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SECRET SHARING SYSTEM, SECRET SHARING APPARATUS, SECRET SHARING METHOD, SECRET SORTING METHOD, SECRET SHARING PROGRAM - NIPPON TELEGRAPH AND TELEPHONE CORPORATION
SECRET SHARING SYSTEM, SECRET SHARING APPARATUS, SECRET SHARING METHOD, SECRET SORTING METHOD, SECRET SHARING PROGRAM
United States Patent Application 20130182836
A secret sharing system according to the present invention includes N secret sharing apparatuses. The secret sharing system according to the present invention includes fragment replacement means and reshare means. The fragment replacement means selects a number, smaller than N, of secret sharing apparatuses, generates a bijection π of {1, . . . , K}→{1, . . . , K}among the selected secret sharing apparatuses and designates a fragment aπ(k)i recorded in a selected secret sharing apparatus as a k-th fragment (i represents an identification number that indicates a selected secret sharing apparatus). The reshare means performs reshare of fragments of a numeric value used as replacements by the fragment replacement means to determine new fragments.
Hamada, Koki (Tokyo, JP)
Ikarashi, Dai (Tokyo, JP)
Chida, Koji (Tokyo, JP)
Takahashi, Katsumi (Tokyo, JP)
13/876110
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20040096057 Modular multiplication of multi-precision numbers May, 2004 Moore
20040184604 Secure method for performing a modular exponentiation operation September, 2004 Joye et al.
Dan Bogdanov, "How to securely perform computations on secret-shared data", 2007, Pages 20-30.
Hiroki Koga, "A General Formula of the (t, n) - Threshold Visual Secret Sharing Scheme", 2002, Springer-Verlag Berlin Heidelberg, Pages 328-345.
Yevgeniy Dodis, Lecture 1: Exposure-Resistent Cryptography, 2007, Pages 1-7.
Pablo Azar, "Secret Sharing and Applications", 2009, Pages 83-86.
1. A secret sharing system comprising N secret sharing apparatuses, wherein it is assumed that N represents an integer equal to or greater than 3, n represents an integer equal to or greater than 1 and equal to or smaller than N, M represents an integer equal to or greater than 1, m represents an integer equal to or greater than 1 and equal to or smaller than M, K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M) are K×M numeric values whose fragments are recorded in the secret sharing apparatuses in a distributed manner, numeric values AK(1), . . . , and AK(M) are a group of k-th numeric values associated with each other, akn(m) is a fragment of a numeric value Ak(m) recorded in an n-th secret sharing apparatus, and i represents an integer equal to or greater than 1 and equal to or smaller than N that indicates secret sharing apparatuses selected from among the N secret sharing apparatuses, and the secret sharing system comprises: selection means that selects a number, equal to or greater than 2 and smaller than N, of secret sharing apparatuses; fragment replacement means that generates a bijection π of {1, . . . , K}→{1, . . . , K}among the secret sharing apparatuses selected by said selection means and designates fragments aπ(k)i(1), . . . , and aπ(k)i(M) of a group of π(k)-th numeric values associated with each other recorded in an i-th selected secret sharing apparatus as fragments of a group of k-th numeric values associated with each other; and reshare means that performs reshare of the fragments aπ(k)i(1), . . . , and aπ(k)i(M) of numeric values Aπ(k)(1), . . . , and Aπ(k)(M) that are used as replacements by said fragment replacement means to determine new fragments bk1(1), . . . , bkN(1), . . . , bk1(M), . . . , and bkN(M) and designates the fragments bk1(1), . . . , bkN(1), . . . , bk1(M), . . . , and bkN(M) as fragments of numeric values Bk(1), . . . , and Bk(M).
2. The secret sharing system according to claim 1, wherein M=1.
3. The secret sharing system according to claim 2, further comprising: initial information distribution means that determines a fragment of each of K numeric values P1, . . . , and PK that are not known to any of N secret sharing apparatuses by a secure computation and records fragments p1n, . . . , and pKn in an n-th secret sharing apparatus; initial multiplication means that determines fragments sk1, . . . , and skN of a numeric value Sk that satisfies a relation that Sk=Pk×Ak(1) for the N secret sharing apparatuses by a secure computation and records the fragments sk1, . . . , and skN in the N secret sharing apparatuses in a distributed manner; checking distribution means that generates fragments qk1, . . . , and qkN of a numeric value Qk that satisfies a relation that Qk=Pπ(k) for k=1 to K by a secure computation and records the fragments qk1, . . . , and qkN in the N secret sharing apparatuses in a distributed manner; checking multiplication means that determines fragments tk1, . . . , and tkN of a numeric value Tk that satisfies a relation that Tk=Qk×Bk(1) for the N secret sharing apparatuses by a secure computation and records the fragments tk1, . . . , and tkN in the N secret sharing apparatuses in a distributed manner; and tamper detection means that checks whether a relation that Tk=Sπ(k) holds or not for k=1 to K.
4. The secret sharing system according to claim 1, wherein it is assumed that N=3, a combination (α, β, γ) is any of combinations (1, 2, 3), (2, 3, 1) and (3, 1, 2), three fragments of the numeric value Ak(m)=akαβ(m)+akβγ(m)+akγα(m) are (akγα(m), akαβ(m)), (akαβ(m), akβγ(m)) and (akβγ(m), akγα(m)), and said three fragments are recorded in three secret sharing apparatuses in a distributed manner, said selection means selects two secret sharing apparatuses, one of the selected secret sharing apparatus being designated as a first secret sharing apparatus, the other of the selected secret sharing apparatus being designated as a second secret sharing apparatus, and the secret sharing apparatus that is not selected being designated as a third secret sharing apparatus, said fragment replacement means designates a fragment of the numeric value Ak(m) recorded in the first secret sharing apparatus as a fragment ak1(m)=(ak31(m), ak12(m)), designates a fragment of the numeric value Ak(m) recorded in the second secret sharing apparatus as a fragment ak2(m)=(ak12(m), ak23(m)), designates a fragment of the numeric value Ak(m) recorded in the third secret sharing apparatus as a fragment ak3(m)=(ak23(m), ak31(m)), generates the bijection π of {1, . . . , K}→{1, . . . , K} in the first secret sharing apparatus or the second secret sharing apparatus, designates fragments aπ(k)1(1), . . . , and aπ(k)1(M) of the group of π(k)-th numeric values associated with each other recorded in the first secret sharing apparatus as fragments of the group of k-th numeric values associated with each other, and designates fragments aπ(k)2(1), . . . , and aπ(k)2(M) of a group of π(k)-th numeric values associated with each other recorded in the second secret sharing apparatus as fragments of the group of k-th numeric values associated with each other, each of said secret sharing apparatuses comprises, as said reshare means, a first random number generation part that generates random numbers bk31(1), . . . , and bk31(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk31(1), . . . , and bk31(M) to the third secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a second random number generation part that generates random numbers bk23(1), and bk23(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk23(1), . . . , and bk23(M) to the third secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a first calculation part that calculates a value xk(m) according to xk(m)=bk31(m)−aπ(k)31(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values xk(1), . . . , and xk(M) to the second secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a second calculation part that calculates a value yk(m) according to yk(m)=bk23(m)−aπ(k)23(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values yk(1), . . . , and yk(M) to the first secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a third calculation part that calculates a value bk12(m) according to bk12(m)=aπ(k)12(m)−xk(m)−yk(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other if the secret sharing apparatus is the first or second secret sharing apparatus, and a fragment update part that designates (bk3l(m), bk12(m)) as a fragment bk1(m) if the secret sharing apparatus is the first secret sharing apparatus, designates (bk12(m), bk23(m)) as a fragment bk2(m) if the secret sharing apparatus is the second secret sharing apparatus, and designates (bk23(m), bk31(m)) as a fragment bk3(m) if the secret sharing apparatus is the third secret sharing apparatus, and each of said secret sharing apparatuses records the fragments bk1(m), bk2(m), and bk3(m) as fragments of the numeric value Bk(m).
5. The secret sharing system according to claim 4, wherein M=1.
6. The secret sharing system according to claim 5, further comprising: initial information distribution means that generates K random values R(1)1, . . . , and R(1)K in the first secret sharing apparatus, generates K random values R(2)1, . . . , and R(2)K in the second secret sharing apparatus, records fragments (r(1)k31, r(1)k12), (r(1)k12, r(1)k23), and (r(1)k23, r(1)k31) of the value R(1)k and fragments (r(2)k31, r(2)k12), (r(2)k12, r(2)k23), and (r(2)k23, r(2)k31) of the value R(2)k in said three secret sharing apparatuses in a secret sharing manner, determines fragments (pk31, pk12), (pk12, pk23), and (pk23, pk31) of a numeric value Pk that satisfies a relation that Pk=R(1)k+R(2)k by a secure computation in said three secret sharing apparatuses, and records the fragments (pk31, pk12), (pk12, pk23), and (pk23, pk31) in said three secret sharing apparatuses in a distributed manner; initial multiplication means that determines fragments (sk31, sk12), (sk12, sk23), and (sk23, sk31) of a value Sk that satisfies a relation that Sk=Pk×Ak(1) by a secure computation in said three secret sharing apparatuses and records the fragments (sk31, sk12), (sk12, sk23), and (sk23, sk31) in said three secret sharing apparatuses in a distributed manner; checking distribution means records other fragments (r′(1)π(k)31, r′(1)π(k)12), (r′(1)π(k)12, r′(1)π(k)23), and (r′(1)π(k)23, r′(1)π(k)31) of said numeric value R(1)π(k) and other fragments (r′(2)π(k)31, r′(2)π(k)12), (r′(2)π(k)12, r′(2)π(k)23), and (r′(2)π(k)23, r′(2)π(k)31) of said numeric value R(2)π(k) in said three secret sharing apparatuses in a distributed manner, determines fragments (qk31, qk12), (qk12, qk23), and (qk23, qk31) of a numeric value Qk that satisfies a relation that Qk=R(1)π(k)+R(2)π(k) by a secure computation using the other fragments in said three secret sharing apparatuses, and records the fragments (qk31, qk12), (qk12, qk23), and (qk23, qk31) in said three secret sharing apparatuses in a distributed manner; and checking multiplication means that determines fragments (tk31, tk12), (tk12, tk23), and (tk23, tk31) of a numeric value Tk that satisfies a relation that Tk=Qk×Bk(1) by a secure computation in said three secret sharing apparatuses and records the fragments (tk31, tk12), (tk12, tk23), and (tk23, tk31) in said three secret sharing apparatuses in a distributed manner, wherein fragments recorded in the first secret sharing apparatus are sk1=(sk31, sk12) and tk1=(tk31, tk12), fragments recorded in the second secret sharing apparatus are sk2=(sk12, sk23) and tk2=(tk12, tk23), and fragments recorded in the third secret sharing apparatus are sk3=(sk23, sk31) and tk3=(tk23, tk31), each secret sharing apparatus comprises a third random number generation part that generates a random number uk and transmits the random number uk to the second secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a fourth random number generation part that generates a random number vk and transmits the random number vk to the first secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a fourth calculation part that calculates a value dk according to dk=sπ(k)12−tk12−uk−vk and transmits the value dk to the third secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a fifth calculation part that calculates a value ek according to ek=sπ(k)12−tk12−uk−vk and transmits the value ek to the third secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a first check part that checks whether a relation that dk=ek holds or not and terminates a processing if the relation does not hold if the secret sharing apparatus is the third secret sharing apparatus, a sixth calculation part that calculates a value fk according to fk=sπ(k)31−tk31+uk and transmits the value fk to the third secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a seventh calculation part that calculates a value gk according to gk=sπ(k)23−tk23+vk and transmits the value gk to the third secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, and a second check part that checks whether a relation that fk+gk+dk=0 holds or not and terminates a processing if the relation does not hold if the secret sharing apparatus is the third secret sharing apparatus.
7. A secret sharing apparatus in a secret sharing system comprising 3 secret sharing apparatuses, wherein it is assumed that M represents an integer equal to or greater than 1, m represents an integer equal to or greater than 1 and equal or smaller than M, K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M) are K×M numeric values whose fragments are recorded in the secret sharing apparatuses in a distributed manner, numeric values AK(1), . . . , and AK(M) are a group of k-th numeric values associated with each other, a fragment of a numeric value Ak(m) to be recorded is ak1(m)=(ak31(m), ak12(m)) if the secret sharing apparatus is selected as a first secret sharing apparatus, a fragment of the numeric value Ak(m) to be recorded is ak2(m)=(ak12(m), ak23(m)) if the secret sharing apparatus is selected as a second secret sharing apparatus, and a fragment of the numeric value Ak(m) to be recorded is ak3(m)=(ak23(m), ak31(m)) if the secret sharing apparatus is selected as a third secret sharing apparatus, the secret sharing apparatus comprises a fragment replacement part that generates a bijection π of {1, . . . , K}→{1, . . . , K} and designates a fragment of a group of π(k)-th numeric values associated with each other as a fragment of a group of k-th numeric values associated with each other if the secret sharing apparatus is selected as the first secret sharing apparatus or the second secret sharing apparatus, a first random number generation part that generates random numbers bk31(1), . . . , and bk31(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk31(1), . . . , and bk31(M) to the third secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a second random number generation part that generates random numbers bk23(1), . . . , and bk23(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk23(1), . . . , and bk23(M) to the third secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a first calculation part that calculates a value xk(m) according to xk(m)=bk31(m)−aπ(k)31(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values xk(1), . . . , and xk(M) to the second secret sharing apparatus if the secret sharing apparatus is the first secret sharing apparatus, a second calculation part that calculates a value yk(m) according to yk(m)=bk23(m)−aπ(k)23(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values yk(1), . . . , and yk(M) to the first secret sharing apparatus if the secret sharing apparatus is the second secret sharing apparatus, a third calculation part that calculates a value bk12(m) according to bk12(m)=aπ(k)12(m)−xk(m)−yk(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other if the secret sharing apparatus is the first or second secret sharing apparatus, and a fragment update part that designates (bk31(m), bk12(m)) as a fragment bk1(m) if the secret sharing apparatus is the first secret sharing apparatus, designates (bk12(m), bk23(m)) as a fragment bk2(m) if the secret sharing apparatus is the second secret sharing apparatus, and designates (bk23(m), bk31(m)) as a fragment bk3(m) if the secret sharing apparatus is the third secret sharing apparatus.
8. The secret sharing apparatus according to claim 7, wherein M=1.
9. A secret sharing method, wherein it is assumed that K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, M represents an integer equal to or greater than 1, m represents an integer equal to or greater than 1 and equal to or smaller than M, A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M) are K×M numeric values, a combination (α, β, γ) is any of combinations (1, 2, 3), (2, 3, 1) and (3, 1, 2), three fragments of the numeric value Ak(m)=akαβ(m)+akβγ(m)+akγα(m) are (akγα(m), akαβ(m)), (akαβ(m), akβγ(m)) and (akβγ(m), akγα(m)), a group of k-th numeric values associated with each other is formed by Ak(1), . . . , and Ak(M), and three secret sharing apparatuses in which said fragments are recorded in a distributed manner are used, and the secret sharing method comprises: a selection step of selecting two secret sharing apparatuses, one of the selected secret sharing apparatuses being designated as a first secret sharing apparatus, the other of the selected secret sharing apparatuses being designated as a second secret sharing apparatus, and the secret sharing apparatus that is not selected being designated as a third secret sharing apparatus; a fragment replacement step of designating a fragment of the numeric value Ak(m) recorded in the first secret sharing apparatus as a fragment ak1(m)=(ak31(m), ak12(m)), designating a fragment of the numeric value Ak(m) recorded in the second secret sharing apparatus as a fragment ak2(m)=(ak12(m), ak23(m)), designating a fragment of the numeric value Ak(m) recorded in the third secret sharing apparatus as a fragment ak3(m)=(ak23(m), ak31(m)), generating a bijection π of {1, . . . , K}→{1, . . . , K} in the first secret sharing apparatus or the second secret sharing apparatus, designating fragments aπ(k)1(1), . . . , and aπ(k)1(M) of a group of π(k)-th numeric values associated with each other recorded in the first secret sharing apparatus as fragments of the group of k-th numeric values associated with each other, and designating fragments aπ(k)2(1), . . . , and aπ(k)2(M) of a group of π(k)-th numeric values associated with each other recorded in the second secret sharing apparatus as fragments of the group of k-th numeric values associated with each other; a first random number generation step in which the first secret sharing apparatus generates random numbers bk31(1), . . . , and bk31(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk31(1), . . . , and bk31(M) to the third secret sharing apparatus; a second random number generation step in which the second secret sharing apparatus generates random numbers bk23(1), . . . , and bk23(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the random numbers bk23(1), . . . , and bk23(M) to the third secret sharing apparatus; a first calculation step in which the first secret sharing apparatus calculates a value xk(m) according to xk(m)=bk31(m)−aπ(k)31(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values xk(1), . . . , and xk(M) to the second secret sharing apparatus; a second calculation step in which the second secret sharing apparatus calculates a value yk(m) according to yk(m)=bk23(m)−aπ(k)23(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values yk(1), . . . , and yk(M) to the first secret sharing apparatus; a third calculation step in which the first secret sharing apparatus and the second secret sharing apparatus calculate a value bk12(m) according to bk12(m)=aπ(k)12(m)−xk(m)−yk(m) for m=1 to M for reshare of the fragments of the group of k-th numeric values associated with each other; and a fragment update step in which the first secret sharing apparatus designates (bk31(m), bk12(m)) as a fragment bk1(m), the second secret sharing apparatus designates (bk12(m), bk23(m)) as a fragment bk2(m), and the third secret sharing apparatus designates (bk23(m), bk31(m)) as a fragment bk3(m).
10. The secret sharing method according to claim 9, wherein M=1.
11. The secret sharing method according to claim 10, further comprising: an initial information distribution step in which the first secret sharing apparatus generates K random values R(1)1, . . . , and R(1)K, the second secret sharing apparatus generates K random values R(2)1, . . . , and R(2)K, said three secret sharing apparatuses record fragments (r(1)k31, r(1)k12), (r(1)k12, r(1)k23), and (r(1)k23, r(1)k31) of the value R(1)k and fragments (r(2)k31, r(2)k12), (r(2)k12, r(2)k23), and (r(2)k23, r(2)k31) of the value R(2)k in a secret sharing manner, and said three secret sharing apparatuses determine fragments (pk31, pk12), (pk12, pk23), and (pk23, pk31) of a numeric value Pk that satisfies a relation that Pk=R(1)k+R(2)k by a secure computation and record the fragments in a distributed manner; an initial multiplication step in which said three secret sharing apparatuses determine fragments (sk31, sk12), (sk12, sk23), and (sk23, sk31) of a value Sk that satisfies a relation that Sk=Pk×Ak(1) by a secure computation and record the fragments (sk31, sk12), (sk12, sk23), and (sk23, sk31) in a distributed manner; a secret sharing update step in which said three secret sharing apparatuses record fragments bk1, bk2, and bk3 obtained by the secret sharing method according to claim as fragments of a numeric value Bk(1); a checking distribution step in which other fragments (r′(1)π(k)31, r′(1)π(k)12), (r′(1)π(k)12, r′(2)π(k)23), and (r′(1)π(k)23, r′(1)π(k)31) of said numeric value R(1)π(k) and other fragments (r′(2)π(k)31, r′(2)π(k)12), (r′(2)π(k)12, r′(2)π(k)23), and (r′(2)π(k)23, r′(2)π(k)31) of said numeric value R(2)π(k) are recorded in said three secret sharing apparatuses in a distributed manner, said three secret sharing apparatuses determine fragments (qk31, qk12), (qk12, qk23), and (qk23, qk31) of a numeric value Qk that satisfies a relation that Qk=R(1)π(k)+R(2)π(k) by a secure computation using the other fragments and record the fragments (qk31, qk12), (qk12, qk23), and (qk23, qk31) in a distributed manner; and a checking multiplication step in which said three secret sharing apparatuses determine fragments (tk31, tk12), (tk12, tk23), and (tk23, tk31) of a numeric value Tk that satisfies a relation that Tk=Qk×Bk(1) by a secure computation and record the fragments (tk31, tk12), (tk12, tk23), and (tk23, tk31) in a distributed manner, wherein fragments recorded in the first secret sharing apparatus are sk1=(sk31, sk12) and tk1=(tk31, tk12), fragments recorded in the second secret sharing apparatus are sk2=(sk12, sk23) and tk2=(tk12, tk23), and fragments recorded in the third secret sharing apparatus are sk3=(sk23, sk31) and tk3=(tk23, tk31), and the secret sharing method further comprises: a third random number generation step in which the first secret sharing apparatus generates a random number uk and transmits the random number uk to the second secret sharing apparatus; a fourth random number generation step in which the second secret sharing apparatus generates a random number vk and transmits the random number vk to the first secret sharing apparatus; a fourth calculation step in which the first secret sharing apparatus calculates a value dk according to dk=sπ(k)12−tk12−uk−vk and transmits the value dk to the third secret sharing apparatus; a fifth calculation step in which the second secret sharing apparatus calculates a value ek according to ek=sπ(k)12−tk12−uk−vk and transmits the value ek to the third secret sharing apparatus; a first check step in which the third secret sharing apparatus checks whether a relation that dk=ek holds or not and terminates a processing if the relation does not hold; a sixth calculation step in which the first secret sharing apparatus calculates a value fk according to fk=sπ(k)31−tk31+uk and transmits the value fk to the third secret sharing apparatus; a seventh calculation step in which the second secret sharing apparatus calculates a value gk according to gk=sπ(k)23−tk23+vk and transmits the value gk to the third secret sharing apparatus; and a second check step in which the third secret sharing apparatus checks whether a relation that fk+gk+dk=0 holds or not and terminates a processing if the relation does not hold.
12. The secret sharing method according to claim 9, wherein the fragments bk1(m), bk2(m), and bk3(m) obtained in said fragment update step are regarded as new fragments ak1(m), ak2(m), and ak3(m), and said secret sharing method is repeated until said secret sharing apparatuses are selected in all predetermined combinations in said fragment replacement step.
13. A secret sorting method that uses a secret sharing method according to claim 12, comprising: a comparison step in which three secret sharing apparatuses selects two numeric values from a plurality of numeric values whose fragments are recorded in a distributed manner by the secret sharing method according to claim 12 and compare the two numeric values in terms of magnitude by a secure computation; and an exchange step in which each secret sharing apparatus exchanges fragments of the numeric values based on a result of said comparison step.
14. A non-transitory computer readable medium including computer executable instructions that make a computer function as a secret sharing apparatus in a secret sharing system according to claim 1.
The present invention generally relates to an encryption application. In particular, it relates to a secret sharing system, a secret sharing apparatus, a secret sharing method, a secret sorting method and a secret sharing program for performing a functional calculation without disclosing input data.
As a method of yielding a particular calculation result without reconstructing an encrypted numeric value, there is a method referred to as secure computation (such as the method described in Non-patent literature 1). According to the method described in Non-patent literature 1, fragments of a numerical value are distributed among three secure computation apparatuses, which can hold the results of calculations such as addition, subtraction, constant sum, multiplication, constant multiplication, logical operations (NOT, AND, OR and exclusive-OR) and data format conversion (integer-to-binary) without reconstructing the original numeric value.
Non-patent literature 1: Koji Chida, Dai Ikarashi, Katsumi Takahashi, “Efficient 3-Party Secure Function Evaluation and Its Application”, 48-th IPSJ SIG Technical Report, CSEC, pp. 1-7, Mar. 4, 2010.
However, the conventional technique has a problem that a plurality of pieces of data cannot be randomly replaced while concealing the association of the pieces of data. An object of the present invention is to provide a secure computation technique of outputting data that cannot be associated with a plurality of pieces of input data.
The present invention relates to a secret sharing. In general, in a (k, n) secret sharing, a secret sharing system has two parameters k and n and divides a value to be concealed into n fragments in such a manner that gathering less than k of the n fragments does not lead to leakage of information concerning the original value but gathering k or more of the n fragments permits reconstruction of the original value. A secret sharing system according to the present invention comprises N secret sharing apparatuses R1, . . . , and RN. It is assumed that N represents an integer equal to or greater than 3, n represents an integer equal to or greater than 1 and equal to or smaller than N, M represents an integer equal to or greater than 1, m represents an integer equal to or greater than 1 and equal to or smaller than M, K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M) are K×M numeric values whose fragments are recorded in the secret sharing apparatuses in a distributed manner, numeric values AK(1), . . . , and AK(M) are a group of k-th numeric values associated with each other, and akn(m) is a fragment of the numeric value Ak(m) recorded in an n-th secret sharing apparatus. The secret sharing system according to the present invention comprises selection means, fragment replacement means and reshare means. The selection means selects a number, equal to or greater than 2 and smaller than N, of secret sharing apparatuses. The fragment replacement means generates a bijection π of {1, . . . , K}→{1, . . . , K} among the selected secret sharing apparatuses and designates fragments aπ(k)i(1), . . . , and aπ(k)i(M) of a group of π(k)-th numeric values associated with each other recorded in a selected secret sharing apparatus Ri (i represents an identification number that indicates a selected secret sharing apparatus) as fragments of a group of k-th numeric values associated with each other. The reshare means performs reshare of the fragments aπ(k)i(1), . . . , and aπ(k)i(M) of numeric values Aπ(k)(1), . . . , and Aπ(k)(M) that are used as replacements by said fragment replacement means to determine new fragments bk1(1), . . . , bkN(1), . . . , bk1(M), . . . , and bkN(M) (this process will be referred to as a reshare, hereinafter). In the case where reshare of a group of numeric values is performed while maintaining the association of the numeric values associated with each other, the same bijection π can be used to replace the fragments of the numeric values of the group of numeric values associated with each other.
The secret sharing system according to the present invention can further comprise initial information distribution means, initial multiplication means, checking distribution means, checking multiplication means and tamper detection means. The initial information distribution means determines fragments p1n, . . . , and pKn of each of K numeric values P1, . . . , and PK that are not known to any of the secret sharing apparatuses R1, . . . , and RN by a secure computation and records the fragments p1n, . . . , and pKn in a secret sharing apparatus Rn. The initial multiplication means determines fragments sk1, . . . , and skN of a numeric value Sk that satisfies a relation that Sk=Pk×Ak(1) for the secret sharing apparatuses R1, . . . , and RN by a secure computation and records the fragments sk1, . . . , and skN in the secret sharing apparatuses R1, . . . , and RN in a distributed manner. The checking distribution means generates fragments qk1, . . . , and qkN of a numeric value Qk that satisfies a relation that Qk=Pπ(k) for k=1 to K by a secure computation and records the fragments qk1, . . . , and qkN in the secret sharing apparatuses R1, . . . , and RN in a distributed manner. The checking multiplication means determines fragments tk1, . . . , and tkN of a numeric value Tk that satisfies a relation that Tk=Qk×Bk(1) for the secret sharing apparatuses R1, . . . , and RN by a secure computation and records the fragments tk1, . . . , and tkN in the secret sharing apparatuses R1, . . . , and RN in a distributed manner. The tamper detection means checks whether a relation that Tk=Sπ(k) holds or not for k=1 to K.
For example, in a case where the secret sharing system comprises three secret sharing apparatuses, it is assumed that three fragments of an m-th numeric value Ak(m)=akαβ(m)+akβγ(m)+aγα(m) of a group of k-th numeric values associated with each other are (akγα(m), akαβ(m)), (akαβ(m), akβγ(m)) and (akβγ(m), akγα(m)) (a combination (α, β, γ) is any of combinations (1, 2, 3), (2, 3, 1) and (3, 1, 2)). It is further assumed that a fragment recorded in a secret sharing apparatus selected as a first secret sharing apparatus is ak1(m)=(ak31(m), ak12(m)), a fragment recorded in a secret sharing apparatus selected as a second secret sharing apparatus is ak2(m)=(ak12(m), ak23(m)), and a fragment recorded in a secret sharing apparatus selected as a third secret sharing apparatus is ak3(m)=(ak23(m), ak31(m)). Each secret sharing apparatus can comprise a fragment replacement part, a first random number generation part, a second random number generation part, a first calculation part, a second calculation part, a third calculation part and a fragment update part. If the secret sharing apparatus is selected as the first secret sharing apparatus or the second secret sharing apparatus, the fragment replacement part generates a bijection π of {1, . . . , K}→{1, . . . , K} and designates fragments of the numeric values of a group of π(k)-th numeric values associated with each other as fragments of the numeric values of the group of k-th numeric values associated with each other. If the secret sharing apparatus is the first secret sharing apparatus, the first random number generation part generates random numbers bk31(1), . . . , and bk31(M) for reshare of the fragments of the numeric values of the group of k-th numeric values associated with each other resulting from the designation and transmits the random numbers bk31(1), . . . , and bk31(M) to the third secret sharing apparatus. If the secret sharing apparatus is the second secret sharing apparatus, the second random number generation part generates random numbers bk23(1), . . . , and bk23(M) for reshare of the fragments of the numeric values of the group of k-th numeric values associated with each other and transmits the random numbers bk23(1), . . . , and bk23(M) to the third secret sharing apparatus. If the secret sharing apparatus is the first secret sharing apparatus, the first calculation part calculates a value xk(m) according to xk(m)=bk31(m)−aπ(k)31(m) for m=1 to M for reshare of the fragments of the numeric values xk(1), . . . , and xk(M) of the group of k-th numeric values associated with each other and transmits the value to the second secret sharing apparatus. If the secret sharing apparatus is the second secret sharing apparatus, the second calculation part calculates a value yk(m) according to yk(m)=bk23(m)−aπ(k)23(m) for m=1 to M for reshare of the fragments of the numeric values of the group of k-th numeric values associated with each other and transmits the value yk(1), . . . , and yk(M) to the first secret sharing apparatus. If the secret sharing apparatus is the first or second secret sharing apparatus, the third calculation part calculates a value bk12(m) according to bk12(m)=aπ(k)12(m)−xk(m)−yk(m) for m=1 to M for reshare of the fragments of the numeric values of the group of k-th numeric values associated with each other. The fragment update part designates (bk31(m), bk12(m)) as a fragment bk1(m) if the secret sharing apparatus is the first secret sharing apparatus, designates (bk12(m), bk23(m)) as a fragment bk2(m) if the secret sharing apparatus is the second secret sharing apparatus, and designates (bk23(m), bk31(m)) as a fragment bk3(m) if the secret sharing apparatus is the third secret sharing apparatus. The fragment replacement parts of all the secret sharing apparatuses form the fragment replacement means of the secret sharing system. The first random number generation part, the second random number generation part, the first calculation part, the second calculation part, the third calculation part and the fragment update part form the reshare means of the secret sharing system.
With the secret sharing system according to the present invention, any secret sharing apparatus that is not selected by the fragment replacement part does not know the bijection π and therefore cannot know the association between the numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M) and the numeric values B1(1), . . . , BK(1), . . . , B1(M), . . . , and BK(M). According to the present invention, a sorting algorithm based on comparison, such as quick sort, can be achieved by a secure computation without increasing the number of comparisons.
FIG. 1 is a diagram showing an example of a functional configuration of secret sharing systems according to first and second embodiments;
FIG. 2 is a diagram showing a flow of a secret sharing process performed by the secret sharing system according to the first embodiment;
FIG. 3 is a diagram showing a flow of a process of sorting numeric values in the secret sharing systems according to the present invention;
FIG. 4 is a diagram showing a quick sort algorithm;
FIG. 5 is a diagram showing a flow of a secret sharing process performed by the secret sharing system according to the second embodiment;
FIG. 6 is a diagram showing an example of a functional configuration of secret sharing systems according to third and fourth embodiments;
FIG. 7 is a diagram showing an example of a specific configuration of reshare parts according to the third and fourth embodiments;
FIG. 8 is a diagram showing a flow of a secret sharing process performed by the secret sharing system according to the third embodiment;
FIG. 9 is a diagram showing a specific structure of a tamper detection part according to the fourth embodiment; and
FIG. 10 is a diagram showing a flow of a secret sharing process performed by the secret sharing system according to the fourth embodiment.
In the following, embodiments of the present invention will be described in detail. Components having the same functions will be denoted by the same reference numerals, and redundancy of the description thereof will be avoided.
In the description of MEANS TO SOLVE THE PROBLEMS, it has been assumed that numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , AK(M) are K×M numeric values whose fragments are to be distributed among and recorded in secret sharing apparatuses, the numeric values Ak(1), . . . , and Ak(M) are associated to form a k-th numeric value group, and a fragment of a numeric value Ak(m) to be recorded in an n-th secret sharing apparatus is denoted by akn(m). In the description of DETAILED DESCRIPTION OF THE EMBODIMENTS, to facilitate understanding of the present invention, a case where M=1 will be first described, and then a case where M is not limited to 1 will be described. In the description of the case where M=1, Ak(1) is expressed as Ak, and akn(1) is expressed as akn.
FIG. 1 shows an example of a functional configuration of a secret sharing system according to the first embodiment. FIG. 2 shows a flow of a secret sharing process performed by the secret sharing system according to the first embodiment. The secret sharing system according to this embodiment comprises N secret sharing apparatuses 1001, . . . , and 100N and selection means 105 connected to a network 1000 (N represents an integer equal to or greater than 3, and n represents an integer equal to or greater than 1 and equal to or smaller than N). A1, . . . , and AK are K numeric values whose fragments are to be distributed among and recorded in the secret sharing apparatuses 100n (K represents an integer equal to or greater than 2), a numeric value Ak is a k-th numeric value (k represents an integer equal to or greater than 1 and equal to or smaller than K), and akn represents a k-th fragment to be recorded in the secret sharing apparatus 100n. The numeric values A1, . . . , and AK are a group of numeric values to be concealed, for example, a group of numeric values to be sorted. The group of numeric values to be sorted can be a group of numeric values Ak each representing the annual income of a particular person. The selection means 105 may be provided in any of the secret sharing apparatuses or provided as a separate apparatus.
The secret sharing system according to this embodiment comprises the selection means, fragment replacement means and reshare means. The secret sharing apparatus 100n comprises at least a fragment replacement part 110n, a reshare part 120n and a recording part 190n. The recording part 190n records fragments a1n, . . . , and aKn, for example. The recording part 190n also records information concerning what number fragment of the numeric value Ak is the fragment akn recorded in itself.
The selection means 105 selects a number, smaller than N, of secret sharing apparatuses (S105). For example, if the secret sharing requires N′ of the N fragments to reconstruct the numeric value, it is enough that the fragment replacement means selects a number, equal to or greater than N′ and smaller than N, secret sharing apparatuses.
The fragment replacement means comprises at least fragment replacement parts 1101, . . . , and 110N. A bijection π of {1, . . . , K}→{1, . . . , K} is generated among fragment replacement parts 110i of secret sharing apparatuses 100i (i represents the identification number of the selected secret sharing apparatus) selected by the selection means 105, and a fragment aπ(k)i recorded in a recording part 190i of the selected secret sharing apparatus 100i is designated as the k-th fragment (S110). The bijection π may be a mapping of numbers 1 to K randomly rearranged. The bijection π is desirably a uniformly randomly rearranged mapping and can be generated by Fisher-Yates shuffle (Reference Literature 1: Richard Durstenfeld, “Algorithm 235: Random permutation”, Communications of the ACM archive, Volume 7, Issue 7, 1964), for example. The bijection π may be generated among the selected secret sharing apparatuses 100i or may be generated by one of the selected secret sharing apparatuses 100i and shared among the selected secret sharing apparatuses 100i.
The reshare means comprises at least reshare parts 1201, . . . , 120N. The reshare means performs reshare of the fragments aπ(k)i (a fragment aπ(k)i is the k-th replaced fragment) of the numeric value Aπ(k) that are replacements used by the fragment replacement means to determine new fragments bk1, . . . , and bkN, and assumes the fragments bk1, . . . , bkN as fragments of a numeric value Bk (S120). That is, a relation that Aπ(k)=Bk holds. However, the secret sharing apparatuses that are not selected do not know the bijection π and therefore the relation that Aπ(k)=B(k). The recording part 190n of each secret sharing apparatus 100n records not only the fragment bkn but also information that the k-th fragment bkn recorded in itself is a fragment of the numeric value Bk. Furthermore, if the numeric values B1, . . . , and BK are regarded as new numeric values A1, . . . , and AK, and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111 and S112).
The secret sharing system according to the present invention shuffles the fragments among a limited number of secret sharing apparatuses. Therefore, the secret sharing apparatus that are not selected by the fragment replacement part do not know the bijection π and therefore the association between the numeric values A1, . . . , and Ak and the numeric values B1, . . . , Bk. In other words, if one wants to conceal the association between the numeric values A1, . . . , and AK and the numeric values B1, . . . , and BK from a particular secret sharing apparatus, the secret sharing apparatuses to be selected can determined so that the fragment replacement part does not select that secret sharing apparatus. Furthermore, if the above process is repeated by modifying the combination of the secret sharing apparatuses selected by the fragment replacement part so that every secret sharing apparatus has experience of not being selected at least once, the numeric values B1, . . . , and BK that cannot be associated with the numeric values A1, . . . , and AK by all the secret sharing apparatuses can be obtained.
In the above description of limited shuffling, reshare has not been described in detail. In this section, a reshare method will be described. The reshare method involves the update method disclosed in Section 3.3 of Reference Literature 2 (Amir Herzberg, Stanislaw Jarecki, Hugo Krawczyk, and Moti Yung, “Proactive secret sharing or: How to cope with perpetual leakage”, In Don Coppersmith, editor, CRYPTO 1995, volume 963 of LNCS, pages 339-352. Springer, 1995) and the regeneration method disclosed in Section 6.1 of Reference Literature 3 (Haiyun Luo and Songwu Lu, “Ubiquitous and robust authentication services for ad hoc wireless networks”, In UCLA-CSD-TR-200030, 2000). New fragments are generated among the secure computation apparatuses selected by the selection means 105 according to the update method disclosed in Reference Literature 2, and then, new fragments for the secure computation apparatuses that are not selected by the selection means 105 are generated according to the regeneration method disclosed in Reference Literature 3.
An algorithm, which is an application of the update method disclosed in Reference Literature 2 to the present invention, will be described below. It is assumed that the selection means 105 has selected N′ secret sharing apparatuses. It is further assumed that i and j represent numbers that identify the selected secret sharing apparatuses (any of the N′ numbers selected from among 1 to N) and are not equal to each other (j≠i). It is further assumed that values z1, . . . , and zN are predetermined values and shared among all the secret sharing apparatuses.
(1) All the secret sharing apparatuses 100i generate N′−1 random numbers ui,1, ui,2, . . . , and ui,N′-1.
(2) All the secret sharing apparatuses 100i determine Zi(z): 0=+ui,1z+ui,2z2+ . . . +ui,N′-1zN′-1.
(3) All the secret sharing apparatuses 100i transmit the value of Zi(zj) to all the other selected secret sharing apparatuses 100j (there are N′−1 secret sharing apparatuses 100j that are not selected).
(4) All the secret sharing apparatuses 100i denote the sum of all the values of Zj(zi) received from the other selected secret sharing apparatuses 100j (there are N′−1 secret sharing apparatuses 100j that are not selected) as Z(zi), and determine new fragments bki using the replacement fragments aπ(k)i according to the following formula:
bki=aπ(k)i+Z(zi)
Next, an algorithm, which is an application of the regeneration method disclosed in Reference Literature 3 to the present invention, will be described. It is assumed that the “h” represents a number that identifies a secret sharing apparatus that is not selected (any of the N′-N numbers that are not selected from among 1 to N). It is further assumed that a relation that Lij(z)=(z−zj)/(zi−zj) holds, and Li(z) is the product of the values of Lij(z) for all the numbers j.
(5) All the secret sharing apparatuses 100i generate random numbers vi,j(h) for all the combinations of the numbers j greater than i (i<j) and the secret sharing apparatuses 100h that are not selected.
(6) All the secret sharing apparatuses 100i transmit the random numbers vi,j(h) to the secret sharing apparatuses 100j.
(7) For all the secret sharing apparatuses 100h that are not selected, all the secret sharing apparatuses 100i denote the sum of all the random numbers vi,j(h) for i and j that satisfy a relation that j<i as V(h+) and the sum of all the random numbers vi,j(h) for i and j that satisfy a relation that i<j as V(h−), determine a value whi according to the following formula:
whi=bkiLi(zh)+V(h+)−V(h−)
and transmit the value whi to the secret sharing apparatuses 100h.
(8) All the secret sharing apparatuses 100h regard the sum of all the received values whi as their respective new fragments bkh.
As described above, through the steps (1) to (4), all the selected secret sharing apparatuses record new fragments. Through the steps (5) to (8), all the secret sharing apparatus that are not selected record new fragments.
If the steps (3) and (6) are performed at the same time, the process can be sped up. More specifically, the steps (1), (2) and (5) can be first performed, then the steps (3) and (6) can be performed at the same time, and then the steps (4), (7) and (8) can be performed.
FIG. 3 shows a flow of a process of sorting numeric values in the secret sharing system according to the first embodiment. Through the process described above, new numeric values A1, . . . , and AK that cannot be associated with the initial numeric values A1, . . . , and AK have been obtained (S101). In the case where sorting is also to be performed, the secret sharing apparatus 100n further comprises a comparison part 210n and an exchange part 220n. Comparison parts 2101, . . . , and 210N select two numeric values and compare the two numeric values in terms of magnitude by a secure computation (S210).
Based on the result of comparison by the comparison parts 2101, . . . , and 210N, exchange parts 2201, . . . and 220N exchange fragments of zero sets, one set or a plurality of sets of numeric values (S220). Then, until the sorting process is completed for all the numeric values, Steps S210 and S220 (required processings such as comparison, exchange and combination modification) are repeated (S211, S212).
The result of the comparison in Step S210 is information required for all the secret sharing apparatuses to perform the subsequent processing, and therefore all the secret sharing apparatuses know the information. However, since all the secret sharing apparatuses process the new numeric values A1, . . . , and AK that cannot be associated with the initial numeric values A1, . . . , and AK due to the processing of Step S101, the information concerning the initial numeric values A1, . . . , and AK does not leak. The comparison result is also information that is available by calculation from the output of sorting process, which is public information. Therefore, in the whole protocol according to this embodiment, disclosing the comparison result does not mean leaking more information than required.
More specifically, the quick sort algorithm shown in FIG. 4 can be applied to the operations involved in sorting (Steps S210, S220, S211 and S212). In this case also, the processing of comparing A[i] and A[j] is performed by concealing the values of A[i] and A[j], and the comparison result is public. In the case of this method, the number of comparisons is the same as in the case of the original quick sort and is O(N·logN) on average. Besides, this embodiment can be applied to a sorting algorithm comprising a processing of comparing numeric values in terms of magnitude and a processing of exchanging two elements of an array.
As described above, with the secret sharing system according to this embodiment, the sorting algorithm comprising comparison and element exchange can be achieved by secure computation without increasing the number of comparisons.
[Modification of Limited Shuffling]
Next, the case where M is not limited to 1 will be described. It is assumed that M represents an integer equal to or greater than 1, and m represents an integer equal to or greater than 1 and equal to or smaller than M. It is further assumed that A(1), . . . , and A(M) are vectors each having K elements, and A(m)=(A1(m), . . . , AK(m)). It is further assumed that the elements of the vectors A(1), . . . , and A(M) are associated with each other. In other words, it is assumed that Ak(1), . . . , and Ak(M) are a group of k-th numeric values associated with each other. In this modification, limited shuffling of the group of numeric values is performed while maintaining the association of the group of numeric values associated with each other. It is further assumed that akn(m) represents a fragment of a numeric value Ak(m) recorded in the secret sharing apparatus 100n. Note that the limited shuffling described above is the limited shuffling in the case where M=1, and the following description concerns more general limited shuffling.
The secret sharing system is configured as shown in FIG. 1, and the flow of the secret sharing process is as shown in FIG. 2. The secret sharing apparatus 100n comprises at least the fragment replacement part 110n, the reshare part 120n and the recording part 190n. However, the components are configured as described below and perform the processings described below.
The recording part 190, records fragments a1n(1), . . . , aKn(1), . . . , a1n(M), . . . , and aKn(M), for example. The recording part 190n also records information concerning what number fragment of the numeric value Ak is the fragment akn recorded in itself.
The selection means 105 selects a number, smaller than N, of secret sharing apparatuses (S105). For example, if the secret sharing requires N′ of the N fragments to reconstruct the numeric value, it is enough that the fragment replacement means selects a number, equal to or greater than N′ and smaller than N, secret sharing apparatuses. This processing is the same as that described above.
The fragment replacement means comprises at least fragment replacement parts 1101, . . . , and 110N. A bijection π of {1, . . . , K}→{1, . . . , K} is generated among fragment replacement parts 110i of secret sharing apparatuses 100i (i represents the identification number of the selected secret sharing apparatus) selected by the selection means 105, and fragments aπ(k)i(1) . . . , and aπ(k)i(M) recorded in the recording part 190i of the selected secret sharing apparatus 100i is designated as fragments of the group of k-th numeric values associated with each other (S110).
The reshare means comprises at least reshare parts 1201, . . . , 120N. The reshare means performs reshare of the fragments aπ(k)i(1), . . . , and aπ(k)i(M) (fragments aπ(k)i(1), . . . , and aπ(k)i(M) are the k-th replaced fragments) of the group of numeric values Aπ(k)(1), . . . , and Aπ(k)(M) that are replacements used by the fragment replacement means to determine new fragments bk1(1), . . . , bkN(1), . . . , bk1(M), . . . , and bkN(M), and assumes the fragments bk(1), . . . , bkN(1), . . . , bk(M), . . . , and bkN(M) as fragments of numeric values Bk(1), . . . , and Bk(M) (S120). That is, a relation that Aπ(k)(m)=Bk(m) holds. However, the secret sharing apparatuses that are not selected do not know the bijection π and therefore the relation that Aπ(k)(m)=Bk(m). The recording part 190n of each secret sharing apparatus 100n records not only the fragment bkn(m) but also information that the k-th fragment bkn(m) recorded in itself is a fragment of the numeric value Bk(m). Furthermore, if the numeric values B1(1), . . . , BK(1), . . . , B1(M), . . . , and BK(M) are regarded as new numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M), and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111 and S112).
As described above, if the limited shuffling in which the association of the elements of the vectors is maintained is used, random replacement in the column direction can be performed by regarding each row as one element (a group of numeric values associated with each other) in secret sharing of data in the form of a table, for example.
A secret sharing system according to a second embodiment is configured as shown in FIG. 1. A secret sharing apparatus 100n according to this embodiment further comprises the components shown by dotted lines. FIG. 5 shows a flow of a secret sharing process performed by the secret sharing system according to the second embodiment. The secret sharing system according to this embodiment comprises N secret sharing apparatuses 1001, . . . , and 100N and selection means 105 connected to a network 1000 (N represents an integer equal to or greater than 3, and n represents an integer equal to or greater than 1 and equal to or smaller than N). A1, . . . , and AK are K numeric values whose fragments are to be distributed among and recorded in the secret sharing apparatuses 100n (K represents an integer equal to or greater than 2), a numeric value Ak is a k-th numeric value (k represents an integer equal to or greater than 1 and equal to or smaller than K), and akn represents a k-th fragment to be recorded in the secret sharing apparatus 100n.
The secret sharing system according to this embodiment comprises the selection means 105, initial information distribution means, initial multiplication means, fragment replacement means, reshare means, checking distribution means, checking multiplication means, and tamper detection means. The secret sharing apparatus 100n comprises an initial information distribution part 130n, an initial multiplication part 140n, a fragment replacement part 110n, a reshare part 120n, a checking distribution part 150n, a checking multiplication part 160n, a tamper detection part 170n, and a recording part 190n. The recording part 190n records fragments a1n, . . . , aKn, for example. The recording part 190n also records information concerning what number fragment of the numeric value Ak is the fragment akn recorded in itself.
The selection means 105 is the same as that according to the first embodiment. The initial information distribution means comprises initial information distribution parts 1301, . . . , and 130N. The initial information distribution part 1301 of the secure computation apparatus 100i selected by the selection means 105 determines fragments p11, . . . , pK1, . . . , p1n, . . . , pKn, . . . , p1N, . . . , and pKN of K numeric values P1, . . . , and PK that are not known to all the secret sharing apparatuses 1001, . . . , and 100N, and the fragments p1n, . . . , and pKn are recorded in the secret sharing apparatus 100n (S130). Specifically, two or more secret sharing apparatuses are chosen from among the secret sharing apparatuses selected by the selection means 105. Then, based on the values generated by the chosen secret sharing apparatuses, fragments of the values that are not known to any apparatuses can be generated. For example, two secret sharing apparatuses 100i and 100j are chosen (i≠j), and fragments of the numeric value generated by the secret sharing apparatus 100i and fragments of the numeric value generated by the secret sharing apparatus 100j are recorded in a distributed manner. Then, the sum of the two numeric values is determined by a secure computation, and the fragments are recorded in a distributed manner so that the result of the secure computation is concealed. Then, the fragments of the numeric values that are not known to all the secret sharing apparatuses can be recorded in a distributed manner. Although two secure computation apparatuses are chosen in this example, more than two secure computation apparatuses can also be chosen.
The initial multiplication means comprises initial multiplication parts 1401, . . . , and 140N. The initial multiplication parts 140i, . . . , and 140N determine fragments sk1, . . . , and skN of a numeric value Sk that satisfies a relation that Sk=Pk×Ak, and the fragments are distributed among and recorded in the secret sharing apparatuses 1001, . . . , and 100N.
The fragment replacement means and the reshare means are the same as those according to the first embodiment. The checking distribution means comprises checking distribution parts 1501, . . . , and 150N. The checking distribution parts 1501, . . . , and 150N generate fragments qk1, . . . , and qkN of a numeric value Qk that satisfies a relation that Qk=Pα(k) for k=1 to K by a secure computation, and the fragments are distributed among and recorded in the secret sharing apparatuses 1001, . . . , and 100N (S150). Specifically, based on the values generated by the chosen secret sharing apparatuses in Step S130, other fragments of the values that are not known to any apparatuses can be generated. For example, other fragments (new fragments) of the numeric value generated for the numeric value Pπ(k) by the chosen secret sharing apparatus 100i in Step S130 and other fragments (new fragments) of the numeric value generated for the numeric value Pπ(k) by the chosen secret sharing apparatus 100j in Step S130 are recorded in a distributed manner. Then, the sum of the two numeric values is determined by a secure computation, and the fragments are recorded in a distributed manner so that the result of the secure computation is concealed. Then, fragments of the numeric value Qk that satisfies a relation that Qk=Pπ(k) and is not known to all the secret sharing apparatuses can be recorded in a distributed manner. Although two secure computation apparatuses are chosen in this example, more than two secure computation apparatuses can also be chosen as in Step S130.
The checking multiplication means comprises checking multiplication parts 1601, . . . , and 160N. The checking multiplication parts 1601, . . . , and 160N determine fragments tk1, . . . , and tkN of a numeric value Tk that satisfies a relation that Tk=Qk×Bk by a secure computation, and the fragments tk1, . . . , and tkN are distributed among and recorded in the secret sharing apparatuses 1001, . . . , and 100N (S160).
The tamper detection means comprises tamper detection parts 1701, . . . , and 170N. The tamper detection parts 1701, . . . , and 170N checks whether a relation that Tk=Sπ(k) holds for k=1 to K (S170). If tkn≠sπ(k)n, it is determined that there a tamper has occurred, and abnormal termination occurs. If the numeric values B1, . . . , and BK are regarded as new numeric values A1, . . . , and AK, and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111, S112).
The secret sharing system according to the second embodiment has the same effects as the secret sharing apparatus according to the first embodiment and can check whether an illegal operation to transmit a tampered value to other secret sharing apparatuses has occurred in the course of a process of concealing the association between the numeric values A1, . . . , and AK and the numeric values B1, . . . , and BK. In the case where sorting is also to be performed, the secret sharing apparatus 100n further comprises a comparison part 210n and an exchange part 220n. The specific sorting process is the same as that according to the first embodiment.
In the first and second embodiments, it has been assumed that the number of the secret sharing apparatuses is N (N represents an integer equal to or greater than 3). However, in a third embodiment, the number of the secret sharing apparatuses that form the secret sharing system is limited to 3 for more specific description.
FIG. 6 shows an example of a functional configuration of a secret sharing system according to the third embodiment. FIG. 7 shows an example of a specific configuration of a reshare part according to the third embodiment. FIG. 8 shows a flow of a secret sharing process performed by the secret sharing system according to the third embodiment. The secret sharing system according to this embodiment comprises 3 secret sharing apparatuses 100α, 100β, and 100γ and selection means 105. It is assumed that a numeric value Ak that satisfies a relation Ak=akαβ+akβγ+akγα is a k-th numeric value of K numeric values (K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, and (α, β, γ) is any of (1, 2, 3), (2, 3, 1) and (3, 1, 2)), and the three fragments of the numeric value is denoted as (akγα, aαβ), (akαβ, akβγ), and (akβγ, akγα). The selection means 105 may be provided in any of the secret sharing apparatuses or provided as a separate apparatus.
The secret sharing system according to this embodiment comprises the selection means 105, fragment replacement means and reshare means. Each secret sharing apparatus 100n comprises a fragment replacement part 110n, a reshare part 120n and a recording part 190n (n represents any of α, β and γ). The recording part 190n records fragments of numeric values A1, . . . , and AK, for example.
The selection means 105 selects two secret sharing apparatuses. And one of the secret sharing apparatuses selected by the selection means 105 is designated as a first secret sharing apparatus 1001, the other is designated as a second secret sharing apparatus 1002, and the secret sharing apparatus that is not selected is designated as a third secret sharing apparatus 1003 (S105). The k-th fragment recorded in the first secret sharing apparatus 1001 is denoted as ak1=(ak31, ak12), the k-th fragment recorded in the second secret sharing apparatus 1002 is denoted as ak2=(ak12, ak23), and the k-th fragment recorded in the third secret sharing apparatus 1003 is denoted as ak3=(ak23, ak31).
The fragment replacement means comprises at least fragment replacement parts 110α, 100β and 100γ. The fragment replacement means generates a bijection π of {1, . . . , K}→{1, . . . , K} in the first secret sharing apparatus 1001 or the second secret sharing apparatus 1002, designates a fragment aπ(k)1 recorded in the first secret sharing apparatus 1001 as the k-th fragment, and designates a fragment aπ(k)2 recorded in the second secret sharing apparatus 1002 as the k-th fragment (S110). As described above in the first embodiment, the bijection π may be a mapping of numbers 1 to K randomly rearranged. The bijection π is desirably a uniformly randomly rearranged mapping and can be generated by Fisher-Yates shuffle, for example.
The reshare means comprises at least reshare parts 120α, 120β and 120γ. As shown in FIG. 7, the reshare part 120n comprises a first random number generation part 121n, a second random number generation part 122n, a first calculation part 123n, a second calculation part 124n, a third calculation part 125n and a fragment update part 126n.
A first random number generation part 1211 of the first secret sharing apparatus 1001 generates a random value bk31 for reshare of the k-th fragment and transmits the value to the third secret sharing apparatus 1003 (S121). A second random number generation part 1222 of the second secret sharing apparatus 1002 generates a random value bk23 for reshare of the k-th fragment and transmits the value to the third secret sharing apparatus 1003 (S122). A first calculation part 1231 of the first secret sharing apparatus 1001 calculates a value xk according to xk=bk31−aπ(k)31 for reshare of the k-th fragment and transmits the value xk to the second secret sharing apparatus 1002 (S123).
A second calculation part 1242 of the second secret sharing apparatus 1002 calculates a value yk according to yk=bk23−aπ(k)23 for reshare of the k-th fragment and transmits the value yk to the first secret sharing apparatus 1001 (S124). A third calculation part 1251 of the first secret sharing apparatus 1001 and a third calculation part 1252 of the second secret sharing apparatus 1002 each calculate a value bk12 according to bk12=aπ(k)12−xk−yk for reshare of the k-th fragment (S125). A fragment update part 1261 of the first secret sharing apparatus 1001 designates (bk31, bk12) as a fragment bk1, a fragment update part 1262 of the second secret sharing apparatus 1002 designates (bk12, bk23) as a fragment bk2, and a fragment update part 1263 of the third secret sharing apparatus 1003 designates (bk23, bk31) as a fragment bk3 (S126). The recording part 190n of each secret sharing apparatus 100n records not only the fragment bkn but also information that the k-th fragment bkn recorded in itself is a fragment of the numeric value Bk. As in the first embodiment, the fragments bk1, bk2 and bk3 are fragments of the numeric value Bk. That is, Steps S121 to S125 correspond to Step S120.
Furthermore, if the numeric values B1, . . . , and BK are regarded as new numeric values A1, . . . , and AK, and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111 and S112). Furthermore, if the above process is repeated by modifying the combination of the secret sharing apparatuses selected by the fragment replacement part so that every secret sharing apparatus has experience of not being selected at least once, the numeric values B1, . . . , and BK that cannot be associated with the numeric values A1, . . . , and AK by all the secret sharing apparatuses can be obtained. In this embodiment, every secret sharing apparatus can have experience of not being selected at least once if the combinations of the secret sharing apparatuses selected by the fragment replacement means are {100α, 100β}, {100β, 100γ} and {100γ, 100α}.
Thus, the secret sharing system according to the third embodiment has the same effects as the secret sharing system according to the first embodiment. In the case where sorting is also to be performed, the secret sharing apparatus 100n further comprises a comparison part 210n and an exchange part 220n. The specific sorting process is the same as that according to the first embodiment.
It is assumed that M represents an integer equal to or greater than 1, and m represents an integer equal to or greater than 1 and equal to or smaller than M. It is further assumed that A(1), . . . , and A(M) are vectors each having K elements, and A(m)=(A1(m), . . . , AK(m)). It is further assumed that the elements of the vectors A(1), . . . , and A(M) are associated with each other. In other words, Ak(1), . . . , and Ak(M) are a group of k-th numeric values associated with each other. In this modification, limited shuffling of the group of numeric values is performed while maintaining the association of the numeric values associated with each other. It is further assumed that Ak(m)=akαβ(m)+akβγ(m)+akγα(m) (k represents an integer equal to or greater than 1 and equal to or smaller than K, m represents an integer equal to or greater than 1 and equal to or smaller than M, and (α, β, γ) is any of (1, 2, 3), (2, 3, 1) and (3, 1, 2)), and three fragments are denoted as (akγα(m), akαβ(m)), (akαβ(m), akβγ(m), and (akβγ(m), akγα(m)). Note that the limited shuffling described above is the limited shuffling in the case where M=1, and the following description concerns more general limited shuffling.
An example of a functional configuration of the secret sharing system is the same as that shown in FIG. 6, an example of a specific configuration of the reshare part is the same as that shown in FIG. 7, and a flow of the secret sharing process is the same as that shown in FIG. 8. The secret sharing system comprises the selection means 105, the fragment replacement means and the reshare means. The secret sharing apparatus 100n comprises at least the fragment replacement part 110n, the reshare part 120n and the recording part 190n (n represents any of α, β and γ). However, the components are configured as described below and perform the processings described below.
The selection means 105 selects two secret sharing apparatuses. And one of the secret sharing apparatuses selected by the selection means 105 is designated as a first secret sharing apparatus 1001, the other is designated as a second secret sharing apparatus 1002, and the secret sharing apparatus that is not selected is designated as a third secret sharing apparatus 1003 (S105). The fragment of the numeric value Ak(m) recorded in the first secret sharing apparatus 1001 is denoted as ak1(m)=(ak31(m), ak12(m)), the fragment of the numeric value Ak(m) recorded in the second secret sharing apparatus 1002 is denoted as ak2(m)=(ak12(m), ak23(m)), and the fragment of the numeric value Ak(m) recorded in the third secret sharing apparatus 1003 is denoted as ak3(m)=(ak23(m), ak31(m)).
The fragment replacement means comprises at least the fragment replacement parts 110α, 100β and 100γ. The fragment replacement means generates a bijection π of {1, . . . , K}→{1, . . . , K} in the first secret sharing apparatus 1001 or the second secret sharing apparatus 1002, designates fragments aπ(k)1(1), . . . , and aπ(k)1(M) recorded in the first secret sharing apparatus 1001 as fragments of the group of the k-th numeric values associated with each other and designates fragments aα(k)2(1), . . . , and aπ(k)2(M) recorded in the second secret sharing apparatus 1002 as fragments of the group of the k-th numeric values associated with each other (S110).
The reshare means comprises at least reshare parts 120α, 120β and 120γ. As shown in FIG. 7, the reshare part 120n comprises the first random number generation part 121n, the second random number generation part 122n, the first calculation part 123n, the second calculation part 124n, the third calculation part 125n and the fragment update part 126n.
The first random number generation part 1211 of the first secret sharing apparatus 1001 generates random values bk31(1), . . . , and bk31(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values bk31(1), . . . , and bk31(M) to the third secret sharing apparatus 1003 (S121). The second random number generation part 1222 of the second secret sharing apparatus 1002 generates random values bk23(1), . . . , and bk23(M) for reshare of the fragments of the group of k-th numeric values associated with each other and transmits the values bk23(1), . . . , and bk23(M) to the third secret sharing apparatus 1003 (S122). The first calculation part 1231 of the first secret sharing apparatus 1001 calculates a value xk(m) according to xk(m)=bk31(m)−aπ(k)31(m) for m=1 to M for reshare of the fragments of the group of the k-th numeric values associated with each other and transmits the values xk(1), . . . , and xk(M) to the second secret sharing apparatus 1002 (S123).
The second calculation part 1242 of the second secret sharing apparatus 1002 calculates a value yk(m) according to yk(m)=bk23(m)−aπ(k)23(m) for m=1 to M for reshare of the fragments of the group of the k-th numeric values associated with each other and transmits the values yk(1), . . . , and yk(M) to the first secret sharing apparatus 1001 (S124). The third calculation part 1251 of the first secret sharing apparatus 1001 and the third calculation part 1252 of the second secret sharing apparatus 1002 each calculate a value bk12(m) according to bk12(m)=aπ(k)12(m)−xk(m)−yk(m) for m=1 to M for reshare of the fragments of the group of the k-th numeric values associated with each other (S125). The fragment update part 1261 of the first secret sharing apparatus 1001 designates (bk31(m), bk12(m)) as a fragment bk1(m), the fragment update part 1262 of the second secret sharing apparatus 1002 designates (bk12(m), bk23(m)) as a fragment bk2(m), and the fragment update part 1263 of the third secret sharing apparatus 1003 designates (bk23(m), bk31(m)) as a fragment bk3(m) (S126). The recording part 190n of each secret sharing apparatus 100n records not only the fragment bkn(m) but also information that the k-th fragment bkn(m) recorded in itself is a fragment of the numeric value Bk(m). As in the first embodiment, the fragments bk1(m), bk2(m), and bk3(m) are fragments of the numeric value Bk(m). That is, Steps S121 to S126 correspond to Step S120.
Furthermore, if the numeric values B1(1), . . . , BK(1), . . . , B1(M), . . . , and BK(M) are regarded as new numeric values A1(1), . . . , AK(1), . . . , A1(M), . . . , and AK(M), and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111 and S112).
In a fourth embodiment, again, the number of the secret sharing apparatuses that form the secret sharing system is limited to 3 for more specific description. In addition, the fourth embodiment described below concerns an example in which an illegal operation detection function is provided as in the second embodiment.
A configuration of a secret sharing system according to the fourth embodiment is also shown in FIG. 6. A secret sharing apparatus 100n according to this embodiment additionally comprises the components shown by dotted lines. FIG. 9 shows a specific structure of a tamper detection part. FIG. 10 shows a flow of a secret sharing process performed by the secret sharing system according to the fourth embodiment. The secret sharing system according to this embodiment comprises 3 secret sharing apparatuses 100α, 100β, and 100γ and selection means 105 connected to a network 1000. It is assumed that a numeric value Ak that satisfies a relation Ak=akαβ+akβγ+akγα is a k-th numeric value of K numeric values (K represents an integer equal to or greater than 2, k represents an integer equal to or greater than 1 and equal to or smaller than K, and (α, β, γ) is any of (1, 2, 3), (2, 3, 1) and (3, 1, 2)), and the three fragments of the numeric value is denoted as (akγα, akαβ), (akαβ, akβγ), and (akβγ, akγα).
The secret sharing system according to this embodiment comprises the selection means 105, initial information distribution means, initial multiplication means, fragment replacement means, reshare means, checking distribution means, checking multiplication means and tamper detection means. The secret sharing apparatus 100n comprises an initial information distribution part 130n, an initial multiplication part 140n, a fragment replacement part 110n, a reshare part 120n, a checking distribution part 150n, a checking multiplication part 160n, a tamper detection part 170n and a recording part 190n (n represents any of α, β and γ). The recording part 190n records fragments of numeric values A1, . . . , and AK, for example.
The initial information distribution means comprises initial information distribution parts 130α, 130β and 130γ. The initial information distribution parts 130α, 130β and 130γ determine a fragment pkn of each of K numeric values P1, . . . , and PK that are not known to any of the secret sharing apparatuses 100α, 100β and 100γ by a secure computation, and the fragment pkn is recorded in the secret sharing apparatus 100n (S130). For example, the first secret sharing apparatus 1001 generates K random values R(1)1, . . . , and R(1)K, and the second secret sharing apparatus 1002 generates K random values R(2)1, . . . , and R(2)K. Then, fragments (r(1)k31, r(1)k12), (r(1)k12, r(1)k23), and (r(1)k23, r(1)k31) of the value R(1)k and fragments (r(2)k31, r(2)k12), (r(2)k12, r(2)k23), and (r(2)k23, r(2)k31) of the value R(2)k are recorded in the secret sharing apparatuses 1001, 1002 and 1003 in a secret sharing manner. Then, the secret sharing apparatuses 1001, 1002 and 1003 determine fragments (pk31, pk12), (pk12, pk23), and (pk23, pk31) of a numeric value Pk that satisfies a relation that Pk=R(1)k+R(2)k by a secure computation, and the fragments are distributed among and recorded in the secret sharing apparatuses 1001, 1002 and 1003. Through this process, fragments of a numeric value that is not known to all the secret sharing apparatuses 100α, 100β and 100γ can be recorded in a distributed manner.
The initial multiplication means comprises initial multiplication parts 140α, 140β and 140γ. The initial multiplication parts 140α, 140β and 140γ determine fragments (skγα, skαβ), (skαβ, sβγ), and (sβγ, sγα) of a numeric value Sk that satisfies a relation that Sk=Pk×Ak by a secure computation, and the fragments are distributed among and recorded in the secret sharing apparatuses 100α, 100β and 100γ (S140).
The fragment replacement means and the reshare means are the same as those according to the third embodiment. The fragment replacement means and the reshare means serve to record fragments bk1, bk2, and bk3 in the secret sharing apparatuses 100α, 100β and 100γ as fragments of a numeric value Bk. The checking distribution means comprises checking distribution parts 150α, 150β and 150γ. The checking distribution parts 150α, 150β and 150γ generate fragments (qkγα, qkαβ), (qkαβ, qkβγ), and (qkβγ, qkγα) of a numeric value Qk that satisfies a relation that Qk=Pπ(k) for k=1 to K by a secure computation, and the fragments are distributed among and recorded in the secret sharing apparatuses 100α, 100β and 100γ (S150). For example, other fragments (r′(1)π(k)31, r′(1)π(k)12), (r′(1)π(k)12, r′(1)π(k)23), and (r′(1)π(k)23, r′(1)π(k)31) of the numeric value R(1)π(k) generated by the first secret sharing apparatus 1001 in Step S130 and other fragments (r′(2)π(k)31, r′(2)π(k)12), (r′(2)π(k)12, r′(2)π(k)23), and (r′(2)π(k)23, r′(2)π(k)31) of the numeric value R(2)π(k) generated by the second secret sharing apparatus 1002 are recorded in a secret sharing manner Then, the secret sharing apparatuses 1001, 1002 and 1003 generate fragments (qk31, qk12), (qk12, qk23), and (qk23, qk31) of the numeric value Qk that satisfies a relation that Qk=R(1)π(k)+R(2)π(k) by a secure computation using the other fragments, and the fragments are recorded in the secret sharing apparatuses 1001, 1002 and 1003 in a distributed manner. Through the process described above, fragments of a numeric value Qk that satisfies a relation that Qk=Pπ(k) and is not known to all the secret sharing apparatuses 100α, 100β and 100γ can be recorded in a distributed manner.
The checking multiplication means comprises checking multiplication parts 160α, 160β and 160γ. The checking multiplication parts 160α, 160β and 160γ determine fragments (tkγα, tkαβ), (tkαβ, tkβγ), and (tkβγ, tkγα) of a numeric value Tk that satisfies a relation that Tk=Qk×Bk by a secure computation, and the fragments are recorded in the secret sharing apparatuses 100α, 100β and 100γ in a distributed manner (S160).
The tamper detection means comprises tamper detection parts 170α, 170β and 170γ. As shown in FIG. 9, the tamper detection part 170n comprises a third random number generation part 171n, a fourth random number generation part 172n, a fourth calculation part 173n, a fifth calculation part 174n, a first check part 175n, a sixth calculation part 176n, a seventh calculation part 177n and a second check part 178n. The tamper detection means performs a processing as described below depending on which of the first secret sharing apparatus 1001, the second secret sharing apparatus 1002 and the secret sharing apparatus 1003 the secret sharing apparatuses 100α, 100β and 100γ operate as.
A third random number generation part 1711 of the first secret sharing apparatus 1001 generates a random number uk and transmits the random number to the second secret sharing apparatus 1002 (S171). A fourth random number generation part 1722 of the second secret sharing apparatus 1002 generates a random number vk and transmits the random number vk to the first secret sharing apparatus 1001 (S172). The fourth calculation part 1731 of the first secret sharing apparatus 1001 calculates a value dk according to dk=sπ(k)12−tk12−uk−vk and transmits the value dk to the third secret sharing apparatus 1003 (S173).
A fifth calculation part 1742 of the second secret sharing apparatus 1002 calculates a value ek according to ek=sπ(k)12−tk12−uk−vk and transmits the value ek to the third secret sharing apparatus 1003 (S174). A first check part 1753 of the third secret sharing apparatus 1003 checks whether a relation dk=ek holds and terminates the processing if the relation does not holds (S175).
A sixth calculation part 1761 of the first secret sharing apparatus 1001 calculates a value fk according to fk=sπ(k)31−tk31+uk and transmits the value fk to the third secret sharing apparatus 1003 (S176). A seventh calculation part 1772 of the second secret sharing apparatus 1002 calculates a value gk according to gk=sπ(k)23−tk23+vk and transmits the value gk to the third secret sharing apparatus 1003 (S177). A second check part 1783 of the third secret sharing apparatus 1003 checks whether a relation that fk+gk+dk=0 holds and terminates the processing if the relation does not hold (S178). If the numeric values B1, . . . , and BK are regarded as new numeric values A1, . . . , and AK, and the combination of the secret sharing apparatuses selected by the fragment replacement means is modified, the process described above can be repeated (S111 and S112).
The secret sharing system according to the fourth embodiment has the same effects as the secret sharing apparatus according to the third embodiment and can check whether an illegal operation to transmit a tampered value to other secret sharing apparatuses has not occurred in the course of a process of concealing the association between the numeric values A1, . . . , and AK and the numeric values B1, . . . , and BK. In the case where sorting is also to be performed, the secret sharing apparatus 100n further comprises a comparison part 210n and an exchange part 220n. The specific sorting process is the same as that according to the first embodiment.
[Secret Calculation]
In the above description, it has been assumed that the secure computation is not limited to a particular method, and no specific example has been shown. In the following, a specific example of a basic secure computation that can be used in each component of the secret sharing systems according to the third and fourth embodiments will be described. In the following description, it will be assumed that fragments of a numeric value A recorded in the secret sharing apparatuses 100α, 100β and 100γ in a distributed manner are denoted as (aγα, aαβ), (aαβ, aβγ), and (aβγ, aγα), fragments of a numeric value B recorded in the secret sharing apparatuses 100α, 100β and 100γ in a distributed manner are denoted as (bγα, bαβ), (bαβ, bβγ), and (bβγ, bγα), and fragments of a numeric value C recorded in the secret sharing apparatuses 100α, 100β and 100γ in a distributed manner are denoted as (cγα, cαβ), (cαβ, cβγ), and (cβγ, cγα).
Secret Sharing of Numeric Value A
(1) Random numbers aαβ, aβγ are generated.
(2) A value aγα is calculated according to aγα=A−aαβ−aβγ, (aγα, aαβ), (aαβ, aβγ), and (aβγ, aαα) are designated as fragments of the value A, and the fragments (aγα, aαβ), (aαβ, aβγ), and (aβγ, aγα) are distributed among and recorded in the secret sharing apparatuses 100α, 100β and 100γ.
Reconstruction of Numeric Value A
(1) The secret sharing apparatus 100α transmits the value aγα to the secret sharing apparatus 100β and transmits the value aαβ to the secret sharing apparatus 100γ. The secret sharing apparatus 100β transmits the value aαβ to the secret sharing apparatus 100γ and transmits the value aβγ to the secret sharing apparatus 100α. The secret sharing apparatus 100γ transmits the value aβγ to the secret sharing apparatus 100α and transmits the value aγα to the secret sharing apparatus 100β.
(2) The secret sharing apparatus 100α calculates a value aαβ+aβγ+aγα to reconstruct the numeric value A if the value aβγ received from the secret sharing apparatus 100β and the value aβγ received from the secret sharing apparatus 100γ agree with each other. The secret sharing apparatus 100β calculates the value aαβ+aβγ+aγα to reconstruct the numeric value A if the value aγα received from the secret sharing apparatus 100γ and the value aγα received from the secret sharing apparatus 100α agree with each other. The secret sharing apparatus 100γ calculates the value aαβ+aβγ+aγα, to reconstruct the numeric value A if the value aαβ received from the secret sharing apparatus 100α and the value aαβ received from the secret sharing apparatus 100β agree with each other.
Secret Calculation of C=A+B
(1) The secret sharing apparatus 100α calculates the fragment (cγα, cαβ)=(aγα+bγα, aαβ+bαβ) and records the fragment (cγα, cαβ), the secret sharing apparatus 100β calculates the fragment (cαβ, cβγ)=(aαβ+bαβ, aβγ+bβγ) and records the fragment (cαβ, cβγ), and the secret sharing apparatus 100γ calculates the fragment (cβγ, cγα)=(aβγ+bβγ, aγα+bγα) and records the fragment (cβγ, cγα).
Secret Calculation of C=A−B
(1) The secret sharing apparatus 100α calculates the fragment (cγα, cαβ)=(aγα−bγα, aαβ−bαβ) and records the fragment (cγα, cαβ), the secret sharing apparatus 100β calculates the fragment (cαβ, cβγ)=(aαβ−bαβ, aβγ−bβγ) and records the fragment (cαβ, cβγ), and the secret sharing apparatus 100γ calculates the fragment (cβγ, cγα)=(aβγ−bβγ, aγα−bγα) and records the fragment (cβγ, cγα).
Secret Calculation of C=A+S (S Represents a Known Constant)
(1) The secret sharing apparatus 100α calculates the fragment (cγα, cαβ)=(aγα+S, aαβ) and records the fragment (cγα, cαβ), and the secret sharing apparatus 100γ calculates the fragment (cβγ, cγα)=(aγγ, aγα+S) and records the fragment (cβγ, cγα). The secret sharing apparatus 100β does not perform any processing.
Secret Calculation of C=AS (S Represents a Known Constant)
(1) The secret sharing apparatus 100α calculates the fragment (cγα, xαβ)=(aγαS, aαβS) and records the fragment (cγα, cαβ), the secret sharing apparatus 100β calculates the fragment (cαβ, cβγ)=(aαβS, aβγS) and records the fragment (cαβ, cβγ), and the secret sharing apparatus 100γ calculates the fragment (cβγ, cγα)=(aβγS, aγαS) and records the fragment (cβγ, cγα).
Secret Calculation of C=AB
(1) The secret sharing apparatus 100α generates random numbers r1, r2, and cγα and calculates a value cαβ according to cαβ=(aγα+aαβ)(bγα+bαβ)−r1−r2−cγα. Then, the secret sharing apparatus 100α transmits (r1, cαβ) to the secret sharing apparatus 100β and (r2, cγα) to the secret sharing apparatus 100γ.
(2) The secret sharing apparatus 100β calculates a value y according to y=aαβbβγ+aβγbαβ+r1 and transmits the value y to the secret sharing apparatus 100γ.
(3) The secret sharing apparatus 100γ calculates a value z according to z=aβγbγα+aγαbβγ+r2 and transmits the value z to the secret sharing apparatus 100α.
(4) The secret sharing apparatus 100β and the secret sharing apparatus 100γ each calculates a value cβγ according to cβγ=y+z+aβγbβγ.
(5) The secret sharing apparatus 100α records the fragment (cγα, cαβ), the secret sharing apparatus 100β records the fragment (cαβ, cβγ), and the secret sharing apparatus 100γ records the fragment (cβγ, cγα).
The various processings described above can be performed not only sequentially in the order described above but also in parallel with each other or individually as required or depending on the processing power of the apparatus that performs the processings. Furthermore, of course, other various modifications can be appropriately made to the processings without departing from the spirit of the present invention.
In the case where the configurations described above are implemented by a computer, the specific processings of the functions of the apparatuses are described in a program. The computer executes the program to implement the processings described above.
The program that describes the specific processings can be recorded in a computer-readable recording medium. The computer-readable recording medium may be any type of recording medium, such as a magnetic recording device, an optical disk, a magneto-optical recording medium and a semiconductor memory.
The program may be distributed by selling, transferring or lending a portable recording medium, such as a DVD and a CD-ROM, in which the program is recorded, for example. Alternatively, the program may be distributed by storing the program in a storage device in a server computer and transferring the program from the server computer to other computers via a network.
The computer that executes the program first temporarily stores, in a storage device thereof, the program recorded in a portable recording medium or transferred from a server computer, for example. Then, when performing the processings, the computer reads the program from the storage device and performs the processings according to the read program. In an alternative implementation, the computer may read the program directly from the portable recording medium and perform the processings according to the program. As a further alternative, the computer may perform the processings according to the program each time the computer receives the program transferred from the server computer. As a further alternative, the processings described above may be performed on an application service provider (ASP) basis, in which the server computer does not transmit the program to the computer, and the processings are implemented only through execution instruction and result acquisition. The programs according to the embodiments of the present invention include a quasi-program, which is information to be processed by a computer (such as data that is not a direct instruction to a computer but has a property that defines the processings performed by the computer).
Although a predetermined program is executed on a computer to implement the apparatus according to the present invention in the embodiments described above, at least part of the specific processing may be implemented by hardware.
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