Pith. sign in

REVIEW 4 major objections 5 minor 75 references

Integer Factorization: Another perspective

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that factoring an RSA modulus is equivalent to finding small roots of one of four bivariate quadratics, with Coppersmith's method recovering the factors in polynomial time when the root-size bound is met.

desk verdict Theorem 3.11's Coppersmith reduction fails because n(x0,y0)=n can never vanish modulo M>n; the paper is mostly tautological restatement rather than a new approach. read the letter →

arxiv 2507.07055 v1 pith:M7KCFVMX submitted 2025-07-09 math.NT cs.CR

Gilda Rech Bansimba , Regis Freguin Babindamana This is my paper · ORCID
classification math.NTcs.CR MSC 11A5111Y0594A60
keywords integerfactorizationRSAbivariatepolynomialCoppersmith'smethodsemiprimematrixdecompositionGröbnerbasisLebesgueintegralreformulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that integer factorization can be re-encoded in three unfamiliar settings: a rectangle with known area, a 2×2 integer matrix determinant, and a two-variable quadratic equation. Its strongest claim is that when a semiprime's factor coordinates satisfy a small-root bound, Coppersmith's bivariate method recovers the factors in polynomial time, making factorization equivalent to a small-root search. This matters because RSA security relies on factorization being hard; if the equivalence held, the foundations of the cryptosystem's one-way assumption would be in question. The authors present the three viewpoints as ways to bring integral calculus, Gröbner bases, and lattice methods to bear on a classical problem.

What carries the argument

The central object is the bivariate quadratic $n(x,y)=36xy+6|x+y|+1$ or $36xy+6|x-y|-1$ (equivalently one of the four sign variants), which encodes $n=(6x+1)(6y+1)$, $n=(6x-1)(6y-1)$, or the mixed cases. Since every prime greater than 3 has the form $6k\pm1$, this one polynomial family covers every RSA semiprime. The argument's engine is Coppersmith's theorem for small roots of bivariate polynomial equations, which promises polynomial-time recovery of a solution $(x_0,y_0)$ when the product of radius bounds $XY$ is below $M^{1/2-\varepsilon}$; the auxiliary reformulations are the double-integral identity from Fubini's theorem and the matrix identity $N=PQ$ with $\det N=n$, $\det P=p$, $\det Q=q$.

What would settle it

Take a small semiprime such as $n=91=(6\cdot1+1)(6\cdot2+1)$ and any integer $M>91$: the congruence $n(1,2)\equiv0\pmod M$ fails because $n(1,2)=91$, so the polynomial equation at the factorization point has no root modulo $M$ and Theorem 3.11 cannot be instantiated. A direct check of this congruence for arbitrary $M>n$ is enough to expose the incompatibility.

Watch

Extended reading notes

Core claim

The paper's central constructive claim is that every RSA modulus $n=pq$ with $p,q>3$ is the value of one of four bivariate quadratic polynomials $n(x,y)=36xy\pm6(x\pm y)\pm1$, because every such prime is $6k\pm1$. It then applies Coppersmith's bivariate small-roots theorem: if the true solution is $(x_0,y_0)$ with $|x_0|<X$, $|y_0|<Y$, and $XY<M^{1/2-\varepsilon}$ for a modulus $M>n$, then the congruence $n(x,y)\equiv0\pmod M$ can be solved and the integer roots recovered in polynomial time. On the paper's own terms, this makes factoring $n$ algorithmically identical to finding small roots of $n(x,y)$; the rectangle and matrix viewpoints are presented as equivalent reformulations that prepare the same reduction.

Load-bearing premise

The load-bearing premise is that the true solution satisfies both the small-root bound $XY < M^{1/2-\varepsilon}$ and the congruence $n(x_0,y_0)\equiv0\pmod M$ with $M>n$; for typical RSA semiprimes the coordinates are too large (roughly $\sqrt{n}/6$), and the congruence can only hold when $M$ divides $n$.

Editorial extensions

If this is right

  • If the Coppersmith reduction is sound, semiprimes whose factor coordinates satisfy $XY<M^{1/2-\varepsilon}$ would be factored in polynomial time in the size of the inputs.
  • Every RSA semiprime appears as one of the four quadratic forms $36xy\pm6(x\pm y)\pm1$, so the search for factors is exactly a search for integer points on those surfaces.
  • The determinant reformulation implies that factoring $n$ is equivalent to splitting a $2\times2$ integer matrix into two integer matrices whose determinants multiply to $n$; Gröbner-basis and diagonalization methods can be directed at that system.
  • The integral reformulation says that any procedure producing the upper limits of the double integral equal to $n$ outputs the two primes directly.
  • If a modulus is a triangular number, $8n+1$ is a square and the smaller factor is read off in constant or logarithmic time; this special-case 'RSA broken' claim does not extend to arbitrary moduli.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implicit route the authors do not develop is to choose the auxiliary modulus $M$ as a divisor of $n$ rather than a number greater than $n$; then the congruence $n(x,y)\equiv0\pmod M$ holds at the true factorization point, and a Coppersmith search over one factor becomes a concrete thing to test on small moduli.
  • Because $x=(p\pm1)/6$ and $y=(q\pm1)/6$, moduli with unbalanced factors have one tiny coordinate, and moduli with very close factors have small $XY$ relative to $n$; those parameter regimes are where the small-root condition might be satisfiable.
  • The authors' future-work hint about hyperbolas suggests a lattice-geometry reading: the solutions of $n(x,y)=C$ lie on a hyperbola, and divisors of $n-1$ constrain the coordinates, so the problem becomes one of constrained lattice-point finding that Coppersmith's method or continued-fraction techniques could attack.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes three reformulations of integer factorization: (1) viewing a semiprime n = pq as the area of a rectangle whose side lengths are recovered from integral bounds; (2) expressing n as the determinant of a 2-by-2 integer matrix and attempting recovery through Gröbner basis or matrix diagonalization; and (3) representing n as one of four bivariate polynomials n(x,y) = 36xy ± 6(x±y) ± 1 and claiming that finding the small roots (x,y) is equivalent to factorization through Coppersmith's method. The background sections survey classical factoring algorithms and the paper claims several 'new vulnerabilities,' including a constant-time or logarithmic-time break when a certain discriminant is square.

Significance. If the central Coppersmith equivalence were valid, the paper would imply a major breakthrough in the factoring of RSA moduli. However, the load-bearing theorems contain internal inconsistencies and are not supported by the cited results. The paper does provide a broad survey of classical factorization algorithms and some worked numerical examples, and the integral reformulations are pedagogically clear, but these positive features do not establish any new algorithmic capability.

major comments (4)
  1. [§3.3, Theorem 3.11] Theorem 3.11's reduction to Coppersmith's method is vacuous because its congruence premise is unsatisfiable. For any integer solution (x0,y0) with n(x0,y0)=n and M>n, one has n(x0,y0) ≡ n mod M, which is not 0. Coppersmith's modular-root theorem requires the polynomial to vanish modulo M at the root, so the condition 'n(x,y) ≡ 0 mod M' cannot be met for any RSA-type modulus M>n. The proof, which is a one-sentence citation to [10], therefore does not establish polynomial-time recovery of the factors. This invalidates the abstract's central claim that factorization is equivalent to finding small roots of the bivariate polynomial.
  2. [§3.3, Theorem 3.10] The condition XY < W^{1/3} stated in Theorem 3.10 is not attainable for RSA moduli. With X and Y on the order of sqrt(n)/6, the weight W = max(1, 6X, 6Y, 36XY) is on the order of n, so the condition demands XY ~ n/36 < n^{1/3}, which fails for all n > 1. The paper itself concedes after Theorem 3.10 that this bound 'cannot be achieved in practice,' and Theorem 3.11, presented as the residue-ring fix, fails for the reason given above. Thus no algorithmic consequence follows for the factoring problem the paper targets.
  3. [§3.1, Theorem 3.5] Theorem 3.5 contains load-bearing arithmetic errors. The discriminant of y^2+y-2n=0 is 1+8n, not 1+23n as written, and the corresponding condition should be y^2-8n-1=0 rather than the stated y^2-23n-1=0. Example 3.6 is also inconsistent: for n=25651, 8n+1=205209 is not a perfect square, yet the text claims p=227 is a factor found from this square. The example for n=22008842474653 similarly states '8n+1 = 176070739797225 is a square' but that number is also not a square. These errors invalidate the announced 'new vulnerability' for triangular RSA moduli as stated.
  4. [§3.2, Theorem 3.7 and system (5)] Theorem 3.7's proof assumes without justification that one can choose integers a,b with gcd(a,b)=n and hence ad-bc=n via Bézout's identity; this is equivalent to the factorization problem, not a consequence of Bézout's theorem. Moreover, the Gröbner-basis system (5) explicitly includes the equation ad-bc-n=0, which is the target factorization equation and is not derived from the matrix-entry equations (2). The system therefore does not reduce factorization to an algebraically simpler problem; it merely restates the target as one of the polynomial equations.
minor comments (5)
  1. [Throughout] There are numerous typos and inconsistent notations, including 'coppersmith' and 'coopersmith' for Coppersmith, 'Le' in Theorem 3.5, 'Triavially' in Theorem 3.7, and inconsistent capitalization of 'Gröbner'. The equation numbering in §3.2 is duplicated: (2), (3), (4), and (5) are each used more than once.
  2. [Abstract and §2.2] The abstract contains an incomplete sentence: 'its proved that Shor's algorithm could on a quantum computer.' Also, the description of Shor's algorithm in Table 1 lists its complexity as O(b^3), but no reference or derivation is given in the text.
  3. [§3.1, Example 3.6] The last item of Example 3.6 reads '8n+1 = is a square' with the value missing, and the preceding item gives a non-square value for 8n+1 while still asserting it is a square.
  4. [§3.3, Theorem 3.10] The polynomial representation in Theorem 3.10 uses a Kronecker-delta expansion and defines n10=n01=6, but the four forms listed in Theorem 3.9 have different signs and the absolute-value expression n(x,y)=36xy+6|x±y|±1 is not a polynomial in the variables x and y. The notation should be clarified.
  5. [References] The reference list contains duplicate entries: [54] and [55] are the same McKee reference, and [66] and [67] are the same Goldwasser–Killian reference. Also, the list is unusually long relative to the paper's new content, and some entries (e.g., [37], [38]) are self-citations to closely related work that is not discussed in the text.

Circularity Check

3 steps flagged · score 7.0 of 10

Theorem 3.11's Coppersmith reduction is vacuous: by construction n(x0,y0)=n, so the required congruence n(x,y)≡0 mod M with M>n is unsatisfiable; the central equivalence collapses into the definition of n(x,y).

  1. self definitional [§3.3, Theorem 3.9, Remark 3.1, Theorem 3.11]
    "Consider the polynomial 𝑛(𝑥,𝑦)∈ Z[𝑥,𝑦] as defined above. Let 𝑀 be a RSA type modulus greater than 𝑛,𝑋 and 𝑌 such that for all (𝑥,𝑦) solutions of 𝑛(𝑥,𝑦),|𝑥| < 𝑋 and |𝑦| < 𝑌 . If 𝑋𝑌 < 𝑀 1/2−𝜖 for some 0 < 𝜖 < 1/2 then we can compute 𝑛(𝑥,𝑦)≡ 0 mod 𝑀 and find solutions of 𝑛(𝑥,𝑦) over Z."

    By Theorem 3.9, n(x,y) is defined as one of (6x±1)(6y±1). At the factor solution (x0,y0), n(x0,y0)=pq=n by that defining identity. Since 0<n<M, reducing modulo M gives n(x0,y0)≡n≠0 mod M. Hence the hypothesis n(x,y)≡0 mod M is unsatisfiable at the only root that would recover p and q; the hypothesis is equivalent to M|n, which contradicts M>n. The proof is a one-sentence citation to Coppersmith [10] and introduces no modified polynomial such as n(x,y)−n. The claimed reduction to Coppersmith's method is therefore not an algorithm but a definitional contradiction built into the choice of n(x,y).

  2. renaming known result [§3.1, Proposition 3.3]
    "=(6𝛼± 1)( 6𝛽± 1) =𝑦(𝛼)𝑦(𝛽), from lemma 3.1 =𝑝𝑞 =𝑛"

    The lower bounds ±1/6 and the constant integrand 6 are chosen precisely so that the double integral evaluates to (6α±1)(6β±1). Thus the unknowns α,β are just the variables of the standard representation p=6α±1, q=6β±1, and 'finding integral bounds' is the factorization equation itself in new notation. The rectangle/perimeter formulation similarly renames p and q as the sides of a rectangle of known area n without adding any independent constraint such as a known perimeter. The equivalence is a notational relabeling, not a derivation of p and q from easier data.

1 more flagged steps
  1. self definitional [§3.2, Theorem 3.7 and Gröbner system (5)]
    "𝑎𝑑−𝑏𝑐−𝑛 = 0 (13)"

    The Gröbner system (5) includes the equation ad−bc−n=0, which is the determinant condition defining the matrix N in Theorem 3.7. Since Theorem 3.7 starts by choosing a,b,c,d with ad−bc=n, the matrix-decomposition problem is the equation n=pq transcribed as det N=det P·det Q. Solving the system is therefore equivalent to factoring n by construction: the target equation appears as one of the equations in the system, and no independent source of information about p or q is introduced. The 'matrix decomposition point of view' is a restatement of the factorization problem rather than a reduction of it.

full rationale

Theorem 3.11's Coppersmith reduction is empty because n(x,y) is defined as the factorization product (6x±1)(6y±1), so at the factor solution n(x0,y0)=n; since M>n, the congruence n≡0 mod M is unsatisfiable. Similarly, Proposition 3.3's integral reformulation selects ±1/6 bounds and constant 6 so that the integral equals (6α±1)(6β±1), i.e., the target factorization itself, and the matrix system (5) contains ad−bc−n=0 as one of its equations, directly restating the determinant condition. No independent constraint on p,q is introduced in any of these reformulations, so the claimed equivalences reduce by construction to the factorization equation they purport to solve. The score reflects that the central Coppersmith claim is vacuous while the other two perspectives are merely renaming, not independent results. No self-citation chain or fabricated circularity is needed; the equations themselves exhibit the reduction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new entities, but it relies on several unexamined premises: the arbitrary choice of the matrix N, the assumed ability to meet the Coppersmith bound, and the classical but unstated special-form condition for triangular numbers. The free parameters are not fitted to data but are chosen to make the reformulations identities.

free parameters (2)
  • M (modulus in Theorem 3.11)
    M is specified only as 'greater than n'. To satisfy XY < M^{1/2-ε} with x,y of order sqrt(n)/6 would require M >> n^2, which is not a typical RSA setting and is not discussed.
  • Integral lower bounds α0, β0 = ±1/6
    Chosen so that the double integral evaluates to (6α±1)(6β±1)=pq. This encodes the factors p and q directly, making the reformulation definitional.
assumptions (4)
  • standard math Fubini-Tonelli theorem
    Used in Proposition 3.3 to separate the double integral into a product of single integrals.
  • standard math Coppersmith's bivariate small-root theorem
    Invoked in Theorems 3.10 and 3.11 without verifying its applicability conditions, including whether the polynomial has a root modulo M and whether the bound is met.
  • domain assumption n is a semiprime, n=pq with p,q primes
    The whole paper restricts to RSA moduli; the reformulations only apply to semiprimes.
  • ad hoc to paper Existence of a,b,c,d with ad-bc=n and gcd(a,b)=n
    Theorem 3.7 uses Bezout to construct a matrix N with determinant n, but the choice of N is arbitrary and the Gröbner system that follows contains the target equation ad-bc-n=0, so no reduction is achieved.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Integer Factorization: Another perspective." pith.science (2026). https://pith.science/paper/M7KCFVMX

@misc{pith2026250707055,
  author       = {Pith},
  title        = {Pith review of: Integer Factorization: Another perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7KCFVMX}},
  note         = {Machine review of arXiv:2507.07055}
}
abstract

Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods like the General Number Field Sieve, no known algorithm can break the problem in polynomial time, while its proved that Shor's algorithm could on a quantum computer. In this paper, we recall some factorization algorithms and then approach the problem under different angles. Firstly, we take the problem from the ring $\displaystyle\left(\mathbb{Z}, \text{+}, \cdot\right)$ to the Lebesgue space $\mathcal{L}^{1}\left(X\right)$ where $X$ can be $\mathbb{Q}$ or any given interval setting. From this first perspective, integer factorization becomes equivalent to finding the perimeter of a rectangle whose area is known. In this case, it is equivalent to either finding bounds of integrals or finding primitives for some given bounds. Secondly, we take the problem from the ring $\displaystyle\left(\mathbb{Z}, \text{+}, \cdot\right) $ to the ring of matrices $\left( M_{2}\text{(}\mathbb{Z}\text{)}, \ \text{+} \ \cdot\right)$ and show that this problem is equivalent to matrix decomposition, and therefore present some possible computing algorithms, particularly using Gr\"obner basis and through matrix diagonalization. Finally, we address the problem depending on algebraic forms of factors and show that this problem is equivalent to finding small roots of a bivariate polynomial through coppersmith's method. The aim of this study is to propose innovative methodological approaches to reformulate this problem, thereby offering new perspectives.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

75 extracted references · 74 canonical work pages

  1. [10]

    Coppersmith, Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities, Journal of Cryptology, 10(4), 1997, pp

    D. Coppersmith, Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities, Journal of Cryptology, 10(4), 1997, pp. 233–260

    work page 1997

  2. [3]

    A. O. L. Atkin and F. Morain. Finding suitable curves for the elliptic curve method of factorization. Mathematics of Computation, 60:399–405, 1993

    work page 1993

  3. [4]

    I., Ariffin, M

    Abubakar, S. I., Ariffin, M. R. K. and Asbullah, M. A. (2019).A New Improved Bound for Short Decryption Exponent on RSA Modulus N = pq using Wiener’s Method , Malaysian Journal of Mathematical Sciences 13(S) April: 89–99

    work page 2019

  4. [5]

    and Saxena N

    Agrawal M., Kayal N. and Saxena N. PRIMES in P. Ann. of Math., 160, 781–793 (2004)

    work page 2004

  5. [6]

    A comparison with the Gaussian Integer Method, Mathematics of Computation, Volume 72, Number 242, Pages 953–967

    Antoine Joux, Reynald Lercier Improvements to the General Number Field Sieve for Descrete Logarithms in Prime Fields. A comparison with the Gaussian Integer Method, Mathematics of Computation, Volume 72, Number 242, Pages 953–967

    work page

  6. [7]

    Lenstra and H

    Arjen K. Lenstra and H. W. Lenstra, Jr. (ds.) The development of the number field sieve, Lecture Notes in Math. (1993) 1554. Springer-Verlag

    work page 1993

  7. [8]

    Sugli integrali multipli

    Fubini, Guido (1907), "Sugli integrali multipli", Rom. Acc. L. Rend. (5), 16 (1): 608–614, JFM 38.0343.02

    work page 1907

  8. [9]

    Barry Mazur, Modular curves and the Eisenstein ideal,IHES Publ. Math. 47 (1977), 33-186

    work page 1977

Show all 75 references
  1. [11]

    (2007) Faster Addition and Doubling on Elliptic Curves

    Bernstein D.J., Lange T. (2007) Faster Addition and Doubling on Elliptic Curves. In: Kurosawa K. (eds) Advances in Cryptology – ASIACRYPT 2007. ASIACRYPT

    2007

  2. [12]

    In: Boztaş, S., Lu, H.-F

    Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: Boztaş, S., Lu, H.-F. (eds.) AAECC 2007. LNCS, vol. 4851, pp. 20–27. Springer, Heidelberg (2007)

    2007

  3. [13]

    In: Vaudenay, S

    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405.Springer, Heidelberg (2008)

    2008

  4. [14]

    In: Fossorier, M.P.C., Hoholdt, T., Poli, A

    Brier, E., Joye, M.: Fast point multiplication on elliptic curves through isogenies. In: Fossorier, M.P.C., Hoholdt, T., Poli, A. (eds.) AAECC 2003. LNCS, vol. 2643, pp. 43–50. Springer, Heidelberg (2003)

    2003

  5. [15]

    and Tonien, J

    Bunder, M. and Tonien, J. (2017).A New Attack on the RSA Cryptosystem Based on Continued Fractions, Malaysian Journal of Mathematical Sciences 11(S) August: 45 - 57 (2017)

    2017

  6. [16]

    Carl Friedrich Gauss, Disquisitiones Arithmeticae, Springer, https://doi.org/10.1007/978-1-4939-7560-0

    doi

  7. [17]

    EUROCRYPT ’84’, LNCS 209, pp.169-182, 1985

    Carl Pomerance, The quadratic sieve factoring algorithm,Advances in Cryptology, Proc. EUROCRYPT ’84’, LNCS 209, pp.169-182, 1985. Springer-Verlag, Berlin Heidelberg 1985

    1985

  8. [18]

    proceedings of Symposia in Applied Mathematics, Volume 48, 1994

    Carl Pomerance The Number Field Sieve . proceedings of Symposia in Applied Mathematics, Volume 48, 1994

    1994

  9. [19]

    Chenglian Liu and Ziwei Ye, Estimating the Φ(𝑛) of Upper/Lower Bound in its RSA Cryptosystem, https://eprint.iacr.org/2012/666.pdf

    2012

  10. [20]

    In: Ohta, K., Pei, D

    Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)

    1998

  11. [21]

    Montréal, CA: McGill University

    Connell, Ian (1999), Elliptic Curve Handbook. Montréal, CA: McGill University

    1999

  12. [22]

    Gilat, Gauss’s Lemma and the Irrationality of Roots , Revisited, Math

    D. Gilat, Gauss’s Lemma and the Irrationality of Roots , Revisited, Math. Mag. 85 (2012) 114-116

    2012

  13. [23]

    D. J. Bernstein, P. Birkner, and T. Lange. Starfish on strike. In Progress in Cryp- tology - LATINCRYPT 2010, Proceedings, pages 61–80, 2010

    2010

  14. [24]

    Gordon, Asurvey of fast exponentiation methods , Journal of Algorithms 27 (1998), no

    D.M. Gordon, Asurvey of fast exponentiation methods , Journal of Algorithms 27 (1998), no. 1, 129–146

    1998

  15. [25]

    Dale Husemöller, Elliptic Curves, second Edition, Springer-Verlag New York, Inc. 2004

    2004

  16. [26]

    Bernstein, Peter BIRKNER, Tanja LANGE and Christiane

    Daniel J. Bernstein, Peter BIRKNER, Tanja LANGE and Christiane. Peters, ECM Using Ewards curves Mathematics of Computation Vol. 82, No. 282 (APRIL 2013), pp. 1139-1179

    2013

  17. [27]

    Johannis Pellii; & speciatim de Problematis imperfecte determinatis (On Algebra by Dr

    De Algebra D. Johannis Pellii; & speciatim de Problematis imperfecte determinatis (On Algebra by Dr. John Pell and especially on an incompletely determined problem), pp. 234–236

  18. [28]

    De Weger, B. (2002). Cryptanalysis of rsa with small prime difference . Applicable Algebra in Engineering, Communication and Computing, 13(1):17–28

    2002

  19. [29]

    Théorie génerale des équations algébriques

    E. Bezout, "Théorie génerale des équations algébriques" , Paris (1779)

  20. [30]

    EECM: ECM using Edwards curves, software: https://eecm.cr.yp.to/mpfq.html, 2017

    2017

  21. [31]

    ElGamal, T. (1985). A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Blakley, G.R., Chaum, D. (eds) Advances in Cryptology. CRYPTO 1984. Lecture Notes in Computer Science, vol 196. Springer, Berlin, Heidelberg

    1985

  22. [32]

    Lemmermeyer, Residue Class Rings Lecture Notes, https://www.mathi.uni- heidelberg.de/ flemmermeyer/pell/bfc03.pdf

    F. Lemmermeyer, Residue Class Rings Lecture Notes, https://www.mathi.uni- heidelberg.de/ flemmermeyer/pell/bfc03.pdf

  23. [33]

    Fermat (1894), Oeuvres de Fermat, 2, p. 256

  24. [34]

    Bansimba, Régis F

    Gilda R. Bansimba, Régis F. Babindamana,Construction of a New Hyperbola Group Structure. International Journal of Algebra 2019 Vol. 13, 2019, no. 8, 349 - 363

    2019

  25. [35]

    Bansimba, Regis F

    Gilda R. Bansimba, Regis F. Babindamana, Basile G. R. Bossoto. Some Arithmetical properties on Hyperbola, JP Journal of Algebra, Number Theory and Applications, Vol. 50, no. 1, P. 45-100, 2021

    2021

  26. [36]

    Lattice Points on the Fermat Factoriza- tion Method

    Gilda R. Bansimba, Basile G. R. Bossoto, "Lattice Points on the Fermat Factoriza- tion Method", Journal of Mathematics, vol. 2022, Article ID 6360264, 18 pages,

    2022

  27. [37]

    Bansimba, Regis Freguin Babindamana, Basile Guy R

    Gilda R. Bansimba, Regis Freguin Babindamana, Basile Guy R. Bossoto A New Hyperbola based Approach to factoring Integers [math.NT], https://doi.org/10.48550/arXiv.2304.07474

    doi arXiv

  28. [38]

    Babindamana, Basile Guy R

    Gilda Rech Bansimba, Regis F. Babindamana, Basile Guy R. Bossoto, A Continued Fraction-Hyperbola based Attack on RSA cryptosystem [cs.CR], https://doi.org/10.48550/arXiv.2304.03957

    doi arXiv

  29. [39]

    Bansimba,PytLoc: a python tool that locks and obfuscate python source files using a one time pad lock , Python Package Index https://pypi.org/project/pytLoc/

    Gilda R. Bansimba,PytLoc: a python tool that locks and obfuscate python source files using a one time pad lock , Python Package Index https://pypi.org/project/pytLoc/

  30. [40]

    H. W. Lenstra, Factoring integers with elliptic curves,Annals of Mathematics. 126 (3): 649–673

  31. [41]

    Edwards, A Normal Form for Elliptic Curves, Bulletin of the American Mathematical Society, Volume 44, Number 3, July 2007, Pages 393–422

    Harold M. Edwards, A Normal Form for Elliptic Curves, Bulletin of the American Mathematical Society, Volume 44, Number 3, July 2007, Pages 393–422

    2007

  32. [42]

    Harris, C.R., Millman, K.J., van der Walt, S.J. et al. Array programming with NumPy. Nature 585, 357–362 (2020)ℎ𝑡𝑡𝑝𝑠 ://𝑟𝑑𝑐𝑢.𝑏𝑒/𝑐𝑅𝑄𝐻𝑉

    2020

  33. [43]

    Huseyin Hisil, Kenneth Koon-Ho Wong, Gary Carter, Ed Dawson, Twisted Ed- wards curves revisited, in Asiacrypt 2008

    2008

  34. [44]

    Matplotlib: A 2D Graphics Environment

    J. D. Hunter, "Matplotlib: A 2D Graphics Environment", Comput- ing in Science & Engineering, vol. 9, no. 3, pp. 90-95, 2007. ℎ𝑡𝑡𝑝𝑠 : //𝑧𝑒𝑛𝑜𝑑𝑜.𝑜𝑟𝑔/𝑟𝑒𝑐𝑜𝑟𝑑/6513224#.𝑌𝑡𝑄𝑀𝑚𝑆 −𝑙2𝐼𝐴 Integer Factorization: Another perspective Research Article ()

    2007

  35. [45]

    J. Pollard. Théorèmes of factorization and primality testing. Proceedings of Cam- bridge Philosophical Society, 76: 521-528, 1974

    1974

  36. [46]

    Computational aspects of the Aryabhata algorithm

    Kak, Subhash (1986), "Computational aspects of the Aryabhata algorithm" , Indian Journal of History of Science, 21 (1): 62–71

    1986

  37. [47]

    Two Simple Lower Bounds for Euler’s Function , vol

    Kendall, D.G., Osborn, H.B. Two Simple Lower Bounds for Euler’s Function , vol. 17. Texas Journal of Science (1965)

    1965

  38. [48]

    Kevin Ford, Florian Luca, and Carl Pomerance, The Image of Carmichael’s 𝜆−Function

  39. [49]

    75 (256): 2037–2047

    Kleinjung, Thorsten, On polynomial selection for the general number field sieve , Mathematics of Computation. 75 (256): 2037–2047

  40. [50]

    Suite des réflexions sur la résolution algébrique des équations. Section troisieme. De la résolution des équations du cinquieme degré & des degrés ultérieurs

    Lagrange, Joseph-Louis (1771). "Suite des réflexions sur la résolution algébrique des équations. Section troisieme. De la résolution des équations du cinquieme degré & des degrés ultérieurs". Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin: 138–...

  41. [51]

    L. E. Dickson, History of the Theory of Numbers, Vol. 2,Chelsea Publishing Co., 1920

    1920

  42. [52]

    and Pomerance, C.:Primality testing with Gaussian periods

    Lenstra, H.W. and Pomerance, C.:Primality testing with Gaussian periods. Techni- cal report, http://www.math.dartmouth.edu/ carlp/aks041411.pdf (2011)

    2011

  43. [53]

    Another proof of Cauchy’s group Theorem

    McKay, James H. (1959), "Another proof of Cauchy’s group Theorem", American Mathematical Monthly, 66: 119, doi:10.2307/2310010

    1959 doi

  44. [54]

    Speeding Fermat’s factoring method

    McKee, J (1999). "Speeding Fermat’s factoring method". Mathematics of Compu- tation (68): 1729–1737

    1999

  45. [55]

    McKee, J (1999) Speeding Fermat’s factoring method,Mathematics of Computation (68): 1729–1737

    1999

  46. [56]

    Koblitz, Elliptic curve cryptosystems

    N. Koblitz, Elliptic curve cryptosystems. Math. Comp. 48. (1987), p. 203–209

    1987

  47. [57]

    Nitaj, A. (2013). Diophantine and lattice cryptanalysis of the rsa cryptosystem . In Artificial Intelligence, Evolutionary Computing and Metaheuristics, pages 139–168. Springer

    2013

  48. [58]

    (2014) Fast Point Multi- plication Algorithms for Binary Elliptic Curves with and without Precomputation

    Oliveira T., Aranha D.F., López J., Rodríguez-Henríquez F. (2014) Fast Point Multi- plication Algorithms for Binary Elliptic Curves with and without Precomputation. In: Joux A., Youssef A. (eds) Selected Areas in Cryptography – SAC 2014. SAC

    2014

  49. [59]

    Zimmermann et al

    P. Zimmermann et al. GMP-ECM (elliptic curve method for integer factorization). Software: https://gforge.inria.fr/projects/ecm/, 2012

    2012

  50. [60]

    Summing up the Euler Φ function

    Paul Loomis, M.P., Polhill, J. Summing up the Euler Φ function. The College Mathematics Journal 39(1), 34–42 (2008)

    2008

  51. [61]

    Paul Zimmermann, Factorization of RSA-250 , https://lists.gforge.inria.fr/pipermail/cado-nfs-discuss/2020- February/001166.html

    2020

  52. [62]

    Speeding the Pollard and elliptic curve methods of factor- ization,Mathematics of computation, 48(177):243–264, 1987

    Peter L Montgomery. Speeding the Pollard and elliptic curve methods of factor- ization,Mathematics of computation, 48(177):243–264, 1987

    1987

  53. [63]

    R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM , vol. 21, no. 2, pp. 120–126, 1978

    1978

  54. [64]

    Finding ECM-friendly curves through a study of Galois properties

    Razvan Barbulescu, Joppe Bos, Cyril Bouvier, Thorsten Kleinjung, Peter Mont- gomery. Finding ECM-friendly curves through a study of Galois properties. ANTS-X 10th Algorithmic Number Theory Symposium - 2012

    2012

  55. [65]

    Brent, Recent Progress and Prospects for Integer Factorization Algo- rithms, Oxford University Computing Laboratory , Wolfson Building, Parks Road, http://www.comlab.ox.ac.uk

    Richard P. Brent, Recent Progress and Prospects for Integer Factorization Algo- rithms, Oxford University Computing Laboratory , Wolfson Building, Parks Road, http://www.comlab.ox.ac.uk

  56. [66]

    Goldwasser, J

    S. Goldwasser, J. Killian, Primality testing using elliptic curves, Proceedings of the 18th Annual ACM Symposium of computing, ACM, New york, 1986, pp. 316-329

    1986

  57. [67]

    Goldwasser, J

    S. Goldwasser, J. Killian, Primality testing using elliptic curves, Proceedings of the 18th Annual ACM Symposium of computing , ACM, New york, 1986, pp. 316-329

    1986

  58. [68]

    Lang (1966), Introduction to Diophantine Approximations, Addison-Wesley Pub

    S. Lang (1966), Introduction to Diophantine Approximations, Addison-Wesley Pub. Co

    1966

  59. [69]

    An arithmetic function arising from the Φ function

    Shapiro, H. An arithmetic function arising from the Φ function. The American Mathematical Monthly 50, 18–30 (1943)

    1943

  60. [70]

    Taverne, J., Faz-Hernández, A., Aranha, D.F., Rodríguez-Henríquez, F., Hankerson, D., López, J.: Speeding scalar multiplication over binary elliptic curves using the new carry-less multiplication instruction. J. Cryptogr. Eng. 1, 187–199 (2011)

    2011

  61. [71]

    CADO-NFS, An Implementation of the Num- ber Field Sieve Algorithm , Release 2.3.0, 2017,http://cado-nfs.gforge.inria.fr/

    The CADO-NFS Development Team. CADO-NFS, An Implementation of the Num- ber Field Sieve Algorithm , Release 2.3.0, 2017,http://cado-nfs.gforge.inria.fr/

    2017

  62. [72]

    Miller, Use of elliptic curves in cryptography

    V. Miller, Use of elliptic curves in cryptography. Lecture Notes in Computer Science 85 (1985), p. 417–426

    1985

  63. [73]

    H., Andreas A

    Vagn L. H., Andreas A. On quadratic curves over finite fields , 2018, https://arxiv.org/pdf/1802.10486.pdf

    2018 arXiv

  64. [74]

    Wiener, M. J. (1990).Cryptanalysis of short rsa secret exponents, IEEE Transactions on Information theory, 36(3):553–558

    1990

  65. [2007]

    Springer, Berlin, Heidelberg

    Lecture Notes in Computer Science , vol 4833. Springer, Berlin, Heidelberg

  66. [2014]

    Springer

    Lecture Notes in Computer Science , vol 8781. Springer

  67. [2022]

    https://doi.org/10.1155/2022/6360264

    2022 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.