Strong fully homomorphic white-box and method for using same
A fully homomorphic white-box implementation of one or more cryptographic operations is presented. This method allows construction of white-box implementations from general-purpose code without necessitating specialized knowledge in cryptography, and with minimal impact to the processing and memory requirements for non-white-box implementations. This method and the techniques that use it are ideally suited for securing “math heavy” implementations, such as codecs, that currently do not benefit from white-box security because of memory or processing concerns. Further, the fully homomorphic white-box construction can produce a white-box implementation from general purpose program code, such as C or C++.
1. A method of computing an algorithm (m,S) having i operations with input m and secret S, comprising:
defining a white-box fully-homomorphic key generation function (P,p)←Gen(1 w ) with public-key P and private-key p that selects random prime numbers p, q, s∈W of similar size, wherein:
B={0,1} b is the domain of order b, of the algorithm , where b>=8;
W={0,1} w is a white-box domain of order w, for w>>b;
p>2 b is a white-box fully-homomorphic private key;
N=pq;
k=s(p−1);
P=(N,k) is a white-box fully-homomorphic public key;
defining a white-box fully-homomorphic encoding function Enc(P,m) m rk+1 (mod N) that generates a random integer r∈W, then performs an encoding of the input m∈B;
defining a white-box fully-homomorphic decoding function Dec(p,c):=c(mod p) that decodes c by computing c modulo p;
defining i transform key pairs (T i ,t i ), at least on part from P and p, wherein T i is the i th transform public key and t i is the i th transform private key;
selecting prime numbers t i , and q i ∈W of similar size to p, with t i >2 b ;
computing N i =p i p −1 q 2 such that N i and N are pairwise co-prime;
computing e i =p i p −1 where T i =(N i ,e 1 );
accepting an encoded input c 0 , where c 0 =Enc(P,m);
generate an encoded output c′ by performing, for each of the i operations:
accepting an encoded transform public key T i =Enc(T i ,S);
performing the i th operation on the encoded input c i−1 and the encoded transform public key in modulo N i to obtain an encoded output c i ;
reencoding c i with transform T i without any interim decoding operation;
decoding the encoded output c′ with the private key p to recover an output m′ according to m′=Dec(p,c′), such that m′= (m,S).
2. The method of claim 1 , wherein the algorithm is a decryption algorithm including at least one of a Rivest-Shamir-Aldeman (RSA) algorithm, an elliptic curve cryptography (ECC) algorithm, an advanced encryption standard (AES) algorithm, and a triple data standard (TDES) algorithm.
3. The method of claim 2 , wherein:
the algorithm includes an RSA decryption algorithm RSADecrypt;
the accepted encoded input is c 0 =Enc(P,M), wherein M=RSAEncrypt(RSAPLK,m) is an RSA encrypted version of the input message m encoded with the white-box fully-homomorphic public key P, where (RSAPVK, RSAPLK) is an RSA private/public keypair corresponding to the RSADecrypt and RSAEncrypt algorithms;
the accepted encoded transform public key is T i =Enc(T i , RSAPVK), wherein RSAPVK is the RSA private key encoded with the white-box fully-homomorphic public key P;
the method further comprising performing one or more operations including an RSADecrypt implementation, with encoded input c i−1 and the encoded transform public key T i =Enc(T i , RSAPVK) used to compute each encoded output c i ; and
decoding the encoded output c′ with the private key p to recovers the output message m′ according to m′=Dec(p, c′).
4. The method of claim 1 , wherein w is selected for semantic security.
5. The method of claim 1 , further comprising:
securely encoding the input message m according to c 0 =Enc(P,m); and
securely encoding the transform public key according to T i =Enc(T i ,S).