A system for generating a digital signature comprising: a first security device configured to generate a first private key, the first private key comprising a first private exponent, d 1, computed in accordance with the expression d 1 = (1/ e 1) mod( p 1 - 1)( q 1 - 1), where p 1 and q 1 are randomly chosen prime numbers and e 1 satisfies the expression gcd( p 1 - 1, e 1) = gcd( q 1 - 1, e 1), and store the first private key in a memory of the first security device, the first security device configured to, in response to receiving a message, m, generate a first signature, σ 1, based on the message and the first private key in accordance with the expression σ 1 ( m) = m d 1 mod n 1, where m ∈ Z n 1 = {0,..., n 1 - 1} and n 1 = p 1 q 1; a second security device configured to generate a second private key independently from the first private key, the second private key comprising a second private exponent, d 2, computed in accordance with the expression d 2 = (1/ e 2) mod( p 2 - 1)( q 2 - 1), where p 2 and q 2 are randomly chosen prime numbers and e 2 satisfies the expression gcd( p 2 - 1, e 2) = gcd( q 2 - 1, e 2), and store the a second private key in a memory of the second security device, the second security device configured to, in response to receiving the message, m, generate a second signature, σ 2, based on the message and the second private key in accordance with the expression σ 2 ( m) = m d 2 mod n 2, where m ∈ Z n 2 = {0,..., n 2 - 1} and n 2 = p 2 q 2; and a computing device remote from the first security device and the second security device, the computing device configured to: responsive to receiving a request for the digital signature, send the message to the first security device and the second security device; responsive to receiving the first signature and the second signature, generate a composite cryptographic signature, σ c, based on the first signature and the second signature in accordance with the expression σ c = ( bn 1 σ 1 + an 2 σ 2)mod n 1 n 2, where a and b satisfy the expression an 1 + bn 2 = 1; append the composite cryptographic signature to a digital document; and generate a composite cryptographic public key based on the first and second public keys and comprising a composite public modulus, n c, computed in accordance with the expression n c = n 1 n 2. a first security device configured to generate a first private key, the first private key comprising a first private exponent, d 1, computed in accordance with the expression d 1 = (1/ e 1) mod( p 1 - 1)( q 1 - 1), where p 1 and q 1 are randomly chosen prime numbers and e 1 satisfies the expression gcd( p 1 - 1, e 1) = gcd( q 1 - 1, e 1), and store the first private key in a memory of the first security device, the first security device configured to, in response to receiving a message, m, generate a first signature, σ 1, based on the message and the first private key in accordance with the expression σ 1 ( m) = m d 1 mod n 1, where m ∈ Z n 1 = {0,..., n 1 - 1} and n 1 = p 1 q 1; a second security device configured to generate a second private key independently from the first private key, the second private key comprising a second private exponent, d 2, computed in accordance with the expression d 2 = (1/ e 2) mod( p 2 - 1)( q 2 - 1), where p 2 and q 2 are randomly chosen prime numbers and e 2 satisfies the expression gcd( p 2 - 1, e 2) = gcd( q 2 - 1, e 2), and store the a second private key in a memory of the second security device, the second security device configured to, in response to receiving the message, m, generate a second signature, σ 2, based on the message and the second private key in accordance with the expression σ 2 ( m) = m d 2 mod n 2, where m ∈ Z n 2 = {0,..., n 2 - 1} and n 2 = p 2 q 2; and a computing device remote from the first security device and the second security device, the computing device configured to: responsive to receiving a request for the digital signature, send the message to the first security device and the second security device; responsive to receiving the first signature and the second signature, generate a composite cryptographic signature, σ c, based on the first signature and the second signature in accordance with the expression σ c = ( bn 1 σ 1 + an 2 σ 2)mod n 1 n 2, where a and b satisfy the expression an 1 + bn 2 = 1; append the composite cryptographic signature to a digital document; and generate a composite cryptographic public key based on the first and second public keys and comprising a composite public modulus, n c, computed in accordance with the expression n c = n 1 n 2. responsive to receiving a request for the digital signature, send the message to the first security device and the second security device; responsive to receiving the first signature and the second signature, generate a composite cryptographic signature, σ c, based on the first signature and the second signature in accordance with the expression σ c = ( bn 1 σ 1 + an 2 σ 2)mod n 1 n 2, where a and b satisfy the expression an 1 + bn 2 = 1; append the composite cryptographic signature to a digital document; and generate a composite cryptographic public key based on the first and second public keys and comprising a composite public modulus, n c, computed in accordance with the expression n c = n 1 n 2.