Part 2
Secure Communications

Digital Encryption of Analog Signals -- Without Bandwidth Expansion, Without Digital Compression, Without Loss of Information

INTRA modems also resolve a long standing problem in encryption theory. Suppose [e] is an operator for digital encryption and [d] performs decryption so that [d][e] = [I]. For example, [e] could be the non-linear DES algorithm. With INTRA, any analog signal can be counter-rotated to its DATA representation by the demodulation operator [D]. The original waveform may then be digitally encrypted and rotated back into the SIGNAL coordinate representation for transmission.

A conventional (AMPS) cellular phone provides an example. In operator notation, the baseband portion of the transmitter computes [M][e][D] and the receiver computes [M][d][D]. In this transmitter, the voice is first processed by [D] before encryption; whereas in conventional digital cellular phones the voice is first processed by a digital voice compression algorithm. The baseband signal can modulate a carrier, if desired (AMPS uses FM modulation, for which INTRA is well suited). Note that the combined transmitter and receiver result in an identity [I] if, and only if, [M] and [D] are commuting operators. That is, if [t] and [r] are the carrier send and receive operators, then a signal, s, is recovered since

[M][d][D][r][t][M][e][D]s = [I]s

A conventional digital cellular phone system (GSM, IS-54, IS-95, PCS) cannot achieve identity, [I], because there is no exact inverse operator for digital voice compression. So the INTRA system will have a higher score on voice recognition and intelligibility tests and behave differently to noise on the link. In this new type of digital voice encryptor, arbitrary analog signals are digitally manipulated without bandwidth expansion and without a digital voice compression algorithm. Analog voice (or video) can be further manipulated by appending digital forward error correction (FEC) and digital signaling protocols thus expanding the COM dimensionality.

The effect of the non-linear encryption operator on a noise vector, N, can be examined in the vector space less inside R. Geometrically, a secret vector, V, ---that usually depends on the sequence of transmitted vectors--- is modulo-added to the Information Vector, S, in the transmitter, and the same secret vector is modulo-subtracted in the receiver. The INTRA receiver computes

((S+V)modR+N-V)modR=(S+N)modR=S+N

So unlike systems that use digital compression algorithms, an INTRA system should behave gracefully in noise. Expanding the modem's dimensionality, that is, "band-spreading", can add redundancy for FEC in severe noise.

Modulo arithmetic, which is essential to digital encryption, insures that the encrypted vector plus noise will be in the same M-Dimensional space as the Information Vector. From Shannon's proof of the Vernam Cipher Theorem {originally in a Bell Labs Memorandum, May 10, 1943}, this transmission system is provably as secure as the digital encryption algorithm itself, since the resultant vector can lie anywhere in the in-band space.

Digital encryption without bandwidth expansion is a remarkable prediction. If voice, bandlimited to 300-3300Hz, is passed through an A/D, digitally encrypted by an XOR, then passed through a D/A, the resultant bandwidth will be expanded, because the encrypting bits are completely random (i.e. the encrypted output will include 0-300 Hz). In the past the only solution to this dilemma was to use digital voice compression and a modem.

Compression-less Networking

The encryption results above apply to any analog signal, not just voice, and obviously the principles apply if [e]=[I]. Today we take for granted that efficient digital transmission of voice (or digital video) is preceded by the step of digital compression. This may need to be re-examined. INTRA allows uncompressed analog signals to be networked digitally.

To evaluate the above digital cellular phone technique, one can compare it to the GSM standard for TDMA cellular in Europe. In GSM a 200 KHz RF bandwidth is used to transmit 271 Kb/s for carrying 8 users plus some overhead associated with TDMA. Each user's vocoder produces 13 Kb/s, to which is added 9.8 KB/s of Forward Error Correction code (FEC). Vocoders do not behave gracefully for bit-errors without FEC. This makes the total voice rate per user 22.8 Kb/s. In effect, 16.8 KHz (=22.8*200/271) is devoted to one 3 KHz voice channel in the GSM digital cellular system.

GSM expands the input analog bandwidth by over a factor of five (=16.8/3). Digitally encrypted INTRA should easily provide digital forward error correction as well as multiple access overhead with much less expansion and still be compression-less. INTRA could easily be three times as bandwidth efficient as GSM, degrade gracefully in noise, and suffer no quality loss since there is no digital compression algorithm. This is independent of the multiple access scheme employed (i.e. TDMA, CDMA, FDMA).

For comparison, the European analog TACS cellular system uses two redundant FM sidebands each with 12.5 Khz bandwidth for each 3 KHz voice channel. In the USA the 30KHz AMPS FM system and the IS-54 digital system have a similar comparison, but with lower voice-intelligibility scores since IS-54 uses a 7.95 Kb/s vocoder. In either case, today's compressed digital voice systems and compressionless analog voice systems are not very bandwidth efficient compared to compression-less INTRA.