586 research outputs found
ΠΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΡΠΉ Π»ΠΈΠ½Π΅ΠΉΠ½ΡΠΉ Π³Π΅Π½Π΅ΡΠ°ΡΠΎΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ Ρ ΠΊΠΎΠ½ΡΡΠΎΠ»Π΅ΠΌ ΠΎΡΠΈΠ±ΠΎΠΊ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ
A parallel linear generator of multi-valued pseudorandom sequences, which operates under conditions of generating hardware errors caused by destructive adversary actions is proposed. The main types of modification of the pseudorandom sequence in case of adversary attack are considered. A distinctive feature of the iterative process of ensuring the reliability of computational operations is the "arithmetic" of computational operations by representing a system of generating recurring logical formulas as a system of many-valued logic algebra functions. The subsequent realization of multivalued logic algebra functions by means of arithmetic polynomials allowed us to parallelize the process of generating multivalued pseudorandom sequences and level out the existing complexity (specificity) of cryptographic transformations of logical data types which limit the use of redundant coding methods. As a result, a solution that allows to apply redundant modular codes to control the accuracy of the computational operations performed by the nodes of pseudorandom sequence generation is proposed. Moreover, unlike the known solutions, the proposed method provides obtaining fragments of a pseudorandom sequence on the basis of one recursive arithmetic formula with parallel calculation errors control. The use of modular forms made it possible to transfer computations from the rational numbers field arithmetic to integer arithmetic of a simple field.
Among the existing variety of codes correcting errors (maximally spaced codes), a special place is occupied by multivalued Reed-Solomon codes. Reed-Solomon codes usage in the formation of pseudorandom sequences allows the formation of code-like structures that monitor and ensure the reliability of computational operations. The calculated probability of failure-free operation of the parallel linear generator of multivalued pseudorandom sequences with an error control function based on the principle of functioning β sliding redundancy is obtained. The achieved results can find wide application at realization of perspective high-efficiency cryptographic information protection facility.ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΡΠΉ Π»ΠΈΠ½Π΅ΠΉΠ½ΡΠΉ Π³Π΅Π½Π΅ΡΠ°ΡΠΎΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ, ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΡΡΡΠΈΠΉ Π² ΡΡΠ»ΠΎΠ²ΠΈΡΡ
Π³Π΅Π½Π΅ΡΠ°ΡΠΈΠΈ Π°ΠΏΠΏΠ°ΡΠ°ΡΠ½ΡΡ
ΠΎΡΠΈΠ±ΠΎΠΊ, ΠΎΠ±ΡΡΠ»ΠΎΠ²Π»Π΅Π½Π½ΡΡ
Π΄Π΅ΡΡΡΡΠΊΡΠΈΠ²Π½ΡΠΌΠΈ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΡΠΌΠΈ Π·Π»ΠΎΡΠΌΡΡΠ»Π΅Π½Π½ΠΈΠΊΠ°. Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½Ρ ΠΎΡΠ½ΠΎΠ²Π½ΡΠ΅ Π²ΠΈΠ΄Ρ ΠΌΠΎΠ΄ΠΈΡΠΈΠΊΠ°ΡΠΈΠΈ ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ ΠΏΡΠΈ Π°ΡΠ°ΠΊΠ°Ρ
Π·Π»ΠΎΡΠΌΡΡΠ»Π΅Π½Π½ΠΈΠΊΠ°. ΠΡΠ»ΠΈΡΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΡΡ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠ³ΠΎ ΠΈΡΠ΅ΡΠ°ΡΠΈΠ²Π½ΠΎΠ³ΠΎ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠ΅Π½ΠΈΡ Π΄ΠΎΡΡΠΎΠ²Π΅ΡΠ½ΠΎΡΡΠΈ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ ΡΠ²Π»ΡΠ΅ΡΡΡ Β«Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠ·Π°ΡΠΈΡΒ» Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ ΠΏΡΡΠ΅ΠΌ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌΡ ΠΏΠΎΡΠΎΠΆΠ΄Π°ΡΡΠΈΡ
ΡΠ΅ΠΊΡΡΡΠ΅Π½ΡΠ½ΡΡ
Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΎΡΠΌΡΠ» ΠΊΠ°ΠΊ ΡΠΈΡΡΠ΅ΠΌΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ Π°Π»Π³Π΅Π±ΡΡ Π»ΠΎΠ³ΠΈΠΊΠΈ. ΠΠΎΡΠ»Π΅Π΄ΡΡΡΠ°Ρ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ Π°Π»Π³Π΅Π±ΡΡ Π»ΠΎΠ³ΠΈΠΊΠΈ ΠΏΠΎΡΡΠ΅Π΄ΡΡΠ²ΠΎΠΌ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΎΠ² ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΠ»Π° ΡΠ°ΡΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΠΈΡΡ ΠΏΡΠΎΡΠ΅ΡΡ Π³Π΅Π½Π΅ΡΠ°ΡΠΈΠΈ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ ΠΈ Π½ΠΈΠ²Π΅Π»ΠΈΡΠΎΠ²Π°ΡΡ ΡΡΡΠ΅ΡΡΠ²ΡΡΡΡΡ ΡΠ»ΠΎΠΆΠ½ΠΎΡΡΡ (ΡΠΏΠ΅ΡΠΈΡΠΈΠΊΡ) ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΠΉ Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΈΠΏΠΎΠ² Π΄Π°Π½Π½ΡΡ
, ΠΎΠ³ΡΠ°Π½ΠΈΡΠΈΠ²Π°ΡΡΠΈΡ
ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΈΠ·Π±ΡΡΠΎΡΠ½ΠΎΠ³ΠΎ ΠΊΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΡ. Π ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ΅ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½ΠΎ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅, ΠΏΠΎΠ·Π²ΠΎΠ»ΡΡΡΠ΅Π΅ ΠΏΡΠΈΠΌΠ΅Π½ΠΈΡΡ ΠΈΠ·Π±ΡΡΠΎΡΠ½ΡΠ΅ ΠΌΠΎΠ΄ΡΠ»ΡΡΠ½ΡΠ΅ ΠΊΠΎΠ΄Ρ Π΄Π»Ρ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ Π±Π΅Π·ΠΎΡΠΈΠ±ΠΎΡΠ½ΠΎΡΡΠΈ ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΠΌΡΡ
Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ ΡΠ·Π»Π°ΠΌΠΈ Π³Π΅Π½Π΅ΡΠ°ΡΠΈΠΈ ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ. ΠΡΠΈΡΠ΅ΠΌ Π² ΠΎΡΠ»ΠΈΡΠΈΠ΅ ΠΎΡ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
ΡΠ΅ΡΠ΅Π½ΠΈΠΉ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΠΌΡΠΉ ΠΌΠ΅ΡΠΎΠ΄ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠΈΠ²Π°Π΅Ρ ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΠ΅ ΡΡΠ°Π³ΠΌΠ΅Π½ΡΠΎΠ² ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΠΎΠΉ ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠΈ Π½Π° ΠΎΡΠ½ΠΎΠ²Π°Π½ΠΈΠΈ ΠΎΠ΄Π½ΠΎΠΉ ΡΠ΅ΠΊΡΡΡΠΈΠ²Π½ΠΎΠΉ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΎΡΠΌΡΠ»Ρ Ρ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΡΠΌ ΠΊΠΎΠ½ΡΡΠΎΠ»Π΅ΠΌ ΠΎΡΠΈΠ±ΠΎΠΊ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ. ΠΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΌΠΎΠ΄ΡΠ»ΡΡΠ½ΡΡ
ΡΠΎΡΠΌ ΠΏΠΎΠ·Π²ΠΎΠ»ΠΈΠ»ΠΎ ΠΏΠ΅ΡΠ΅Π½Π΅ΡΡΠΈ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΠΈΠ· Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΠΈ ΠΏΠΎΠ»Ρ ΡΠ°ΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΡ
ΡΠΈΡΠ΅Π» Π² ΡΠ΅Π»ΠΎΡΠΈΡΠ»Π΅Π½Π½ΡΡ Π°ΡΠΈΡΠΌΠ΅ΡΠΈΠΊΡ ΠΏΡΠΎΡΡΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ.
Π‘ΡΠ΅Π΄ΠΈ ΡΡΡΠ΅ΡΡΠ²ΡΡΡΠ΅Π³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΠΎΠ±ΡΠ°Π·ΠΈΡ ΠΊΠΎΠ΄ΠΎΠ², ΠΈΡΠΏΡΠ°Π²Π»ΡΡΡΠΈΡ
ΠΎΡΠΈΠ±ΠΊΠΈ (ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡΠ½ΠΎ ΡΠ°Π·Π½Π΅ΡΠ΅Π½Π½ΡΡ
ΠΊΠΎΠ΄ΠΎΠ²), ΠΎΡΠΎΠ±ΠΎΠ΅ ΠΌΠ΅ΡΡΠΎ Π·Π°Π½ΠΈΠΌΠ°ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΠ΅ ΠΊΠΎΠ΄Ρ Π ΠΈΠ΄Π° β Π‘ΠΎΠ»ΠΎΠΌΠΎΠ½Π°. ΠΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΊΠΎΠ΄ΠΎΠ² Π ΠΈΠ΄Π° β Π‘ΠΎΠ»ΠΎΠΌΠΎΠ½Π° ΠΏΡΠΈ ΡΠΎΡΠΌΠΈΡΠΎΠ²Π°Π½ΠΈΠΈ ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΠΎΡΠΌΠΈΡΠΎΠ²Π°ΡΡ ΠΊΠΎΠ΄ΠΎΠΏΠΎΠ΄ΠΎΠ±Π½ΡΠ΅ ΡΡΡΡΠΊΡΡΡΡ, ΠΎΡΡΡΠ΅ΡΡΠ²Π»ΡΡΡΠΈΠ΅ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ ΠΈ ΠΎΠ±Π΅ΡΠΏΠ΅ΡΠ΅Π½ΠΈΠ΅ Π΄ΠΎΡΡΠΎΠ²Π΅ΡΠ½ΠΎΡΡΠΈ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΉ. ΠΠΎΠ»ΡΡΠ΅Π½Ρ ΡΠ°ΡΡΠ΅ΡΠ½ΡΠ΅ Π΄Π°Π½Π½ΡΠ΅ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠΈ Π±Π΅Π·ΠΎΡΠΊΠ°Π·Π½ΠΎΠΉ ΡΠ°Π±ΠΎΡΡ ΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΡΠ½ΠΎΠ³ΠΎ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠ³ΠΎ Π³Π΅Π½Π΅ΡΠ°ΡΠΎΡΠ° ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΠΏΡΠ΅Π²Π΄ΠΎΡΠ»ΡΡΠ°ΠΉΠ½ΡΡ
ΠΏΠΎΡΠ»Π΅Π΄ΠΎΠ²Π°ΡΠ΅Π»ΡΠ½ΠΎΡΡΠ΅ΠΉ Ρ ΡΡΠ½ΠΊΡΠΈΠ΅ΠΉ ΠΊΠΎΠ½ΡΡΠΎΠ»Ρ ΠΎΡΠΈΠ±ΠΎΠΊ ΠΏΠΎ ΠΏΡΠΈΠ½ΡΠΈΠΏΡ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ β ΡΠΊΠΎΠ»ΡΠ·ΡΡΠ΅Π΅ ΡΠ΅Π·Π΅ΡΠ²ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅. ΠΠΎΡΡΠΈΠ³Π½ΡΡΡΠ΅ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΡ ΠΌΠΎΠ³ΡΡ Π½Π°ΠΉΡΠΈ ΡΠΈΡΠΎΠΊΠΎΠ΅ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΠΏΡΠΈ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΠΈ ΠΏΠ΅ΡΡΠΏΠ΅ΠΊΡΠΈΠ²Π½ΡΡ
Π²ΡΡΠΎΠΊΠΎΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΡΡΠ΅Π΄ΡΡΠ² ΠΊΡΠΈΠΏΡΠΎΠ³ΡΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ Π·Π°ΡΠΈΡΡ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΈ
STATISTICAL PROPERTIES OF PSEUDORANDOM SEQUENCES
Random numbers (in one sense or another) have applications in computer simulation, Monte Carlo integration, cryptography, randomized computation, radar ranging, and other areas. It is impractical to generate random numbers in real life, instead sequences of numbers (or of bits) that appear to be ``random yet repeatable are used in real life applications. These sequences are called pseudorandom sequences. To determine the suitability of pseudorandom sequences for applications, we need to study their properties, in particular, their statistical properties. The simplest property is the minimal period of the sequence. That is, the shortest number of steps until the sequence repeats. One important type of pseudorandom sequences is the sequences generated by feedback with carry shift registers (FCSRs). In this dissertation, we study statistical properties of N-ary FCSR sequences with odd prime connection integer q and least period (q-1)/2. These are called half-β-sequences. More precisely, our work includes: The number of occurrences of one symbol within one period of a half-β-sequence; The number of pairs of symbols with a fixed distance between them within one period of a half-β-sequence; The number of triples of consecutive symbols within one period of a half-β-sequence.
In particular we give a bound on the number of occurrences of one symbol within one period of a binary half-β-sequence and also the autocorrelation value in binary case. The results show that the distributions of half-β-sequences are fairly flat. However, these sequences in the binary case also have some undesirable features as high autocorrelation values. We give bounds on the number of occurrences of two symbols with a fixed distance between them in an β-sequence, whose period reaches the maximum and obtain conditions on the connection integer that guarantee the distribution is highly uniform.
In another study of a cryptographically important statistical property, we study a generalization of correlation immunity (CI). CI is a measure of resistance to Siegenthaler\u27s divide and conquer attack on nonlinear combiners. In this dissertation, we present results on correlation immune functions with regard to the q-transform, a generalization of the Walsh-Hadamard transform, to measure the proximity of two functions. We give two definitions of q-correlation immune functions and the relationship between them. Certain properties and constructions for q-correlation immune functions are discussed. We examine the connection between correlation immune functions and q-correlation immune functions
Society-oriented cryptographic techniques for information protection
Groups play an important role in our modern world. They are more reliable and more trustworthy than individuals. This is the reason why, in an organisation, crucial decisions are left to a group of people rather than to an individual. Cryptography supports group activity by offering a wide range of cryptographic operations which can only be successfully executed if a well-defined group of people agrees to co-operate. This thesis looks at two fundamental cryptographic tools that are useful for the management of secret information. The first part looks in detail at secret sharing schemes. The second part focuses on society-oriented cryptographic systems, which are the application of secret sharing schemes in cryptography. The outline of thesis is as follows
Computationally efficient search for large primes
To satisfy the speed of communication and to meet the demand for the continuously larger prime numbers, the primality testing and prime numbers generating algorithms require continuous advancement. To find the most efficient algorithm, a need for a survey of methods arises. Concurrently, an urge for the analysis of algorithms\u27 performances emanates. The critical criteria in the analysis of the prime numbers generation are the number of probes, number of generated primes, and an average time required in producing one prime. Hence, the purpose of this thesis is to indicate the best performing algorithm. The survey the methods, establishment of the comparison criteria, and comparison of approaches are the required steps to find the best performing algorithm.
In the first step of this research paper the methods were surveyed and classified using the approach described in Menezes [66]. Wifle chapter 2 sorted, described, compared, and summarized primality testing methods, chapter 3 sorted, described, compared, and summarized prime numbers generating methods. In the next step applying a uniform technique, the computer programs were written to the selected algorithms. The programs were installed on the Unix operating system, running on the Sun 5.8 server to perform the computer experiments. The computer experiments\u27 results pertaining to the selected algorithms, provided required parameters to compare the algorithms\u27 performances. The results from the computer experiments were tabulated to compare the parameters and to indicate the best performing algorithm.
Survey of methods indicated that the deterministic and randomized are the main approaches in prime numbers generation. Random number generation found application in the cryptographic keys generation. Contemporaneously, a need for deterministically generated provable primes emerged in the code encryption, decryption, and in the other cryptographic areas.
The analysis of algorithms\u27 performances indicated that the prime nurnbers generated through the randomized techniques required smaller number of probes. This is due to the method that eliminates the non-primes in the initial step, that pre-tests randomly generated primes for possible divisibility factors. Analysis indicated that the smaller number of probes increases algorithm\u27s efficiency. Further analysis indicated that a ratio of randomly generated primes to the expected number of primes, generated in the specific interval is smaller than the deterministically generated primes. In this comparison the Miller-Rabin\u27s and the Gordon\u27s algorithms that randomly generate primes were compared versus the SFA and the Sequences Containing Primes. The name Sequences Containing Primes algorithm is abbreviated in this thesis as 6kseq. In the interval [99000,1000001 the Miller Rabin method generated 57 out of 87 expected primes, the SFA algorithm generated 83 out of 87 approximated primes. The expected number of primes was computed using the approximation n/ln(n) presented by Menezes [66]. The average consumed time of originating one prime in the [99000, 100000] interval recorded 0.056 [s] for Miller-Rabin test, 0.0001 [s] for SFA, and 0.0003 [s] for 6kseq. The Gordon\u27s algorithm in the interval [1,100000] required 100578 probes and generated 32 out of 8686 expected number of primes.
Algorithm Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers invented by Doctor Verkhovsky [1081 verifies and generates primes and quasi primes using special mathematical constructs. This algorithm indicated best performance in the interval [1,1000] generating and verifying 3585 variances of provable primes or quasi primes. The Parametric Representation of Composite Twins algorithm consumed an average time per prime, or quasi prime of 0.0022315 [s]. The Parametric Representation of Composite Twins and Generation of Prime and Quasi Prime Numbers algorithm implements very unique method of testing both primes and quasi-primes. Because of the uniqueness of the method that verifies both primes and quasi-primes, this algorithm cannot be compared with the other primality testing or prime numbers generating algorithms.
The ((a!)^2)*((-1^b) Function In Generating Primes algorithm [105] developed by Doctor Verkhovsky was compared versus extended Fermat algorithm. In the range of [1,10001 the [105] algorithm exhausted an average 0.00001 [s] per prime, originated 167 primes, while the extended Fermat algorithm also produced 167 primes, but consumed an average 0.00599 [s] per prime.
Thus, the computer experiments and comparison of methods proved that the SFA algorithm is deterministic, that originates provable primes. The survey of methods and analysis of selected approaches indicated that the SFA sieve algorithm that sequentially generates primes is computationally efficient, indicated better performance considering the computational speed, the simplicity of method, and the number of generated primes in the specified intervals
Optimizations of Isogeny-based Key Exchange
Supersingular Isogeny Diffie-Hellman (SIDH) is a key exchange scheme that is believed to
be quantum-resistant. It is based on the difficulty of finding a certain isogeny between given
elliptic curves. Over the last nine years, optimizations have been proposed that significantly
increased the performance of its implementations. Today, SIDH is a promising candidate in
the US National Institute for Standards and Technologyβs (NISTβs) post-quantum cryptography
standardization process.
This work is a self-contained introduction to the active research on SIDH from a high-level,
algorithmic lens. After an introduction to elliptic curves and SIDH itself, we describe the
mathematical and algorithmic building blocks of the fastest known implementations.
Regarding elliptic curves, we describe which algorithms, data structures and trade-offs regard-
ing elliptic curve arithmetic and isogeny computations exist and quantify their runtime cost in
field operations. These findings are then tailored to the situation of SIDH. As a result, we give
efficient algorithms for the performance-critical parts of the protocol
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