Cryptographic device having reduced vulnerability to side-channel attack and method of operating same

A method for operating a cryptographic device, said method comprising the steps of:

• storing a key e and an encryption function E(x,y), where E(x,y) is an encryption of an arbitrary message x with an arbitrary key y and E(x,Y) equals xy mod(n), where n is a secret number equal to p times q, where p and q are secret prime numbers;
• storing a first, resistant function F1(y,t) and providing a message recipient with a second function F2(x,t), where t is an arbitrary integer, said first function F1(y,t) equals (y)(1/t mod(φ(n))), where φ(n) is the number of positive integers less than or equal to n that are relatively prime to n, said second function F2(x,t) equals xt mod(n), and F2(E(x, F1(y,t)), t)=E(x,y);
• inputting a message m for encryption;
• selecting a random number r
• generating a temporary key F1(e,r);
• generating E(m, F1(e,r));
• outputting E(m, F1(e,r)), said message recipient being provided with said random number r;
• whereby
• said message recipient can generate F2(E(m, F1(e,r)), r)=E(m,e) as an encryption c of said message m with said key e, and then decrypt said encryption c;
• and whereby
• signals generated by said device which are susceptible to side-channel attack will vary with said random number r thus reducing the vulnerability of said key e to side channel analysis and F2(E(m, F1(e,r)), r)=(me/r mod(n))r mod(n), where the exponent e/r is taken to be (e)(1/r mod(φ(n))), equals me mod(n), as said encryption c of said message m;
• whereby a recipient of said encryption c can recover said message m as cd mod(n), where (e)(d)=1 mod(φ(n)).

A method for operating a cryptographic device, said method comprising the steps of:

• storing a key d and an decryption function D(x,y), where D(x,y) is a decryption of an arbitrary message x with an arbitrary key y and a decryption key d;
• storing a second function F2(x,t), where t is an arbitrary integer, F2(x,t) equals xt mod(n) and F2(E(x, F1(y,t)), t)=E(x,y), wherein F1(y,t) equals (y)(1/t mod(φ(n))), E(x,y) is an encryption function corresponding to decryption function D(x,y) and said encryption function E(x,y) and said decryption function D(x,y) equal xy mod(n), where n is a secret number equal to p times g, where p and a are secret prime numbers, and where φ(n) is the number of positive integers less than or equal to n that are relatively prime to n;
• receiving a modified message E(m, F1(e,r)), where e is an encryption key corresponding to said encryption key d, and r is a random number, and F1 is a resistant function;
• obtaining r;
• generating F2(E(m, F1(e,r)), r)=E(m,e) as an encryption c of said message m with said key e;
• and
• decrypting encryption c;
• whereby
• signals generated in producing said modified message which are susceptible to side-channel attack will vary with said random number r thus reducing the vulnerability of said key e to side channel analysis and F2(E(m, F1(e,r)), r)=(me/r mod(n))r mod(n) where the exponent e/r is taken to be (e)(1/r mod(φ(n))), equals me mod(n), as said encryption c of said message m;
• whereby a recipient of said encryption c can recover said message m as cd mod(n), where (e)(d)=1 mod(φ(n)).

A cryptographic device, comprising:

• a data store storing a key e for an encryption function E(x,y), where E(x,y) is an encryption of an arbitrary message x with an arbitrary key y and said encryption function E(x,y) equals xy mod(n);
• an input;
• an output;
• a processor controlling said cryptographic device, said processor being programmed to:
• receive a message m for encryption from said input;
• a select a random number r;
• generate a temporary key F1(e,r), where F1 is a resistant function and F1(y,t) equals (y)(1/t mod(φ(n))) where t is an arbitrary integer and n is a secret number equal to p times q, where p and q are secret prime numbers, and where φ(n) is the number of positive integers less than or equal to n that are relatively prime to n;
• generate E(m, F1(e,r));
• generate F2(E(m, F1(e,r)), r), where F2(x,t) equals xt mod(n) and
• F2(E(x, F1(y,t)), n)=E(x,y), whereby F2(E(m, F1(e,r)), r)=E(m,e) as an encryption c of said message m with said key c;
• and
• transmit said encryption c to said output;
• whereby
• signals generated by said device which are susceptible to side-channel attack will vary with said random number r thus reducing the vulnerability of said key e to side channel analysis and F2(E(m, F1(e,r)), r)=(me/r mod(n))r mod(n), where the exponent e/r is taken to be (e)(1/r mod(φ(n))), equals me mod(n), as said encryption c of message m;
• whereby a recipient of said encryption c can recover said message m as cd mod(n), where (e)(d)=1 mod(φ(n)).