CN109190414B - Fully homomorphic confusion method for multiplier - Google Patents

Abstract

The invention discloses a fully homomorphic obfuscating method for a multiplier, which comprises the steps of firstly carrying out fully homomorphic encryption on a multiplier and a multiplicand of the multiplier, then carrying out multiplication operation by using encrypted data, judging whether an input key is equal to a set correct key value or not when a key is input at a second input port of the multiplier, and carrying out decryption according to a judgment conclusion to obtain final product output; the advantages are that the multiplier and the multiplicand are encrypted in the same state, and the product is output in a mixed mode, so that the original data of the multiplier and the multiplicand are prevented from being stolen, an IP core of the multiplier is protected, and the safety of the multiplier in an integrated circuit is improved.

Description

Fully homomorphic confusion method for multiplier

Technical Field

The present invention relates to a homomorphic aliasing method, and more particularly, to a homomorphic aliasing method for a multiplier.

Background

As part of the arithmetic unit, multipliers are essential in digital circuit design. In a hardware circuit design, a multiplier may directly use a multiplication sign, and when logic synthesis is performed, an Intellectual Property (IP) core of the multiplier in a process library is called to complete multiplication. The multiplier, as an IP core in an integrated circuit, is reusable, so that it may be subject to various attacks, such as IP core piracy. The existing IP core protection is to hide the Function of a circuit by hardware confusion, i.e. changing the design of the IP core, for example, Zhang and others propose a hardware confusion method combining a Physical Unclonable Function (PUF) and a finite state machine to effectively protect the IP core of a Field-Programmable Gate Array (FPGA) Device, and implement a forced payment permission of Pay-Per-Device.

However, the traditional IP protection method rarely involves protection of the multiplier IP core, and also does not involve protection of the original input data of the multiplier, so when the multiplier performs multiplication, the multiplier and the multiplicand are operated by inputting the original data into the multiplier, and the original data is easily utilized by attacks such as hardware trojans and the like to maliciously modify a circuit or a design, thereby causing the whole circuit to abnormally operate or steal the whole circuit information. Therefore, how to effectively protect the operational data of the multiplier in the integrated circuit and the security of the IP core itself has become an urgent problem to be solved.

Disclosure of Invention

The invention aims to solve the technical problem of providing a homomorphic confusion method for a multiplier, which can carry out homomorphic encryption on a multiplier and a multiplicand and output product confusion, thereby avoiding the stealing of the original data of the multiplier and the multiplicand, protecting an IP core of the multiplier and improving the safety of the multiplier in an integrated circuit.

The technical scheme adopted by the invention for solving the technical problems is as follows: a method of homomorphic obfuscation for a multiplier comprising the steps of:

(1) the first input port of the multiplier is marked as a ', the second input port is marked as b', the product output port is marked as c ', the binary multiplier is marked as a', and a ″, is marked as a_na_n-1…a₂a₁N is the number of bits of the binary multiplier, the binary multiplicand is denoted as b ", b ″_m b_m-1…b₂b₁M is the number of bits of the binary multiplicand, a bit binary key input into the multiplier through a second input port is set and is marked as KS, wherein l is an integer larger than 1;

(2) randomly generating two decimal variables r which are more than or equal to-2 and less than or equal to 2 by adopting a random function₁And r₂；

(3) Setting a binary digit of one bit in the multiplier, and using the binary digit of one bit as a correct secret key of the multiplier;

(4) and (3) recording the decimal number corresponding to the a 'as a, recording the decimal number corresponding to the b' as b, and performing fully homomorphic encryption on the a and the b, wherein the specific encryption process is as follows:

A. if m is equal to n, the data obtained by encrypting a is recorded as c₁And recording the data obtained by encrypting b as c₀The encryption is carried out according to the following steps:

a-1, determining whether n is equal to 1: if n is equal to 1, let c₁＝a+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+2r₁)*(b+2r₂)<p/2, is the sign of the multiply operation; if n is greater than 1, let c₁＝a+p*q+4ⁿr₁，c₀＝b+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+4ⁿr₁)*(b+4ⁿr₂)<p/2；

A-2, mixing₁And c₀Inputting the result into a multiplier to perform multiplication, and recording the result as c, c ═ c₁*c₀；

A-3, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the correct key of the multiplier:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether n is equal to 1: if n is equal to 1, modulo p and then modulo 2 by c to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at this time, out is (c mod p) mod 2, and mod is a modulo symbol; if n is more than 1, c is firstly modulo p and then 4 is carried outⁿTaking the modulus to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at the moment, out is (c mod p) mod 4ⁿ；

When the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, c firstly modulates p, then modulates 2 and finally inverts the n according to the bit to obtain decrypted data out output, wherein the decrypted data out is the product output by the multiplier, and at the moment, out is — ((c mod p) mod 2) and-is an inverted symbol; if n is greater than 1, c is modulo p and then 4ⁿAnd performing modulus operation and bit-wise negation to obtain decrypted data out, wherein the decrypted data out is the product of the output of the multiplier, and at the moment, out is — ((c mod p) mod 4ⁿ)；

B. If n > m, the quotient of dividing n by m is recorded as k, the remainder is recorded as h, and a' is updated: if h is not equal to 0, the left most bit in a ″M-h 0 is complemented to obtain a ═ a_(k+1)m…a_n+2a_n+1a_n a_n-1…a₂a₁Wherein a is_(k+1)m…a_n+2a_n+10 … 00, if h is 0, a "remains unchanged; the quotient of dividing the current a' digit by m is marked as j, and a_m…a₃a₂a₁The corresponding decimal is denoted D1, a_2m…a_m+3a_m+2a_m+1The corresponding decimal is denoted D2, and so on, a_jm…a_(j-1)m+3a_(j-1)m+2a_(j-1)m+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

b-1, recording data obtained by encrypting Di as c_iI is 1, 2, …, j, and the data obtained by encrypting b is recorded as c₀；

B-2, judging whether m is equal to 1: if m is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(b+2r₂)<p/2; if m is greater than 1, let c_i＝Di+p*q+4^mr₁，c₀＝b+p*q+4^mr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4^mr₁)*(b+4^mr₂)<p/2；

B-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

B-4, determining whether the one-bit binary key KS input by the second input port B' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, taking the modulus of C1, C2, C3, … and Cj respectively for p and then for 2 to obtain decrypted data, recording Ci as C' i,where c' i ═ Ci mod p) mod 2, mod is the modulo sign; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4^mObtaining a modulus, obtaining decrypted data, and recording the data after Ci decryption as c 'i, wherein c' i is (Ci mod p) mod 4^mMod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, modulo p and modulo 2 are respectively performed by C1, C2, C3, … and Cj, and finally bitwise negation is performed to obtain decrypted data, and Ci decrypted data is recorded as C 'i, namely C' i is ═ to ((Ci mod p) mod 2), and mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4^m) Mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ … + c'1, outputting data out, which is the product of the multiplier output;

C. if n is<m, the quotient of m divided by n is recorded as k, the remainder is recorded as h, and b' is updated: if h is not equal to 0, n-h 0 s are complemented on the high position of b ″, namely the leftmost position, so that b ″, which is equal to b_(k+1)n…b_m+2b_m+1b_mb_m-1…b₂b₁，b_(k+1)n…b_m+2b_m+10 … 00, if h equals 0, then b "remains unchanged; the quotient of dividing the current b' digit by n is denoted as j, and b_n…b₃b₂b₁The corresponding decimal number is D1, b_2n…b_n+3b_n+2b_n+1The corresponding decimal is denoted D2, and so on, b_jn…b_(j-1)n+3b_(j-1)n+2b_(j-1)n+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

c-1, recording data obtained by encrypting Di as C_iI is 1, 2, …, j, and data obtained by encrypting a is recorded as c₀；

C-2, judging whether n is equal to 1: if n is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝a+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(a+2r₂)<p/2; if n is greater than 1, let c_i＝Di+p*q+4ⁿr₁，c₀＝a+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4ⁿr₁)*(a+4ⁿr₂)<p/2；

C-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

C-4, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key is equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p and then modulo 2 are respectively taken for C1, C2, C3, … and Cj to obtain decrypted data, and Ci is decrypted to be recorded as C 'i, wherein C' i is (Ci mod p) mod 2, and mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿTaking the modulus to obtain decrypted data c '1, c '2, c '3, …, c ' k, wherein c ' i is (Ci mod p) mod 4ⁿMod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p, modulo 2, and finally bitwise negation are performed on C1, C2, C3, …, and Cj, respectively, to obtain decrypted data, and the decrypted data Ci is recorded as C 'i, that is, C' i ═ to ((Ci mod p) mod 2), mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4ⁿ) Mod is a modulo sign, then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, data out is output, which is the product of the multiplier output.

Compared with the prior art, the method has the advantages that the multiplier and the multiplicand of the multiplier are encrypted in a fully homomorphic way, and the encrypted data are used for multiplication, so that the original data of the multiplier and the multiplicand can be prevented from appearing; for side-channel attack, the multiplier chip processes the data after the multiplicand and the multiplicand are encrypted during working and is not the original data of the multiplicand and the multiplicand any more, the leaked power consumption or running time is related to the data after the multiplicand and the multiplicand are encrypted, the meaning represented by the original data of the multiplicand and the multiplicand cannot be reflected, the side-channel attack can be effectively prevented, the product output is controlled through the set correct key of the multiplier at the output port of the multiplier, the multiplied data can be decrypted in the same state and output only when the input one-bit binary key KS is the same as the set correct key of the multiplier, when an attacker does not know the correct key of the correct multiplier, the one-bit binary key KS is different from the set correct key of the multiplier, the multiplied data is decrypted in the same state and then output in reverse, and output confusion is realized, the correct key of the set multiplier is mastered by a designer and cannot be acquired by an attacker, so that the protection of hardware intellectual property is improved, and the problem of IP embezzlement can be effectively solved.

Drawings

FIG. 1 is a waveform diagram of an output of a multiplier operated with raw data;

FIG. 2 is a waveform diagram of the output of a multiplier operated by the inventive all-homomorphic aliasing method for multipliers.

Detailed Description

The invention is described in further detail below with reference to the accompanying examples.

Example (b): a method of homomorphic obfuscation for a multiplier comprising the steps of:

(1) the first input port of the multiplier is marked as a ', the second input port is marked as b', the product output port is marked as c ', the binary multiplier is marked as a', and a ″, is marked as a_na_n-1…a₂a₁N is the number of bits of the binary multiplier, the binary multiplicand is denoted as b ", b ″_m b_m-1…b₂b₁M is the number of bits of the binary multiplicand, a bit binary key input into the multiplier through a second input port is set and is marked as KS, wherein l is an integer larger than 1;

(2) randomly generating two decimal variables r which are more than or equal to-2 and less than or equal to 2 by adopting a random function₁And r₂；

(3) Setting a binary digit of one bit in the multiplier, and using the binary digit of one bit as a correct secret key of the multiplier;

(4) and (3) recording the decimal number corresponding to the a 'as a, recording the decimal number corresponding to the b' as b, and performing fully homomorphic encryption on the a and the b, wherein the specific encryption process is as follows:

A. if m is equal to n, the data obtained by encrypting a is recorded as c₁And recording the data obtained by encrypting b as c₀The encryption is carried out according to the following steps:

a-1, determining whether n is equal to 1: if n is equal to 1, let c₁＝a+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+2r₁)*(b+2r₂)<p/2, is the sign of the multiply operation; if n is greater than 1, let c₁＝a+p*q+4ⁿr₁，c₀＝b+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+4ⁿr₁)*(b+4ⁿr₂)<p/2；

A-2, mixing₁And c₀Inputting the result into a multiplier to perform multiplication, and recording the result as c, c ═ c₁*c₀；

A-3, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the correct key of the multiplier:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether n is equal to 1: if n is equal to 1, modulo p and then modulo 2 by c to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at this time, out is (c mod p) mod 2, and mod is a modulo symbol; if n is more than 1, c is firstly modulo p and then 4 is carried outⁿTaking the modulus to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at the moment, out is (c mod p) mod 4ⁿ；

When the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, c is firstly modulo p, then modulo 2 and finally inverted according to bits to obtain decrypted data out output, the decrypted data out is the product of the output of the multiplier, and at the moment, out is — ((c mod p) mod 2) -is an inverted symbol; if n is greater than 1, c is modulo p and then 4ⁿAnd performing modulus operation and bit-wise negation to obtain decrypted data out, wherein the decrypted data out is the product of the output of the multiplier, and at the moment, out is — ((c mod p) mod 4ⁿ)；

B. If n > m, the quotient of dividing n by m is recorded as k, the remainder is recorded as h, and a' is updated: if h is not equal to 0, m-h 0's are complemented on the high position of a ″, namely the leftmost position, so that a ″, a ═ a_(k+1)m…a_n+2a_n+1a_n a_n-1…a₂a₁Wherein a is_(k+1)m…a_n+2a_n+10 … 00, if h is 0, a "remains unchanged; the quotient of dividing the current a' digit by m is marked as j, and a_m…a₃a₂a₁The corresponding decimal is denoted D1, a_2m…a_m+3a_m+2a_m+1The corresponding decimal is denoted D2, and so on, a_jm…a_(j-1)m+3a_(j-1)m+2a_(j-1)m+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

b-1, recording data obtained by encrypting Di as c_iI is 1, 2, …, j, and the data obtained by encrypting b is recorded as c₀；

B-2, judging whether m is equal to 1: if m is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(b+2r₂)<p/2; if m is greater than 1, let c_i＝Di+p*q+4^mr₁，c₀＝b+p*q+4^mr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4^mr₁)*(b+4^mr₂)<p/2；

B-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

B-4, determining whether the one-bit binary key KS input by the second input port B' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, modulo p and then modulo 2 are respectively taken for C1, C2, C3, … and Cj to obtain decrypted data, and Ci decrypted data is recorded as C 'i, wherein C' i is (Ci mod p) mod 2, and mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4^mObtaining a modulus, obtaining decrypted data, and recording the data after Ci decryption as c 'i, wherein c' i is (Ci mod p) mod 4^mMod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, modulo p and modulo 2 are respectively performed by C1, C2, C3, … and Cj, and finally bitwise negation is performed to obtain decrypted data, and Ci decrypted data is recorded as C 'i, namely C' i is ═ to ((Ci mod p) mod 2), and mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4^m) Mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ … + c'1, outputting data out, which is the product of the multiplier output;

C. if n is<m, the quotient of m divided by n is recorded as k, the remainder is recorded as h, and b' is updated: if h is not equal to 0, n-h 0 s are complemented on the leftmost side of the b 'to obtain b'＝b_(k+1)n…b_m+2b_m+1b_mb_m-1…b₂b₁，b_(k+1)n…b_m+2b_m+10 … 00, if h equals 0, then b "remains unchanged; the quotient of dividing the current b' digit by n is denoted as j, and b_n…b₃b₂b₁The corresponding decimal number is D1, b_2n…b_n+3b_n+2b_n+1The corresponding decimal is denoted D2, and so on, b_jn…b_(j-1)n+3b_(j-1)n+2b_(j-1)n+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

c-1, recording data obtained by encrypting Di as C_iI is 1, 2, …, j, and data obtained by encrypting a is recorded as c₀；

C-2, judging whether n is equal to 1: if n is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝a+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(a+2r₂)<p/2; if n is greater than 1, let c_i＝Di+p*q+4ⁿr₁，c₀＝a+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4ⁿr₁)*(a+4ⁿr₂)<p/2；

C-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

C-4, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key is equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p and modulo 2 are respectively taken for C1, C2, C3, … and Cj to obtain decrypted data, and the decrypted data is recorded as C 'i, wherein C' i is (Ci mod p) mod2, mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿTaking the modulus to obtain decrypted data c '1, c '2, c '3, …, c ' k, wherein c ' i is (Ci mod p) mod 4ⁿMod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p, modulo 2, and finally bitwise negation are performed on C1, C2, C3, …, and Cj, respectively, to obtain decrypted data, and the decrypted data Ci is recorded as C 'i, that is, C' i ═ to ((Ci mod p) mod 2), mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4ⁿ) Mod is a modulo sign, then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, data out is output, which is the product of the multiplier output.

The output waveform of the multiplier operated by using the original data is shown in fig. 1, and the output waveform of the multiplier operated by using the homomorphic aliasing method for the multiplier of the invention is shown in fig. 2. In fig. 1, a "is a multiplier, b" is a multiplicand, and out is the product of the multiplier outputs. In FIG. 2, a "is the multiplier, b" is the multiplicand, out is the product of the multiplier outputs, r₁And r₂In FIG. 2, a binary number representation, c₁In FIG. 2, c is represented by hexadecimal number₀In FIG. 2, hexadecimal numbers are used for representation, c in FIG. 2, p and q in FIG. 2 are used for representation, a "and b"Using the same bit width, i.e., m-n-2, KS indicates that the l-bit binary key value input from the second input port b' of the multiplier is correct |! KS indicates that the key value input from the second input port b' of the multiplier is erroneous, where we have selected a 4-bit binary key. As can be seen from the analysis of fig. 1 and 2: when the multiplier adopts the homomorphic obfuscating method for the multiplier to operate, when the input 4-bit binary key is equal to the set correct 4-bit key value, the product out output by the multiplier is consistent with the product output by the multiplier in fig. 1, thereby showing that the homomorphic obfuscating method for the multiplier of the invention has correct logic function; when the input 4-bit binary key is not equal to the set correct 4-bit key value, the product out output by the multiplier is opposite to the product output by the multiplier in fig. 1, thereby showing that the fully homomorphic obfuscating method for the multiplier performs obfuscated output on the result output by the multiplier, and effectively protecting the operational data of the multiplier in the integrated circuit and the security of the IP core thereof.

Claims (1)

1. A method of homomorphic obfuscation for a multiplier comprising the steps of:

(1) the first input port of the multiplier is marked as a ', the second input port is marked as b', the product output port is marked as c ', the binary multiplier is marked as a', and a ″, is marked as a_na_n-1 … a₂a₁N is the number of bits of the binary multiplier, the binary multiplicand is denoted as b ", b ″_mb_m-1 … b₂b₁M is the number of bits of the binary multiplicand, a bit binary key input into the multiplier through a second input port is set and is marked as KS, wherein l is an integer larger than 1;

(2) randomly generating two decimal variables r which are more than or equal to-2 and less than or equal to 2 by adopting a random function₁And r₂；

(3) Setting a binary digit of one bit in the multiplier, and using the binary digit of one bit as a correct secret key of the multiplier;

(4) and (3) recording the decimal number corresponding to the a 'as a, recording the decimal number corresponding to the b' as b, and performing fully homomorphic encryption on the a and the b, wherein the specific encryption process is as follows:

A. if m is equal to n, the data obtained by encrypting a is recorded as c₁And recording the data obtained by encrypting b as c₀The encryption is carried out according to the following steps:

a-1, determining whether n is equal to 1: if n is equal to 1, let c₁＝a+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+2r₁)*(b+2r₂)<p/2, is the sign of the multiply operation; if n is greater than 1, let c₁＝a+p*q+4ⁿr₁，c₀＝b+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(a+4ⁿr₁)*(b+4ⁿr₂)<p/2；

A-2, mixing₁And c₀Inputting the result into a multiplier to perform multiplication, and recording the result as c, c ═ c₁*c₀；

A-3, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the correct key of the multiplier:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether n is equal to 1: if n is equal to 1, modulo p and then modulo 2 by c to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at this time, out is (c mod p) mod 2, and mod is a modulo symbol; if n is more than 1, c is firstly modulo p and then 4 is carried outⁿTaking the modulus to obtain decrypted data out, wherein the decrypted data out is the product output by the multiplier, and at the moment, out is (c mod p) mod 4ⁿ；

When the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, c firstly modulates p, then modulates 2 and finally inverts the n according to the bit to obtain decrypted data out output, wherein the decrypted data out is the product output by the multiplier, and at the moment, out is — ((c mod p) mod 2) and-is an inverted symbol; if n is greater than 1, c is modulo p and then 4ⁿAnd performing modulus operation and bit-wise negation to obtain decrypted data out, wherein the decrypted data out is the product of the output of the multiplier, and at the moment, out is — ((c mod p) mod 4ⁿ)；

B. If n > m, the quotient of dividing n by m is recorded as k, the remainder is recorded as h, and a' is updated: if h is not equal to 0, m-h 0's are complemented on the high position of a ″, namely the leftmost position, so that a ″, a ═ a_(k+1)m … a_n+2a_n+1a_n a_n-1 … a₂a₁Wherein a is_(k+1)m … a_n+2a_n+10 … 00, if h is 0, a "remains unchanged; the quotient of dividing the current a' digit by m is marked as j, and a_m … a₃a₂a₁The corresponding decimal is denoted D1, a_2m … a_m+3a_m+2a_m+1The corresponding decimal is denoted D2, and so on, a_jm … a_(j-1)m+3 a_{(j-1)m+ 2}a_(j-1)m+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

b-1, recording data obtained by encrypting Di as c_iI is 1, 2, …, j, and the data obtained by encrypting b is recorded as c₀；

B-2, judging whether m is equal to 1: if m is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝b+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(b+2r₂)<p/2; if m is greater than 1, let c_i＝Di+p*q+4^mr₁，c₀＝b+p*q+4^mr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4^mr₁)*(b+4^mr₂)<p/2；

B-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

B-4, determining whether the one-bit binary key KS input by the second input port B' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key KS is equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, modulo p and then modulo 2 are respectively taken for C1, C2, C3, … and Cj to obtain decrypted data, and Ci decrypted data is recorded as C 'i, wherein C' i is (Ci mod p) mod 2, and mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4^mObtaining a modulus, obtaining decrypted data, and recording the data after Ci decryption as c 'i, wherein c' i is (Ci mod p) mod 4^mMod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key KS is not equal to the correct key of the multiplier, it is determined whether the value of m is equal to 1: if m is equal to 1, modulo p and modulo 2 are respectively performed by C1, C2, C3, … and Cj, and finally bitwise negation is performed to obtain decrypted data, and Ci decrypted data is recorded as C 'i, namely C' i is ═ to ((Ci mod p) mod 2), and mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if m is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4^m) Mod is a modulo symbol; then out is c' j 2^(j-1)m+c'(j-1)*2^(j-2)m+…+c'2*2^m+ … + c'1, outputting data out, which is the product of the multiplier output;

C. if n is<m, the quotient of m divided by n is recorded as k, the remainder is recorded as h, and b' is updated: if h is not equal to 0, n-h 0 s are complemented on the high position of b ″, namely the leftmost position, so that b ″, which is equal to b_(k+1)n…b_m+2b_m+1b_mb_m-1…b₂b₁，b_(k+1)n…b_m+2b_m+10 … 00, if h equals 0, then b "remains unchanged; the quotient of dividing the current b' digit by n is denoted as j, and b_n…b₃b₂b₁The corresponding decimal number is D1, b_2n…b_n+3b_n+2b_n+1The corresponding decimal is denoted D2, and so on, b_jn…b_(j-1)n+3b_(j-1)n+2b_(j-1)n+1And the corresponding decimal system is marked as Dj, and the encryption is carried out according to the following steps:

c-1, recording data obtained by encrypting Di as C_iI is 1, 2, …, j, and data obtained by encrypting a is recorded as c₀；

C-2, judging whether n is equal to 1: if n is equal to 1, let c_i＝Di+p*q+2r₁，c₀＝a+p*q+2r₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+2r₁)*(a+2r₂)<p/2; if n is greater than 1, let c_i＝Di+p*q+4ⁿr₁，c₀＝a+p*q+4ⁿr₂Wherein p is a positive odd number, q is a positive integer larger than p, and satisfies-p/2<(Di+4ⁿr₁)*(a+4ⁿr₂)<p/2；

C-3, mixing₁，c₂，c₃，…，c_jAre each independently of c₀Inputting into a multiplier to perform multiplication, and c_iAnd c₀The result of the operation is denoted as Ci, i.e., Ci ═ c_i*c₀；

C-4, determining whether the one-bit binary key KS input by the second input port b' of the multiplier is equal to the set correct one-bit binary key value:

when the input one-bit binary key is equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p and then modulo 2 are respectively taken for C1, C2, C3, … and Cj to obtain decrypted data, and Ci is decrypted to be recorded as C 'i, wherein C' i is (Ci mod p) mod 2, and mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^{(j -2)n}+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿTaking the modulus to obtain decrypted data c '1, c '2, c '3, …, c ' k, wherein c ' i is (Ci mod p) mod 4ⁿMod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier;

when the input one-bit binary key is not equal to the correct key of the multiplier, it is determined whether the value of n is equal to 1: if n is equal to 1, modulo p, modulo 2, and finally bitwise negation are performed on C1, C2, C3, …, and Cj, respectively, to obtain decrypted data, and the decrypted data Ci is recorded as C 'i, that is, C' i ═ to ((Ci mod p) mod 2), mod is a modulo symbol; then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, outputting data out, wherein the data out is the product output by the multiplier; if n is more than 1, taking the modulus of C1, C2, C3, … and Cj to p respectively and then taking 4ⁿAnd modulus taking and bit negation are carried out finally to obtain decrypted data, and Ci decrypted data is recorded as c 'i, wherein c' i is- ((Ci mod p) mod 4ⁿ) Mod is a modulo sign, then out is c' j 2^(j-1)n+c'(j-1)*2^(j-2)n+…+c'2*2ⁿ+ c'1, data out is output, which is the product of the multiplier output.

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