We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in...We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.展开更多
This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns. The generating function of the said patterns has been reported earlier by the author [1...This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns. The generating function of the said patterns has been reported earlier by the author [1] [2]. The paper describes how these non-associative and non commutative properties can be established by using the Cayley table on which a binary operation is defined to act on the 123-avoiding and 132-avoiding patterns of Aunu permutations using a pairing scheme. Our results have generated larger matrices from permutations of points of the Aunu patterns of prime cardinality. It follows that the generated symbols can be used in further studies and analysis in cryptography and game theory thereby providing an interdisciplinary approach and applications of these important permutation patterns.展开更多
The positions of braiding carrier in track and column braiding are represented by a diagrammatic braiding plan and a corresponding lattice-array is defined. A set is then formed so that the permutation analysis can be...The positions of braiding carrier in track and column braiding are represented by a diagrammatic braiding plan and a corresponding lattice-array is defined. A set is then formed so that the permutation analysis can be performed to represent the movement of carriers in a braiding process. The process of 4-step braiding is analyzed as an example to describe the application of the proposed method by expressing a braiding cycle as a product of disjoint cycles. As a result, a mapping relation between the disjoint cycles and the movement of carriers is deduced. Following the same analysis principles, a process of 8-step braiding and the corresponding initial state of the lattice-array is developed. A successful permutation analysis to the process manifests the general suitability of the proposed method.展开更多
Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinaliti...Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinalities of the sets Dn(T) where T■B2.Some of the cardinalities encountered involve inverse binomial coefficients,binomial coefficients,Catalan numbers,and Fibonacci numbers.展开更多
This article proposes an enhanced differential fault analysis（DFA） method named as fault-propagation pattern-based DFA（FPP-DFA）.The main idea of FPP-DFA is using the FPP of the ciphertext difference to predict the...This article proposes an enhanced differential fault analysis（DFA） method named as fault-propagation pattern-based DFA（FPP-DFA）.The main idea of FPP-DFA is using the FPP of the ciphertext difference to predict the fault location and the fault-propagation path.It shows that FPP-DFA is very effective on SPN structure block ciphers using bitwise permutation,which is applied to two block ciphers.The first is PRESENT with the substitution permutation sequence.With the fault model of injecting one nibble fault into the r-2nd round,on average 8 and 16 faults can reduce the key search space of PRESENT-80/128 to 214.7 and 221.1,respectively.The second is PRINTcipher with the permutation substitution sequence.For the first time,it shows that although the permutation of PRINTcipher is secret key dependent,FPP-DFA still works well on it.With the fault model of injecting one nibble fault into the r-2nd round,12 and 24 effective faults can reduce the key search space of PRINTcipher-48/96 to 213.7 and 222.8,respectively.展开更多
文摘We present an intuitively satisfying geometric proof of Fermat's result for positive integers that for prime moduli p, provided p does not divide a. This is known as Fermat’s Little Theorem. The proof is novel in using the idea of colorings applied to regular polygons to establish a number-theoretic result. A lemma traditionally, if ambiguously, attributed to Burnside provides a critical enumeration step.
文摘This paper presents the non-associative and non-commutative properties of the 123-avoiding patterns of Aunu permutation patterns. The generating function of the said patterns has been reported earlier by the author [1] [2]. The paper describes how these non-associative and non commutative properties can be established by using the Cayley table on which a binary operation is defined to act on the 123-avoiding and 132-avoiding patterns of Aunu permutations using a pairing scheme. Our results have generated larger matrices from permutations of points of the Aunu patterns of prime cardinality. It follows that the generated symbols can be used in further studies and analysis in cryptography and game theory thereby providing an interdisciplinary approach and applications of these important permutation patterns.
文摘The positions of braiding carrier in track and column braiding are represented by a diagrammatic braiding plan and a corresponding lattice-array is defined. A set is then formed so that the permutation analysis can be performed to represent the movement of carriers in a braiding process. The process of 4-step braiding is analyzed as an example to describe the application of the proposed method by expressing a braiding cycle as a product of disjoint cycles. As a result, a mapping relation between the disjoint cycles and the movement of carriers is deduced. Following the same analysis principles, a process of 8-step braiding and the corresponding initial state of the lattice-array is developed. A successful permutation analysis to the process manifests the general suitability of the proposed method.
基金The National Natural Science Foundation of China （No. 10801020）.
文摘Let Dn be the set of all signed permutations on n = {1,...,n} with even signs,and let Dn(T) be the set of all signed permutations in Dn which avoids a set T of signed patterns.In this paper,we find all the cardinalities of the sets Dn(T) where T■B2.Some of the cardinalities encountered involve inverse binomial coefficients,binomial coefficients,Catalan numbers,and Fibonacci numbers.
基金Supported by the National Natural Science Foundation ofChina (60772082,61173191,and 61272491)
文摘This article proposes an enhanced differential fault analysis（DFA） method named as fault-propagation pattern-based DFA（FPP-DFA）.The main idea of FPP-DFA is using the FPP of the ciphertext difference to predict the fault location and the fault-propagation path.It shows that FPP-DFA is very effective on SPN structure block ciphers using bitwise permutation,which is applied to two block ciphers.The first is PRESENT with the substitution permutation sequence.With the fault model of injecting one nibble fault into the r-2nd round,on average 8 and 16 faults can reduce the key search space of PRESENT-80/128 to 214.7 and 221.1,respectively.The second is PRINTcipher with the permutation substitution sequence.For the first time,it shows that although the permutation of PRINTcipher is secret key dependent,FPP-DFA still works well on it.With the fault model of injecting one nibble fault into the r-2nd round,12 and 24 effective faults can reduce the key search space of PRINTcipher-48/96 to 213.7 and 222.8,respectively.
