The exact manner of BKZ algorithm for higher block sizes cannot be studied by practical running, so Simulation of BKZ can be used to predict the total cost and output quality of BKZ algorithm. Sampling method of enumeration solution vector v is one of the main components of designing BKZ-Simulation and can be divided into two phases: sampling norm of solution vector v and sampling corresponding coefficient vectors. This paper introduces a simple and efficient idea for sampling the norm of enumeration solution v for any success probability of enumeration bounding functions, while to the best of our knowledge, no such sampling method for norm of enumeration solution is proposed in former studies. Next, this paper analyzes the structure and probability distribution of coefficient vectors (corresponding with enumeration solution v), and consequently introduces the sampling methods for these coefficient vectors which are verified by our test results, while no such a deep analysis for sampling coefficient vectors is considered in design of former BKZ-Simulations. Moreover, this paper proposes an approximation for cost of enumerations pruned by optimal bounding functions.

Background and Objectives: Since exact manner of BKZ algorithm for higher block sizes cannot be studied by practical running, therefore Simulation of BKZ is used to predict the total cost of BKZ and quality of output basis. This paper revises some main components of BKZ-Simulation for better predictions.Methods: At first, by definition of full-enumeration success probability, the optimal enumeration radius is formally defined. Next, this paper defines three more pruning types, besides the well-known pruning by bounding function in GNR-enumerations, and consequently uses these four pruning types collectively in revision of success probability estimation. Also, by using these four pruning types and the process of updating-radius, this paper revises the estimation of enumeration cost. Finally, this paper introduces a simple technique to generate partially better bounding functions. Results: For block sizes of 50≤β≤240, better domains of radius parameters in GNR enumeration are formally introduced. Also, our revised estimation of success probability (for GNR bounding function) in our test results shows non-negligible gap from former estimations in some main former studies. Moreover, our results show that the cost results by our proposed estimator of GNR-enumeration cost are closer to the cost results determined in experimental running of enumeration, than the cost results by Chen-Nguyen estimator.Conclusion: This paper revises the estimators of cost and success probability for GNR-Enumeration, and justifies the value of these revised estimators by sufficient test results (in actual running and Simulation of BKZ). Also our novel definition of optimal enumeration radius can be used effectively in actual running and Simulation of BKZ.

Former studies on progressive-BKZ almost focus on increasing block sizes. Our work in IJCNIS 9. 9, 2018, introduces a new version of progressive-BKZ based on increasing success probabilities, while its results are not sufficiently hopeful! This paper introduces two algorithms of “BKZ with optimal progressive block sizes” and “BKZ with optimal progressive success probabilities”, while their time-complexities are proved to be optimal. However these two proposed algorithms are designed to solve exact-SVP for an input basis, they can be used as an SVP-solver in the body of another BKZ algorithm for practical attacks! Also, our proposed algorithms can be used as reasonable representatives of two approaches of “increasing block sizes” and “increasing success probabilities” in the progressive-BKZ family to be compared with each other for the first time. For dimension n≥90, BKZ with optimal progressive success probabilities shows better runtime than corresponding instances of BKZ with optimal progressive block sizes, so that for Gentry-Halevi’s main lattice challenges, these speedups include: “2^14. 1 times for Toy challenge of n=512”, “2^67. 2 times for Small challenge of n=2048”, “2^235. 5 times for Medium challenge of n=8192” and “2^1207. 7 times for large challenge of n=32768”. Also, the time cost of BKZ with optimal progressive success probabilities and optimal progressive block sizes as two exact-SVP solvers are compared with some main claimants of exact-SVP solvers such as sieve algorithm, extreme-pruned enumeration, full-enumeration, and so on, for the dimensions of 100≤β≤240, and our results show hopeful time cost against these claimants. Moreover, two Cost-Models are approximated for these two optimal progressive BKZ.

BKZ 2. 0 algorithm is one of the claimant lattice reduction algorithms which incorporates extreme pruning as its main phase. The non-extreme pruning and extreme pruning in the original paper by Gamma-Nguyen-Regev (in 2010) respectively are claimed to reach the maximum speedup of 2^(β/4) and 2^(β/2) over full-enumeration. For 37≤β, this paper verifies the claimed speedup of 2^(β/4) by an optimal non-extreme pruned enumeration, while for a practical block size of 100≤β≤250, the upper-bound for speedup of extreme-pruned enumeration when blocks are preprocessed by BKZ with enumeration (as SVP-solver) is estimated from 2^(β/6. 6) to 2^(β/4. 4), and when blocks are preprocessed by BKZ with sieving is estimated from 2^(β/9. 8) to 2^(β/3. 4)! By using these upper-bounds for speedup by extreme-pruning, all former security analyses which use the claimed speedup of 2^(β/2), should be revised (or rejected). Also, this paper proposes a revised version of BKZ 2. 0, so that for a practical block size of 100≤β≤250 and an infinite number of rounds N≈∞, the lower-bound of its speedup over BKZ 2. 0 is estimated by a factor of ρ_0 in range of 2^12≤ρ_0≤2^15. 5 when blocks are preprocessed by BKZ with enumeration, and also lower-bound of its speedup is estimated in the range of 0≤ρ_0≤2^20. 5 when blocks are preprocessed by BKZ with sieving. Moreover, for a finite number of rounds N, speedup of our revised BKZ 2. 0 over original BKZ 2. 0 can be estimated by O(min⁡(N, ρ_0 ) ).

In this paper the results obtained by the numerical method of modeling of Ekibastuz coal burning in BKZ-420 combustion chamber of Kazakhstan Power Plant are presented. They are devoted to the numerical Simulation of combustion processes in the furnace boiler BKZ-420. Boiler’s steam generates capacity equal 420 T/h.Boiler has six vertical pulverized coal burners arranged in two levels with three burners on the front wall of the boiler. High ash, low-grade coal from Ekibastuz burned in the furnace. Its ash content is 40%, volatile –24%, humidity–5%, highest calorific value is 16750 kJ/kg. Milling dispersity of coal was equal to R90= 15%.It was shown in this research that the most intense burning is observed in the central part of the chamber where the flow temperature reaches about 980 °C and it is seen that the temperature reaches a peak in the cross sections of the burners location. The combustion reaction there occurs more intensively.