243, 1907. doi:10.1090/S0002-9904-1907-01453-2. [4] Scott Contini, Ernie Croot, and Igor E. Shparlinski. Complexity of inverting the Euler function. Mathematics of Computation, 75(254):983–996, 2006. doi:10.1090/S0025-5718-0601826-6. [5] Kevin Ford. The distribution of totients. The Ramanujan Journal, 2:67–151, 1998. doi:10.1023/A:1009761909132. [6] Kevin Ford. The number of solutions of ϕ(x) = m. Annals of Mathematics, 150(1):283–311, 1999. doi:10.2307/121103. [7] Derrick H. Lehmer. On Euler’s totient function. Bulletin of the American Mathematical Society, 38(10):745–751, 1932. doi:10.1090/S0002-9904-1932-05521-5. [8] C. T. Ng, M. S. Barketau, T. C. E. Cheng, and M. Y. Kovalyov. “product partition” and related problems of scheduling and systems reliability: Computational complexity and approximation. European Journal of Operational Research, 207(2):601–604, 2010. doi:10.1016/j.ejor.2010.05.034. [9] Claude E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27(3):379–423, 1948. doi:10.1002/j.1538-7305.1948.tb01338.x. [10] Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997. doi:10.1137/S0097539795293172.

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