Encryption method, decryption method, encryption/decryption method, cryptographic communications system, and computer usable medium

A decryption method for decrypting a ciphertext C, which is encrypted according to an encryption method shown below, the encryption method including the steps of:

• providing a plaintext vector m=(m0, m1, . . . , mk−1), which divides a plaintext into K parts,
• providing a base vector B=(B0, B1, . . . , Bk−1) by setting Bi (0≦i≦K−1) to Bi=b0 b1 . . . bi using an integer bi, and
• producing a ciphertext C=m0b0+m1b1+ . . . +mK−1BK−1 using the plaintext vector m and base vector B, wherein the ciphertext C=m0v0B0+m1v1B1+mK−1vK−1BK−1 is further produced by using a random number vector v=(v0, v1, . . . vk−1);
• the decryption method comprising the steps of:
• determining plaintext vector m=(m0, m1, . . . , mk−1) from the ciphertext C according to the algorithm below:
• Step 0
• C0=C/b0
• m0=C0 (mod b1)
• Step i (i=1, 2, . . . , K−2)
• Ci=(Ci−1−mi−1)/bi
• mi=Ci (mod bi+1)
• Step K−1
• mK−1=(CK−2−mK−2)/bK−1;
• and
• obtaining the original plaintext from the plaintext vector m.

An encryption method comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mk−1), which divides a plaintext into K parts;
• providing base vector B=(B0, B1, . . . , Bk−1) by setting Bi (0≦i≦K−1) to Bi=v1b0bi . . . bi using integer bi and random number vi;
• and
• producing ciphertext C=m0b0+m1b1+ . . . +mK−1BK−1 using the plaintext vector m and base vector B.

An encryption/decryption method for converting a plaintext to a ciphertext, and for converting said ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mk−1), which divides a plaintext into K parts;
• providing base vector B=(B0, B1, . . . , Bk−1) by setting said Bi (0≦i≦K−1) to B1=b0bi . . . bi using integer b1;
• selecting w, which satisfies w<P (P:
• prime number), and determining public-key vector c=(c0, c1, . . . , cK−1) according to Formula (a);
• ci=wBi (mod P) (a)
• preparing ciphertext C shown in Formula (b) according to the scalar product of plaintext vector m and public-key vector c;
• c=m0b0+m1c1+ . . . +mK−1cK−1 (b)
• determining an interim decrypted text M for ciphertext C as described in Formula (c);
• M=w−1C(mod P) (c)
• determining plaintext vector m=(m0, m1, . . . , mk−1) by decrypting this interim decrypted text M according to the algorithm below:
• Step 0
• M0=M/b0
• m0≡M0(mod bi)
• Step i (i=1, 2, . . . , K−2)
• M1=Mi−1−mi−1)/bi
• mi≡Mi(mod Bi+1)
• Step K−1
• mK−1=MK−2−mk−2)/bK−1;
• and
• obtaining the original plaintext from the plaintext vector m.

An encryption/decryption method for converting a plaintext to a ciphertext, and for converting said ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mK−1), which divides a plaintext into K (K is a number to the power of 2) parts;
• providing base vector B=(B0, B1, . . . BK−1) by setting said Bi (0≦i≦K−1) to Bi=b0bi . . . bi using integer b1;
• selecting w, which satisfies w<P (P:
• prime number), and determining public-key vector
• c=(c0, c1, . . . , cK−1) according to Formula (d);
• ci≡wBi(mod P) (d)
• preparing ciphertext C shown in Formula (e) according to the scalar product of plaintext vector m and public-key vector c;
• c=m0b0+m1c1+ . . . +mK−1cK−1 (e)
• determining an interim decrypted text M for ciphertext C as described in Formula (f);
• M≡w−1C(mod P) (f)
• determining plaintext vector m=(m0, m1, . . . , mK−1) by decrypting this interim decrypted text M according to the algorithms below:
• Two-part algorithm:
• First Step
• ML≡M(mod BK/2)
• Second Step
• MR=(M−ML)/BK/2
• Highspeed algorithm:
• The two-part algorithm is applied once again to ML, MR. The two-part algorithm is once again applied to each of the four parts of the interim decrypted text. This kind of process is repeated;
• and
• obtaining the original plaintext from the plaintext vector m.

An encryption/decryption method for converting plaintext to a ciphertext, and converting said ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mK−1), which divides a plaintext into K parts;
• providing base vector B=(B0, B1, . . . , BK−1) by setting said Bi (0≦i≦K−1) according to Formula (g);
• Bi=v0b0b1 . . . bi (g)
• Provided that:
• vi:
• random number
• bi:
• integer
• gcd (vi, bi+1)=1
• selecting w, which satisfies w<P (P:
• prime number), and determining public-key vector c=(c0, c1, . . . , cK−1) according to Formula (h);
• Ci=wBi(mod P) (h)
• preparing ciphertext C shown in Formula (i) according to the scalar product of plaintext vector m and public-key vector c;
• C=m0b0+m1c1+ . . . +mK−1cK−1 (i)
• determining an interim decrypted text M for ciphertext C as described in Formula (j);
• M=w−1C(mod P) (j)
• determining plaintext vector m=(m0, m1, . . . , mK−1) by decrypting this interim decrypted text M according to the algorithm below:
• Step 0
• M0=C/b0
• m0≡M0v0−1(mod b1)
• Step i (i=1, 2, . . . , K−2)
• Mi=Mi−1−mi−1vi−1)/bi
• mi=Mivi−1(mod bi+1)
• Step K−1
• MK−1=MK−2−mK−2vK−2)/bK−1
• mK−1=MK−1/vK−1;
• and
• obtaining the original plaintext from the plaintext vector m.

An encryption/decryption method for converting plaintext to a ciphertext, and converting said ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mK−1), which divides a plaintext into K parts;
• providing prime numbers P, Q;
• providing two base vectors B=(B0, B1, . . . , BK−1) by setting first base vector BPi (0≦i≦K−1) to BPi=bP0 bP1 . . . bPi using integer bPi and second base vector BQi (0≦i≦K−1) to BQi=bQ0bQ1 . . . bQi using integer bQi;
• using the Chinese Remainder Theorem to introduce a minimum integer Bi so that the remainders resulting from P, Q constitute BPi BQi, respectively;
• selecting w, which satisfies w<N (N=PQ), and determining public-key vector c=(c0, c1, . . . , cK−1) according to Formula (k);
• ci=wBi(mod N) (k)
• preparing ciphertext C shown in Formula (l) according to the scalar product of plaintext vector m and public-key vector c;
• C=m0c0+m1c1+ . . . +mK−1cK−1 (l)
• determining interim decrypted texts Mp, MQ with moduli P, Q for ciphertext C as described in Formulas (m), (n) respectively;
• MP=w−1C(mod P) (m)
• MQ=w−1C(mod Q) (n)
• determining plaintext vector m=(m0, m1, . . . , mK−1) by decrypting these interim decrypted texts MP, MQ according to the algorithm below:
• Step 0
• MP0=MP/bP0
• MQ0=MQ/bQ0
• M0(P)=MP0(mod bP1)
• M0(Q)=MQ0(mod bQ1)
• m0 is determined according to the Chinese Remainder Theorem,
• Step i (i=1, 2, . . . , K−2)
• MPi=MPi−1−mi−1/bPi
• MQi=(MQi−1−mi−1/bQi
• Mi(P)=MPi(mod bPi+1)
• Mi(Q)=MQi(mod bQi+1)
• m1 is determined according to the Chinese Remainder Theorem,
• Step K−1
• mK−1=MPK−2−mK−2/bPK−1
• or
• mK−1=MQK−2−mK−2/bQK−1;
• and
• obtaining the original plaintext from the plaintext vector m.

A cryptographic communications system for carrying out data communications between a plurality of entities using ciphertext, the cryptographic communications system comprising:

• an encryptor for preparing a ciphertext from a plaintext using the following encryption method;
• the encryption method including the steps of:
• providing a plaintext vector m=(m0, m1, . . . , mK−1), which divides a plaintext into K parts,
• providing a base vector B=(B0, B1, . . . , BK−1) by setting Bi (0≦i≦K−1) to Bi=b0b1 . . . bi using an integer bi, and
• producing a ciphertext C=m0B0+m1B1+ . . . +mK−1 BK−1 using the plaintext vector m and base vector B, wherein the ciphertext C=m0v0B0+m1v1B1+ . . . +mK−1vK−1BK−1 is produced by further using a random number vector v=(v0, v1, . . . , vK−1);
• a communications channel for sending a prepared ciphertext from one entity to another entity;
• and
• a decryptor for decrypting a sent ciphertext to an original plaintext.

A cryptographic communications system for carrying out data communications between a plurality of entities using ciphertext, the cryptographic communications system comprising:

• an encryptor for preparing a ciphertext from a plaintext using the following encryption method;
• the encryption method including the steps of:
• providing plaintext vector m=(m0, m1, . . . , mK−1), which divides a plaintext into K parts,
• providing base vector B=(B0, B1, . . . , BK−1) by setting Bi (0≦i≦K−1) to Bi=vib0b1 . . . bi using integer bi and random number vi and
• producing ciphertext C=m0B0+m1B1+ . . . +mK−1BK−1 using the plaintext vector m and base vector B;
• a communications channel for sending a prepared ciphertext from one entity to another entity;
• and
• a decryptor for decrypting a sent ciphertext to an original plaintext.

An encryption method for producing a ciphertext from a plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts;
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1), which is set to Bi=b0b1 . . . bi using integer bi (0≦i≦K−1);
• and
• producing ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 from the plaintext vector m, arbitrary random number vector r, and base vector B.

A decryption method for decrypting ciphertext C, which is encrypted according to an encryption method shown below,

• the encryption method including the steps of:
• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts,
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1),
• providing a base vector B=(B0, B1, . . . , BK−1),
• which is set to using integer Bi=b0b1 . . . bi using integer Bi (0≦i≦K−1), and
• producing ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 from the plaintext vector m, arbitrary random number vector r, and base vector B;
• the decryption method comprising the steps of determining plaintext vector m=(m0, m1, . . . , mJ−1) from the ciphertext C according to the algorithm below and obtaining the original plaintext from the plaintext vector m:
• Step 0
• C0=C/b0
• m0≡C0(mod b1)
• Step i (i=1, 2, . . . , J−1)
• Ci=(Ci−1−mi−1/bi
• mi≡Ci(mod bi+1).

An encryption method comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts;
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1), which is set to Bi=v1b0b1 . . . bi using an integer bi and random number vi (0≦i≦K−1), and
• producing ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 from the plaintext vector m, arbitrary random number vector r and base vector B.

An encryption/decryption method, which encrypts a plaintext to a ciphertext, and decrypts an encrypted ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts;
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1) by setting said Bi (0≦i≦K−1) to Bi=b0b1 . . . bi using an integer bi;
• selecting w, which satisfies w<P (P:
• prime number), and determining public-key vector c=(c0, c1, . . . , cK−1) according to Formula (m);
• c=wBi(mod P) (m)
• preparing ciphertext C shown in Formula (n) according to the scalar product of plaintext vector m, random number vector r, and public-key vector c;
• C=m0c0+m1c1+ . . . +mJ−1cJ−1+rJcJ+rJ+1cJ+1+ . . . +rK−1cK−1 (n)
• determining an interim decrypted text M for ciphertext C as described in Formula (o);
• M=w−1C(mod P) (o)
• determining plaintext vector m=(m0, m1, . . . , mJ−1) by decrypting this interim decrypted text M according to the algorithm below:
• Step 0
• M0=M/b0
• m0=M0(mod b1)
• Step i (i=1, 2, . . . , J−1)
• Mi=(Mi−1−mi−1/bi
• mi=Mi(mod bi+1);
• and
• obtaining the original plaintext from the plaintext vector m.

An encryption/decryption method, which encrypts a plaintext to a ciphertext, and decrypts an encrypted ciphertext to an original plaintext, comprising the steps of:

• providing plaintext vector m=(m0, m1, . . . , mJ−1) which divides a plaintext to be encrypted into J parts;
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1) by setting said Bi (0≦i≦K−1) according to Formula (p);
• Bi=vib0b1 . . . bi (p)
• Provided that:
• Vi:
• random number
• bi:
• integer
• gcd (vi, bi+1)=1
• selecting w, which satisfies w<P (P:
• prime number), and determining public-key vector c=(c0, c1, . . . , cK−1) according to Formula (q);
• ci=wBi(mod P) (q)
• preparing ciphertext C shown in Formula (r) according to the scalar product of plaintext vector m, random number vector r, and public-key vector c;
• C=m0c0+m1c1+ . . . +mJ−1cJ−1+rJcJ+rJ+1cJ+1+ . . . +rK−1cK−1 (r)
• determining an interim decrypted text M for ciphertext C as described in Formula (s);
• M≡w−1C(mod P) (s)
• determining plaintext vector m=(m0, m1, . . . , mJ) by decrypting this interim decrypted text M according to the algorithm below:
• Step 0
• M0=C/b0
• m0=M0b0−1(mod b1)
• Step i (i=1, 2, . . . , J−1)
• Mi≡(Mi−1−mi−1vi−1)/bi
• mi=Miv1−1(mod bi+1);
• and
• obtaining the original plaintext from the plaintext vector m.

A cryptographic communications system for carrying out data communications among a plurality of entities using ciphertext, comprising:

• an encryptor for preparing a ciphertext from a plaintext using one of the following two encryption methods:
• the first encryption method including the steps of:
• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts,
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1), which is set to Bi=b0b1 . . . bi using an integer bi (0≦i≦K−1), and producing ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 from the plaintext vector m, arbitrary random number vector r, and base vector B, and the second encryption method including the steps of:
• providing plaintext vector m=(m0, m1, . . . , mJ−1), which divides a plaintext to be encrypted into J parts,
• providing an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• providing a base vector B=(B0, B1, . . . , BK−1), which is set to Bi=v1b0b1 . . . bi using an integer bi using integer and random number vi (0≦i≦K−1), and
• producing ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 from the plaintext vector m,
• arbitrary random number vector r and base vector B;
• a communications channel for sending a prepared ciphertext from one entity to another entity;
• and
• a decryptor for decrypting a sent ciphertext to an original plaintext.

A computer readable medium, which stores a program for producing by computer a ciphertext from a plaintext, the computer usable medium comprising:

• first program coding means for causing said computer to produce a plaintext vector m=(m0, m1, . . . , mJ−1) by dividing a plaintext to be encrypted into J parts;
• second program coding means for causing said computer to produce an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• third program coding means for causing said computer to produce a base vector B=(B0, B1, . . . , BJ−1), which is set to Bi=b0b1 . . . bi using integer bi (0≦i≦K−1);
• and
• fourth program coding means for causing said computer to produce a ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 using said plaintext vector m, random number vector r, and base vector B.

A computer readable medium, which stores a program for producing by a computer a ciphertext from a plaintext, the computer usable medium comprising:

• first program coding means for causing said computer to produce a plaintext vector m=(m0, m1, . . . , mJ−1) by dividing a plaintext to be encrypted into J parts;
• second program coding means for causing said computer to produce an arbitrary random number vector r=(rJ, rJ+1, . . . , rK−1);
• third program coding means for causing said computer to produce a base vector B=(B0, B1, . . . , BJ−1), which is set to Bi=vib0b1 . . . bi using integer bi and random number vi (0≦i≦K−1);
• and
• fourth program coding means for causing said computer to produce a ciphertext C=m0B0+m1B1+ . . . +mJ−1BJ−1+rJBJ+rJ+1BJ+1+ . . . +rK−1BK−1 using said plaintext vector m, random number vector r, and base vector B.