scholarly journals Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime

2020 ◽  
Vol 25 (4) ◽  
pp. 63
Author(s):  
Anthony Overmars ◽  
Sitalakshmi Venkatraman
Keyword(s):  
Cycle Time ◽  
Factorization Method ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Arithmetic Operations ◽  
Search Field ◽  
Sums Of Two Squares ◽  
Factorization Algorithms ◽  
Gaussian Primes

The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares N−1, N/2 . As N becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research.

Sum of Squares Rings

1977 ◽  
Vol 29 (1) ◽  
pp. 155-160 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Number Theory ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Quadratic Residues

One of the nicest results in elementary number theory is the following, giving the relation between quadratic residues and sums of squares.

scholarly journals Sums of squares of integral linear forms

2000 ◽  
Vol 69 (3) ◽  
pp. 298-302 ◽  
Author(s):  
Hideyo Sasaki
Keyword(s):  
Quadratic Form ◽  
Positive Definite ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Linear Forms ◽  
Integral Quadratic Form

AbstractIn this paper we prove that every positive definite n-ary integral quadratic form with 12 < n < 13 (respectively 14 ≦ n ≤ 20) that can be represented by a sum of squares of integral linear forms is represented by a sum of 2 · 3n + n + 6 (respectively 3 · 4n + n + 3) squares. We also prove that every positive definite 7-ary integral quadratic form that can be represented by a sum of squares is represented by a sum of 25 squares.

scholarly journals Bounds for distribution functions of sums of squares and radial errors

1991 ◽  
Vol 14 (3) ◽  
pp. 561-569 ◽  
Author(s):  
Roger B. Nelsen ◽  
Berthold Schweizer
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Sum Of Squares ◽  
Distribution Functions ◽  
Probable Error ◽  
Important Case ◽  
Sums Of Squares ◽  
Radial Error ◽  
Common Distribution ◽  
Common Distribution Function

Bounds are found for the distribution function of the sum of squaresX2+Y2whereXandYare arbitrary continuous random variables. The techniques employed, which utilize copulas and their properties, show that the bounds are pointwise best-possible whenXandYare symmetric about0and yield expressions which can be evaluated explicitly whenXandYhave a common distribution function which is concave on(0,∞). Similar results are obtained for the radial error(X2+Y2)½. The important case whereXandYare normally distributed is discussed, and here best-possible bounds on the circular probable error are also obtained.

scholarly journals Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

2022 ◽  
Vol 116 (2) ◽  
Author(s):  
José F. Fernando
Keyword(s):  
Residue Field ◽  
Classical Problem ◽  
Positive Semidefinite ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Local Rings ◽  
Local Case ◽  
Real Geometry ◽  
Excellent Ring ◽  
The One

AbstractA classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

Spectrophotometric Determination of Concentrations of Raw Milk in Solutions Containing Ingredients of Detergents1

1975 ◽  
Vol 38 (10) ◽  
pp. 586-589 ◽  
Author(s):  
E. L. RUIZ ◽  
D. B. BROOKER ◽  
M. E. ANDERSON ◽  
J. R. FISCHER ◽  
R. T. MARSHALL
Keyword(s):  
Sodium Hydroxide ◽  
Sodium Tripolyphosphate ◽  
Hard Water ◽  
Raw Milk ◽  
Sum Of Squares ◽  
Wetting Agent ◽  
Trisodium Phosphate ◽  
Sums Of Squares ◽  
Practical Significance

This paper reports effects of concentration of milk, concentration of individual ingredients of detergents, and temperature on transmittance of light at 527 nm through solutions of milk, water, and individual detergent ingredients. Milk in soft water caused the major portion of the variation, more than 99% of the total sum of squares, for each ingredient except sodium hydroxide and wetting agent. Milk contributed 95.7 and 17.9% to the sums of squares for the sodium hydroxide and wetting agent, respectively. Concentration of detergent ingredient and temperature were of practical significance only with wetting agent. Effects of milk on transmittance were significantly different among milk concentrations for all detergent ingredients, except trisodium phosphate in hard water at the lowest concentrations of milk. Although precipitates contributed to turbidity of solutions containing trisodium phosphate in hard water, milk was responsible for 95.9% of the variance in the sum of squares. Additionally, in hard water milk accounted for 98.9, 99.1 and 99.7% of the sums of squares in analyses of effects on turbidity of sodium tripolyphosphate, sodium hexametaphosphate, and tetrasodium pyrophosphate, respectively. Turbidity increased with time of holding some phosphate solutions, however, this did not appear to pose serious problems in measuring content of milk.

scholarly journals Relative contribution of expected sum of squares values for soybean genotypes × growing environments interaction

2020 ◽  
pp. 382-390
Author(s):  
Ivan Ricardo Carvalho ◽  
José Antonio Gonzalez da Silva ◽  
Luiz Leonardo Ferreira ◽  
Vinícius Jardel Szareski ◽  
Gustavo Henrique Demari ◽  
...  
Keyword(s):  
Experimental Design ◽  
Grain Yield ◽  
Plant Height ◽  
Sum Of Squares ◽  
Factorial Experiment ◽  
Sums Of Squares ◽  
Stem Number ◽  
Relative Contribution ◽  
Soybean Genotypes ◽  
Expected Values

The objective of this work was to assess the effects and tendencies weighted by genotypes x environments interaction for soybean, as well as to employ a biometric approach through the relative contribution of the sum of squares expected values (RCSS) and to define which levels of the variation sources determine the differential effects of the interaction. The experimental design was randomized blocks arranged in a factorial scheme (four growing environments x 20 soybean genotypes). The relative contribution of expected sums of squares values to soybean genotypes x growing environments interaction defined that the environment Tenente Portela - RS significantly influence plant height, number of pods per plant, number of reproductive nodes in the main stem, number of reproductive nodes in the ramifications, number of grains per plant and grain yield. The variation factor soybean genotypes define that number of pods per plant, number of reproductive nodes in the ramifications, number of grains per plant and grain yield are potentiated by genotype TMG 7161 RR. The biometric approach is efficient to understand the treatment levels and the slicing of simple effects of a factorial experiment, being possible to apply this methodology extensively in soybean.

Conscious and Unconscious Reward Cues Can Affect a Critical Component of Executive Control

2011 ◽  
Vol 58 (5) ◽  
pp. 370-375 ◽  
Author(s):  
Rémi L. Capa ◽  
Gaëlle M. Bustin ◽  
Axel Cleeremans ◽  
Michel Hansenne
Keyword(s):  
Correct Response ◽  
Executive Control ◽  
Critical Component ◽  
Arithmetic Operations ◽  
Component Process

The present study investigates whether updating an important function of executive control can be driven by unconscious reward cues. Participants had to memorize several numbers and update those numbers independently according to a sequence of arithmetic operations. At the beginning of each trial, a reward (1 euro or 5 cents) was presented, either subliminally or supraliminally. Participants could earn the reward if they found the correct response on the updating task. Results showed better performance when a high (conscious or unconscious) reward was at stake compared to a low reward. This suggests that subliminal information can influence a component process of executive control traditionally thought to require consciousness.

Design of spline surface vacuum gripper for pick and place robotic arms

2020 ◽  
Vol 4 (2) ◽  
pp. 48-55
Author(s):  
A. S. Jamaludin ◽  
M. N. M. Razali ◽  
N. Jasman ◽  
A. N. A. Ghafar ◽  
M. A. Hadi
Keyword(s):  
Cycle Time ◽  
Industrial Robot ◽  
Annealing Process ◽  
Vacuum Suction ◽  
Robotic Arms ◽  
Custom Made ◽  
Pick And Place ◽  
Manufacturing Procedure ◽  
Production Flexibility ◽  
The Impact

The gripper is the most important part in an industrial robot. It is related with the environment around the robot. Today, the industrial robot grippers have to be tuned and custom made for each application by engineers, by searching to get the desired repeatability and behaviour. Vacuum suction is one of the grippers in Watch Case Press Production (WCPP) and a mechanism to improve the efficiency of the manufacturing procedure. Pick and place are the important process for the annealing process. Thus, by implementing vacuum suction gripper, the process of pick and place can be improved. The purpose of vacuum gripper other than design vacuum suction mechanism is to compare the effectiveness of vacuum suction gripper with the conventional pick and place gripper. Vacuum suction gripper is a mechanism to transport part and which later sequencing, eliminating and reducing the activities required to complete the process. Throughout this study, the process pick and place became more effective, the impact on the production of annealing process is faster. The vacuum suction gripper can pick all part at the production which will lower the loss of the productivity. In conclusion, vacuum suction gripper reduces the cycle time about 20%. Vacuum suction gripper can help lower the cycle time of a machine and allow more frequent process in order to increase the production flexibility.

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