125 AMERICAN Journal of Public Diplomacy and International Studies www. grnjournal.us AMERICAN Journal of Public Diplomacy and International Studies Volume 01, Issue 09, 2023 ISSN (E): 2993-2157 ELECTRONIC DIGITAL SIGNATURE ALGORITHMS BASED ON THE COMPLEXITY OF THE FACTORIZATION PROBLEM Sharopova Muxayyo Muxtor qizi Asian International University Teacher of the "General Technical Sciences" department mukhayyosharopova4@gmail.com Annotation: This article shows the concepts of electronic digital signature and the main goals of its use. In addition, the science "Cryptography", which is the basis of electronic signature, and the initial encryption algorithms of this science are presented, showing examples of them. raised Electronic digital signature algorithms based on the complexity of the factorization problem. Keywords: cryptography, factoring, digital signature, algorithms, electronic document. Introduction. Electronic digital signature based on RSA public key encryption algorithm Each i - user of the system  ii de , - generates a key pair. For this, by taking sufficiently large p and q -prime numbers (these numbers are kept secret), pqn  -number and the value of the Euler function  n =   11  qp is calculated (this number is also kept secret). Then,    1, nei  the number satisfying the condition, i.e., the number -by the number  n -which id is prime to the number -is ie calculated by this formula.  nde ii mod1 This ( ie ; id ) pair ie is declared as public key and id private key . Then i -user j to -user sends encrypted data with signature as follows: 1. Encryption rule: CnM je mod , where M - open information, S - encrypted information; 2. Decoding rule: MnMnC jjj ded  modmod ; 3. Calculation of ERI:   i d PnMH i mod , where i -the -user's iP -signature is calculated by М the -data's  MH -hash function value; 4. Checking the ERI:      nMHnP iii dee i modmod  MH , if  MH  1MH (here 1M -decrypted information), then the electronic document is valid, otherwise it is invalid, because according to the hash function property 1MM  , their hash values are also equal 5. Confidential data transfer protocol:     nMHM j i ed mod   CnPM je i  mod ; 6. Protocol for receiving confidential transmitted information: http://www/ mailto:mukhayyosharopova4@gmail.com 126 AMERICAN Journal of Public Diplomacy and International Studies www. grnjournal.us nC jd mod    nPM jj de i mod iPM  , preliminary information in general may have been changed, therefore nC jd mod iPM 1 and the resulting hash value is u by signature this expression      nMHnP iii dee i modmod  MH , and if the received information has a hash value  1MH , then  MH  1MH the electronic document is valid, otherwise, it is fake. ESIGN digital signature algorithm. ESIGN is an ERI algorithm developed by scientists from Japan (NTT, Japan). The tolerance of this algorithm is determined by the complexity of the factorization problem, such as the RSA algorithm. In the ESIGN algorithm, a pair of large prime numbers p and q serves as a secret key and is determined by the expression n = p 2 *q. The pair (n, k) serves as the public key. Here k is a security parameter. The formation of ERI according to the ESIGN algorithm and its transmission includes the following sequence of steps: 1) M is a hash function for information: m = H(M); The value of m is in the range from 0 to n-1; 2) a random number x smaller than p*q is generated; 3) a very small integer w is: w ((m - x k ) (mod n))/p*q; 4) ERI S is formed for m using the secret key: S x+((w/kx k-1 (mod p))p*q; 5) information is transmitted through the communication channel M and ERI S. Using the received information M and ERI S, the receiving party performs the following sequence of steps: 1) The hash function for M information is m = H(M): S k for S using public key (n, k). (mod n) is; 3) the number a is equal to or greater than the double of the number of bits divided by 3, much smaller than the integer, and 2 is a ; s k (mod n) is compared with m and m+2 a : m = = s k (mod n); m+2 a = = s k (mod n). If s k (mod n) is equal to or greater than m and s k (mod n) is less than m+2 a , then ERI is valid, otherwise invalid. The fact that this algorithm has the ability to perform calculations related to x and k in advance makes it possible to speed up the ERI formation process. This algorithm is much faster than RSA if it uses keys and signatures of the same size, and has the same security as RSA. ESIGN has received a patent in the USA, Canada, England and several countries. References: 1. Sharopova, M. M. (2023). 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