1. A method of calculating a key, the method comprising:
generating two keys;
calculating one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two keys, such that a first value is calculated for each of the coefficient values, excluding 0; and
performing a key calculation based on the one or more first values.
2. The method as claimed in claim 1, wherein the key calculation is a coordinates operation or an exponentiation operation.
3. The method as claimed in claim 1, wherein:
each of the two keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being an integer that has an absolute value of less than or equal to qw2 and is indivisible by q;
q is a prime number or a power exponent of the prime number;
w is a natural number greater than or equal to 2; and
the key calculation is a coordinates operation based on the one or more first values.
4. The method as claimed in claim 3, wherein the generating of the two keys comprises:
selecting, for each of the two keys, a t number of groups from an m\u2212(w\u22121)*(t\u22121) number of groups, where m and t are positive integers;
substituting each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more integers that have an absolute value of less than or equal to qw2 and are indivisible by q; and
substituting an unselected group with 0.
5. The method as claimed in claim 1, wherein:
each of the two keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being either 0or a positive odd number less than or equal to 2w;
w is a natural number greater than or equal to 2; and
the key calculation is an exponentiation operation, where the one or more first values are exponents.
6. The method as claimed in claim 5, wherein the generating of the two keys comprises:
selecting, for each of the two keys, a t number of groups from an m\u2212(w\u22121)*t number of groups, where m and t are positive integers;
substituting each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more positive odd numbers less than or equal to 2w; and
substituting an unselected group with 0.
7. The method as claimed in claim 1, wherein the generating of the two keys comprises:
splitting a calculation target key to generate the two keys.
8. The method as claimed in claim 7, wherein a number of the coefficients, excluding 0, among the coefficients included in each of the two keys is less than a number of coefficients, excluding 0, among coefficients included in the calculation target key.
9. The method as claimed in claim 1, wherein the generating of the two keys comprises:
selecting the two keys from a predetermined group of keys.
10. A method of generating a shared key, the method comprising:
generating two secret keys;
calculating a first public key based on the two secret keys;
calculating a second public key based on the first public key;
transmitting the first public key and the second public key;
receiving a third public key and a fourth public key; and
generating the shared key based on the two secret keys, the third public key, and the fourth public key.
11. The method as claimed in claim 10, wherein:
each of the two secret keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being an integer that has an absolute value of less than or equal to qw2 and is indivisible by q;
q is a prime number or a power exponent of the prime number; and
w is a natural number greater than or equal to 2.
12. The method as claimed in claim 11, wherein the calculating of the first public key comprises:
calculating one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two secret keys, such that a first value is calculated for each of the coefficient values, excluding 0; and
calculating the first public key by performing a coordinates operation based on the one or more first values.
13. The method as claimed in claim 11, wherein the generating of the two secret keys comprises:
selecting, for each of the two secret keys, a t number of groups from an m\u2212(w\u22121)*(t\u22121) number of groups, where m and t are positive integers;
substituting each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more integers that have an absolute value of less than or equal to qw2 and are indivisible by q; and
substituting an unselected group with 0.
14. The method as claimed in claim 10, wherein:
each of the two secret keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being either 0 or a positive odd number less than or equal to 2w; and
w is a natural number greater than or equal to 2.
15. The method as claimed in claim 14, wherein the calculating of the first public key comprises:
calculating one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two secret keys, such that a first value is calculated for each of the coefficient values, excluding 0; and
calculating the first public key by performing an exponentiation operation, where the one or more first values are exponents.
16. The method as claimed in claim 14, wherein the generating of the two secret keys comprises:
selecting, for each of the two secret keys, a t number of groups from an m\u2212(w\u22121)*t number of groups, where m and t are positive integers;
substituting each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more positive odd numbers less than or equal to 2w; and
substituting an unselected group with 0.
17. The method as claimed in claim 10, wherein:
the transmitting of the first public key and the second public key comprises transmitting the first public key and the second public key to an apparatus; and
the receiving of the third public key and the fourth public key comprises receiving the third public key and the fourth public key from the apparatus.
18. The method as claimed in claim 10, wherein the generating of the two secret keys comprises:
selecting the two secret keys from a predetermined group of keys.
19. The method as claimed in claim 10, wherein the calculating of the second public key comprises:
multiplying the first public key by a randomly selected value.
20. The method as claimed in claim 10, wherein the generating of the shared key comprises:
generating the shared key according to an equation: x1X2+y1Y2, where x1 and y1 are the two secret keys, X2 is the third public key, and Y2 is the fourth public key.
21. A computer readable recording medium encoded with the method of claim 1 and implemented by a computer.
22. A computer readable recording medium encoded with the method of claim 10 and implemented by a computer.
23. An apparatus for calculating a key, the apparatus comprising:
a key generation management unit to generate two keys;
a coefficient value calculator to calculate one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two keys, such that a first value is calculated for each of the coefficient values, excluding 0; and
a key calculator to perform a key calculation based on the one or more first values.
24. The apparatus as claimed in claim 23, wherein the key calculation is a coordinates operation or an exponentiation operation.
25. The apparatus as claimed in claim 23, wherein:
each of the two keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being an integer that has an absolute value of less than or equal to qw2 and is indivisible by q;
q is a prime number or a power exponent of the prime number;
w is a natural number greater than or equal to 2; and
the key calculation is a coordinates operation based on the one or more first values.
26. The apparatus as claimed in claim 25, wherein the key generation management unit comprises:
a group selector to select, for each of the two keys, a t number of groups from an m\u2212(w\u22121)*(t\u22121) number of groups, where m and t are positive integers;
a string substitution unit to substitute each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more integers that have an absolute value of less than or equal to qw2 and are indivisible by q; and
a key generator to generate each of the two keys by substituting an unselected group with 0.
27. The apparatus as claimed in claim 23, wherein:
each of the two keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being either 0 or a positive odd number less than or equal to 2w;
w is a natural number greater than or equal to 2; and
the key calculation is an exponentiation operation, where the one or more first values are exponents.
28. The apparatus as claimed in claim 27, wherein the key generation management unit comprises:
a group selector to select, for each of the two keys, a t number of groups from an m\u2212(w\u22121)*t number of groups, where m and t are positive integers;
a string substitution unit to substitute each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more positive odd numbers less than or equal to 2w; and
a key generator to generate each of the two keys by substituting an unselected group with 0.
29. The apparatus as claimed in claim 23, wherein the key generation management unit splits a calculation target key to generate the two keys.
30. The apparatus as claimed in claim 29, wherein a number of the coefficients, excluding 0, among the coefficients included in each of the two keys is less than a number of coefficients, excluding 0, among coefficients included in the calculation target key.
31. The apparatus as claimed in claim 23, wherein the key generation management unit selects the two keys from a predetermined group of keys.
32. An apparatus for generating a shared key, the apparatus comprising:
a key generation management unit to generate two secret keys;
a first calculator to calculate a first public key based on the two secret keys;
a second calculator to calculate a second public key based on the first public key;
a transmitting and receiving unit to transmit the first public key and the second public key, and to receive a third public key and a fourth public key; and
a shared key generator to generate the shared key based on the two secret keys, the third public key, and the fourth public key.
33. The apparatus as claimed in claim 32, wherein:
each of the two secret keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being an integer that has an absolute value of less than or equal to qw2 and is indivisible by q;
q is a prime number or a power exponent of the prime number; and
w is a natural number greater than or equal to 2.
34. The apparatus as claimed in claim 33, wherein the first calculator:
calculates one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two secret keys, such that a first value is calculated for each of the coefficient values, excluding zero; and
calculates the first public key by performing a coordinates operation based on the one or more first values.
35. The apparatus as claimed in claim 33, wherein the key generation management unit comprises:
a group selector to select, for each of the two secret keys, a t number of groups from an m\u2212(w\u22121)*(t\u22121) number of groups, where m and t are positive integers;
a string substitution unit to substitute each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more integers that have an absolute value of less than or equal to qw2 and are indivisible by q; and
a key generator to generate each of the two secret keys by substituting an unselected group with 0.
36. The apparatus as claimed in claim 32, wherein:
each of the two secret keys includes at most one coefficient, excluding 0, among a consecutive w number of coefficients, the at most one coefficient being either 0 or a positive odd number less than or equal to 2w; and
w is a natural number greater than or equal to 2.
37. The apparatus as claimed in claim 36, wherein the first calculator:
calculates one or more first values based on coefficients having an identical coefficient value among coefficients included in each of the two secret keys, such that a first value is calculated for each of the coefficient values, excluding 0; and
calculates the first public key by performing an exponentiation operation, where the one or more first values are exponents.
38. The apparatus as claimed in claim 36, wherein the key generation management unit comprises:
a group selector to select, for each of the two secret keys, a t number of groups from an m\u2212(w\u22121)*t number of groups, where m and t are positive integers;
a string substitution unit to substitute each of the selected t number of groups with a string, the string listing a w\u22121 number of 0s and one or more positive odd numbers less than or equal to 2w; and
a key generator to generate each of the two secret keys by substituting an unselected group with 0.
39. The apparatus as claimed in claim 32, wherein the transmitting and receiving unit transmits the first public key and the second public key to an apparatus, and receives the third public key and the fourth public key from the apparatus.
40. The apparatus as claimed in claim 32, wherein the key generation management unit selects the two secret keys from a predetermined group of keys.
41. The apparatus as claimed in claim 32, wherein the second calculator calculates the second public key by multiplying the first public key by a randomly selected value.
42. The apparatus as claimed in claim 32, wherein the shared key generator generates the shared key according to an equation: x1X2+y1Y2, where x1 and y1 are the two secret keys, X2 is the third public key, and Y2 is the fourth public key.
43. A system for securing transactions between apparatuses, the system comprising:
a first apparatus that generates a first shared key based on two first secret keys, a third public key, and a fourth public key; and
a second apparatus that generates a second shared key based on two second secret keys, a first public key, and a second public key, wherein
the first public key is calculated based on the two first secret keys, the second public key is calculated based on the first public key, the third public key is calculated based on the two second secret keys, the fourth public key is calculated based on the second public key, and a transaction between the first apparatus and the second apparatus is secure if the first shared key is identical to the second shared key.
44. The system as claimed in claim 43, wherein the first apparatus comprises:
a key generation management unit to generate the two first secret keys;
a first calculator to calculate the first public key based on the two first secret keys;
a second calculator to calculate the second public key based on the first public key;
a transmitting and receiving unit to transmit the first public key and the second public key to the second apparatus, and to receive the third public key and the fourth public key from the second apparatus; and
a shared key generator to generate the first shared key based on the two first secret keys, the third public key, and the fourth public key.
45. The system as claimed in claim 43, wherein the second apparatus comprises:
a key generation management unit to generate the two second secret keys;
a first calculator to calculate the third public key based on the two second secret keys;
a second calculator to calculate the fourth public key based on the third public key;
a transmitting and receiving unit to transmit the third public key and the fourth public key to the first apparatus, and to receive the first public key and the second public key from the first apparatus; and
a shared key generator to generate the second shared key based on the two second secret keys, the first public key, and the second public key.
46. The system as claimed in claim 43, wherein the system further comprises:
a third apparatus to transmit the two first secret keys, the first public key, and the second public key to the first apparatus.
47. The system as claimed in claim 43, wherein the system further comprises:
a third apparatus to transmit the two second secret keys, the third public key, and the fourth public key to the second apparatus.
The claims below are in addition to those above.
All refrences to claim(s) which appear below refer to the numbering after this setence.
1. A wavelength selective switch comprising:
an inputoutput port array in which an input port and an output port are arranged in a first direction;
a dispersive element which receives wavelength-multiplexed light from the input port, disperse the wavelength-multiplexed light in a second direction orthogonal to the first direction for each predetermined wavelength component, and emits dispersed light;
a condensing optical system which condenses the dispersed light emitted from the dispersive element; and
a reflection type deflecting element which receives the dispersed light condensed by the condensing optical system and deflects the dispersed light toward the output port,
wherein the reflection type deflecting element includes a deflecting face which receives the dispersed light condensed by the condensing optical system and deflects the dispersed light to the output port, and a cover which covers the deflecting face,
the deflecting face is inclined with respect to the first direction when the reflection type deflecting element is in a non-operating state, and
an inclination angle \u03b81 of the deflecting face with respect to the first direction when the reflection type deflecting element is in the non-operating state,
an inclination angle \u03b82 of the cover with respect to the first direction, and
an inclination angle \u03b1 to the first direction of the dispersed light incident on the deflecting face, satisfy Expression (1) below:
\u03b82>2\xd7\u03b81+\u03b1\u2003\u2003(1)
2. The wavelength selective switch according to claim 1, wherein the inclination angle of the deflecting face with respect to the first direction when the reflection type deflecting element is in an operating state is smaller than the inclination angle \u03b81.
3. The wavelength selective switch according to claim 1, wherein
the inclination angle \u03b81,
a focal length f of the condensing optical system,
a distance L from the input port to the output port most apart from the input port, and
the inclination angle \u03b1 satisfy Equation (2) below:
L<f\xd7tan(2\xd7(\u03b81+\u03b1))\u2003\u2003(2)
4. The wavelength selective switch according to claim 1, wherein
the inclination angle \u03b82,
a focal length f of the condensing optical system,
a distance L from the input port to the output port most apart from the input port, and
the inclination angle \u03b1 satisfy Equation (3) below:
L<f\xd7tan(2\xd7(\u03b82+\u03b1))\u2003\u2003(3)
5. The wavelength selective switch according to claim 1, wherein
a focal length f of the condensing optical system,
a distance L from the input port to the output port most apart from the input port, and
an emission angle \u03c6 of secondary reflected light which is reflected by the cover, reflected by the deflecting face again and then emitted from the reflection type deflecting element, satisfy Expression (4) below:
L<f\xd7tan \u03c6\u2003\u2003(4)
6. The wavelength selective switch according to claim 1, further comprising a base having a main surface orthogonal to the first direction,
wherein the inputoutput port array, the dispersive element and the condensing optical system are mounted on the main surface, and
the cover is inclined with respect to the first direction so that cover reflected light which is condensed by the condensing optical system and then reflected by the cover, is directed to the main surface.
7. The wavelength selective switch according to claim 6, wherein a position hit by the cover reflected light on the main surface is subjected to rough surface processing or optical absorption processing.
8. The wavelength selective switch according to claim 6, wherein a fiber for transmitting a monitoring-light is provided in a position hit by the cover reflected light on the main surface.
9. The wavelength selective switch according to claim 1, wherein the input port is arranged in a center of the inputoutput port array with respect to the first direction.