Method of RSA signature or decryption protected using a homomorphic encryption
Decryption of an RSA encrypted message encrypted with a public RSA key by receiving encrypted key share components computed by generating a private RSA key d and a RSA modulus integer N, where N and d are integers; splitting the private key into key shares, encrypting with a fully homomorphic encryption (FHE) algorithm each key share component by using a Fully Homomorphic Encryption secret key ps associated with a set Ss to generate the encrypted key share components of said secure RSA key, computing an intermediate value YS for each set SS from said encrypted key share components, such that said computed intermediate value is a part of the RSA decrypted message, under FHE-encrypted form, and decrypting the encrypted message by combining said computed intermediate values for all sets.
1. A method of secure Rivest-Shamir-Adelson (RSA) decryption by a client device of an RSA encrypted message c encrypted with a public RSA key forming a key pair with a secure private RSA key, comprising the steps of:
receiving from a server J generated encrypted key share components ed j (i) with j in [1, J] and i in [0, n/b−1] whereby i and j form an index couple (i,j), the encrypted key share components ed j (i) computed by the server by:
generating a private RSA key d and a RSA modulus integer N, where N and d are integers;
splitting the private key d in J key shares dj of length n, with n=log 2 N, j in [1, J], J being an integer, and such that d=d 1 +d 2 + . . . +d J mod phi(N), where each key share dj is equal to (d j (0) . . . d j (i) . . . d j (n/b−1) ) with each key share component d j (i) in {0 . . . 2{circumflex over ( )}b−1} and i in [0, n/b−1], b being an integer inferior to n and phi the Euler's totient function;
encrypting with a fully homomorphic encryption (FHE) algorithm each key share component d j (i) of the private RSA key d by using a Fully Homomorphic Encryption secret key p s associated with a set S s comprising the index couple (i,j), to generate each encrypted key share component ed j (i) with j in [1, J] and i in [0, n/b−1] whereby i and j form an index couple (i,j) of said secure RSA key,
said set S s being a set of integer couples, among a predetermined integer number u of disjoint sets {S 1 , S 2 . . . , S u } generated such that: U{S s }={(i,j) such that i in [0, n/b−1], j in [1, J]} and each said set among {S 1 , . . . , S u } being associated with a Fully Homomorphic Encryption (FHE) secret key;
computing an intermediate value Y S for each set S S in {S 1 , . . . , S u } from said encrypted key share components ed j (i) whose index couple (i,j) is comprised in S S and from said encrypted message c, such that said computed intermediate value is a part of the RSA decrypted message m, under FHE-encrypted form;
decrypting the encrypted message c by combining said computed intermediate values for all sets S s in {S 1 . . . . , S u }.
2. The method of claim 1 wherein said FHE secret keys for each set S S in {S 1 , . . . , S u } are all identical.
3. The method of claim 1 wherein {S 1 , . . . , S u } comprises at least a first set and a second set such that the FHE secret key associated with the first set is different from the FHE secret key associated with the second set.
4. The method of claim 1 wherein the fully FHE algorithm is a Dijik, Gentry, Halevi, Vaikuntanathan (DGHV) Scheme such that a RSA encrypted message c for a data m in ZN, FHE encrypted with a FHE secret key p, is equal to q·p+N·r+m where q is a random integer, r is a random integer and N is the RSA modulus such that |p|>C S (2 log N+log r) with C S the maximum number of elements of any set S S in {S 1 , . . . , S U }, and such that the RSA encrypted message c is deciphered by computing m=(c mod p) mod N.
5. The method of claim 1 wherein an order in which intermediate values Y S are computed for each set S s in {S 1 , . . . , S u } is random.
6. A client device adapted to securely decrypt an Rivest-Shamir-Adelson (RSA) encrypted message c encrypted with a public RSA key forming a key pair with a secure private RSA key, comprising:
a memory ( 201 , 203 , 204 );
a processor ( 201 ), and
an interface ( 207 ) to receive J key share components ed j (i) with j in [1, J] and i in [0, n/b−1] whereby i and j form an index couple (i,j), the encrypted key share components ed j (i) computed by a server by:
generating a private RSA key d and a RSA modulus integer N, where N and d are integers;
splitting the private key d in J key shares d j of length n, with n=log 2 N, j in [1, J], J being an integer, and such that d=d 1 +d 2 + . . . +d J mod phi(N), where each key share d j is equal to (d j (0) . . . d j (i) . . . d j (n/b−1) ) with each key share component d j (i) in {0 . . . 2{circumflex over ( )}b−1} and i in [0, n/b−1], b being an integer inferior to n and phi the Euler's totient function;
encrypting with a fully homomorphic encryption (FHE) algorithm each key share component d j (i) of the private RSA key d by using a Fully Homomorphic Encryption secret key p s associated with a set S s comprising the index couple (i,j), to generate each encrypted key share component ed j (i) with j in [1, J] and i in [0, n/b−1] whereby i and i form an index couple (i,j) of said secure RSA key,
said set S s being a set of integer couples, among a predetermined integer number u of disjoint sets {S 1 , S 2 . . . , S s , S s+1 , . . . S u } generated such that: U{S S }={(i,j) such that i in [0, n/b−1], j in [1, J]} and each said set among {S 1 , . . . , S u } being associated with a Fully Homomorphic Encryption (FHE) secret key;
the memory comprising instructions to cause the processor to:
compute an intermediate value Y S for each set S S in {S 1 , . . . , S u } from said encrypted key share components ed j (i) whose index couple (i,j) is comprised in S s and from said encrypted message c, such that said computed intermediate value is a part of the RSA decrypted message m, under FHE-encrypted form;
decrypt the encrypted message c by combining said computed intermediate values for all sets S s in {S 1 . . . . , S u }.
7. The client device of claim 6 wherein said FHE secret keys for each set S s in {S 1 , . . . , S u } are all identical.
8. The client device of claim 6 wherein {S 1 , . . . , S u } comprises at least a first set and a second set such that the FHE secret key associated with the first set is different from the FHE secret key associated with the second set.
9. The client device of claim 6 wherein the FHE algorithm is a DGHV Dijik, Gentry, Halevi, Vaikuntanathan (DGHV) Scheme such that a RSA encrypted message c for a data m in ZN, FHE encrypted with a FHE secret key p, is equal to q·p+N·r+m where q is a random integer, r is a random integer and N is the RSA modulus such that |p|>C S (2 log N+log r) with Cs the maximum number of elements of any set S S in {S 1 , . . . , S U }, and such that the RSA encrypted message c is deciphered by computing m=(c mod p) mod N;
The client device of claim 6 wherein the order in which intermediate values Y S are computed for every sets S s in {S 1 , . . . , S u } is random.